- A statement reads 'eight more than the product of a number and 5.' Which expression correctly translates it?
- 8 plus the product of 5 and n
- 8 less the product of 5 and n
- 5 less the product of 8 and n
- 5 times the total of n plus 8
Correct answer: 8 plus the product of 5 and n
The translation is 8 plus the product of 5 and n. 'The product of a number and 5' is the product of 5 and n, and 'eight more than' that product adds 8 to it. 8 less the product of 5 and n subtracts the product from 8, which reverses the order of the subtraction and is not an addition at all. 5 less the product of 8 and n exchanges the roles of the 5 and the 8 and subtracts. 5 times the total of n plus 8 multiplies the 5 by the whole sum, a grouping the phrase never calls for.
- A delivery company charges a $6 base fee plus $2 for each package p. Which equation gives the total charge T?
- T is the sum of 6p and 2
- T is the sum of 6 and 8p
- T is the sum of 6 and 2p
- T is the sum of 8 and 2p
Correct answer: T is the sum of 6 and 2p
The equation is T is the sum of 6 and 2p. The $6 base fee is charged once whatever the load, so it stands alone, while the $2 charge repeats for every one of the p packages, giving 2p. T is the sum of 6p and 2 charges the base fee once per package and the per-package fee only once. T is the sum of 6 and 8p and T is the sum of 8 and 2p each fold the $6 and the $2 into a single $8 figure, so one of the two charges is counted twice.
- A statement says 'three times a number is the same as the number plus 14.' Which equation models it?
- n is the sum of 3n and 14
- 3n is the sum of n and 14
- the sum of 3n and n is 14
- 3n is the sum of 14 and 3
Correct answer: 3n is the sum of n and 14
The model is 3n is the sum of n and 14. 'Three times a number' gives 3n on one side, and 'the number plus 14' gives n and 14 added on the other. n is the sum of 3n and 14 swaps the two sides, so the plain number is made the larger quantity. The sum of 3n and n is 14 treats 14 as the total of both expressions rather than as an amount added to the number. 3n is the sum of 14 and 3 simply adds the two figures printed in the statement and drops the number itself.
- A clerk earns a $40 daily allowance plus $9 per completed report r. Which expression shows total daily earnings?
- the sum of 40r and 9
- the sum of 40 and 9r
- the sum of 9 and 49r
- 49 times r for a day
Correct answer: the sum of 40 and 9r
The expression is the sum of 40 and 9r. The $40 allowance is paid once a day however many reports are filed, so it stands alone, while the $9 is earned on each of the r reports, giving 9r. The sum of 40r and 9 makes the $40 the per-report rate and the $9 the fixed amount, exactly reversing the two. The sum of 9 and 49r folds the $40 into the per-report rate and then adds the $9 a second time. 49 times r for a day merges the allowance and the per-report pay into one rate, so the $40 is paid again for every report.
- A technician needs to solve 5x minus 12 equals 23 to find a part count. What is x?
Correct answer: 7
x equals 7. Adding 12 to both sides gives 5x equals 35, and dividing both sides by 5 yields 7. 2 comes from subtracting 12 instead of adding it, leaving 11 to divide by 5. 4 comes from dividing 23 by 5 and ignoring the 12 altogether. 3 comes from dividing the 35 by the 12 rather than by the coefficient 5.
- A phrase reads 'the difference between a number and 9, divided by 2.' Which expression matches it?
- (9 less n) over 2
- n, less half of 9
- (n plus 9) over 2
- n minus 9, over 2
Correct answer: n minus 9, over 2
The expression is n minus 9, over 2. 'The difference between a number and 9' is n minus 9, and the phrase then places that whole difference over 2. (9 less n) over 2 subtracts in the opposite order, which changes the sign of the result. n, less half of 9 halves only the 9 rather than the whole difference. (n plus 9) over 2 adds the two quantities where the phrase asks for a difference.
- Two parts together number 75. The first part is 4 times the second part, which is x. Which equation finds x?
- 4x is the sum of x and 75
- the sum of 4x and 4 is 75
- the sum of x and 4x is 75
- x is the sum of 4x and 75
Correct answer: the sum of x and 4x is 75
The equation is the sum of x and 4x is 75. The second part is x, the first part is 4 times that, or 4x, and the two parts together come to 75. 4x is the sum of x and 75 reads 'together number 75' as a gap between the parts instead of their total. The sum of 4x and 4 is 75 adds a bare 4 rather than the second part itself. x is the sum of 4x and 75 both reverses the sides and treats the smaller part as the larger one.
- A worker solves the equation 8(x plus 3) equals 64 in a problem. What is x?
Correct answer: 5
x equals 5. Dividing both sides by 8 gives x plus 3 equals 8, and subtracting 3 from both sides yields 5. 6 comes from adding 8 to 64 before dividing and then taking off the 3. 8 comes from dividing 64 by 8 and stopping there, leaving the 3 in place. 9 comes from adding 8 to 64 and dividing, so the 3 is never subtracted at all.
- A statement reads 'the height of a pole is 7 feet less than twice its shadow length s.' Which expression gives the height?
- 7 feet above 2s
- 7s above 2 feet
- 2s minus 7 feet
- 2s under 7 feet
Correct answer: 2s minus 7 feet
The height is 2s minus 7 feet. 'Twice its shadow length' is 2s, and '7 feet less than' that amount takes 7 away from it. 7 feet above 2s adds the 7 where the sentence subtracts it. 7s above 2 feet exchanges the 7 and the 2 and then adds. 2s under 7 feet takes twice the shadow away from 7 instead of taking 7 away from twice the shadow.
- A number n makes the equation (n divided by 3) minus 5 equals 4 true. What is n?
Correct answer: 27
n equals 27. Adding 5 to both sides gives n divided by 3 equals 9, and multiplying both sides by 3 yields 27. 17 comes from multiplying 4 by 3 first and then adding the 5, which applies the operations in the wrong order. 22 comes from taking the 5 away at the end instead of adding it before multiplying. 32 comes from adding the 5 a second time after the multiplication.
- A supplier ships 540 units this quarter, which is 20 percent fewer than last quarter. How many units shipped last quarter?
Correct answer: 675
Last quarter shipped 675 units. This quarter's 540 is 80 percent of last quarter's figure, so dividing 540 by 0.80 gives 675. 567 adds 5 percent to 540, 594 adds 10 percent and 621 adds 15 percent; each treats 540 as the base to grow from rather than as the figure already reduced by 20 percent, and none of the three restores a fifth of the original total.
- A stock priced at $250 rises 8 percent one day and falls 8 percent the next day. What is its price after the two days?
- $248.40
- $262.00
- $270.00
- $290.00
Correct answer: $248.40
The price is $248.40. An 8 percent rise takes $250 to $270, and 8 percent of $270 is $21.60, leaving $248.40. $262.00 takes a flat $8 off the $270 instead of 8 percent of it. $270.00 stops after the rise and never applies the fall. $290.00 adds the two percentages together as a single 16 percent gain, which ignores the drop altogether.
- Membership in a club grew from 320 members to 400 members. What was the percent increase?
Correct answer: 25
The increase was 25 percent. The club gained 80 members, and 80 divided by the original 320 is 0.25. 20 divides the 80-member gain by the new total of 400 rather than by the original figure. 31 and 37 both overstate the rise: 31 percent of 320 would be about 99 members and 37 percent about 118, while the club actually gained only 80.
- After a 40 percent markdown, a coat costs $54. What was its original price?
Correct answer: $90
The original price was $90. A 40 percent markdown leaves 60 percent, so $54 divided by 0.60 gives $90. $76 adds 40 percent to the $54 sale price instead of treating $54 as the reduced figure. $80 and $85 are both too low: 40 percent off $80 is $48 and 40 percent off $85 is $51, neither of which is the $54 quoted.
- A herd of 350 cattle increases by 18 percent. How many cattle are there after the increase?
Correct answer: 413
There are 413 cattle. Eighteen percent of 350 is 63, and 350 plus 63 is 413. 368 adds a flat 18 head rather than 18 percent of the herd. 386 adds only about 10 percent of 350. 431 adds roughly 23 percent, more than the 18 percent stated.
- A monthly rent of $900 is raised by 5 percent. What is the new rent?
Correct answer: $945
The new rent is $945. Five percent of $900 is $45, and adding it to $900 gives $945. $855 takes the $45 off instead of adding it. $885 subtracts a smaller amount and still moves the rent the wrong way. $905 adds a flat $5 rather than 5 percent of the rent.
- A budget of 800 dollars is first cut by 25 percent, then the remaining amount is cut by another 20 percent. What is the final budget?
Correct answer: $480
The final budget is $480. A 25 percent cut of $800 leaves $600, and a further 20 percent cut removes $120, leaving $480. $440 adds the two cuts into a single 45 percent reduction of $800, which takes off more than the two successive cuts do. $520 applies the second cut to only part of the remainder. $560 applies a 30 percent cut once and stops, so the second reduction is never made.
- A tire loses 6 percent of its tread, which equals 0.9 millimeters. What was the original tread depth?
Correct answer: 15 mm
The original tread depth was 15 mm. The 0.9-millimeter loss is 6 percent of the original depth, so dividing 0.9 by 0.06 gives 15. 18 mm divides 0.9 by 0.05, treating the loss as 5 percent. 22 mm divides by 0.04, and 26 mm by roughly 0.035; both use a smaller percentage than the 6 percent stated, so both overstate the original depth.
- A coolant mix uses water and antifreeze in a ratio of 3 to 2. How many quarts of antifreeze are in 25 quarts of mix?
Correct answer: 10
There are 10 quarts of antifreeze. The ratio 3 to 2 makes 5 parts in all, so each part is 5 quarts and the 2 antifreeze parts come to 10 quarts. 15 is the water share, which is 3 parts rather than 2. 17 comes from taking 2 parts out of 3 instead of 2 out of 5. 19 is close to the whole 25-quart mix and cannot be the smaller of the two ingredients.
- If 6 identical bolts weigh 18 ounces, how much do 15 of the same bolts weigh?
Correct answer: 45 oz
Fifteen bolts weigh 45 oz. Each bolt weighs 18 divided by 6, which is 3 ounces, so 15 bolts come to 45 ounces. 27 oz is the weight of 9 bolts, 36 oz the weight of 12, and 54 oz the weight of 18; each uses the wrong number of bolts rather than the 15 asked for.
- A scale model uses 1 centimeter for every 8 meters. A bridge drawn 7 centimeters long represents how many meters?
Correct answer: 56
The bridge represents 56 meters. Each centimeter stands for 8 meters, so 7 centimeters stands for 7 times 8, or 56 meters. 49 multiplies 7 by 7 rather than by 8. 63 multiplies 7 by 9, and 70 multiplies 7 by 10; both use a scale larger than the 8 meters per centimeter given.
- A prize fund is divided among three teams in the ratio 1 to 4 to 5. If the fund is $5,000, how much does the team with the largest share get?
Correct answer: $2,500
The largest share is $2,500. The ratio 1 to 4 to 5 makes 10 parts, so one part is $500 and the 5-part share is $2,500. $1,000 divides the fund by the number of parts in the smallest share rather than by the total number of parts. $1,667 splits the fund equally among the three teams and ignores the ratio. $2,000 is the 4-part share, which belongs to the middle team, not the largest one.
- A proportion states that 9 is to 12 as x is to 20. What is x?
Correct answer: 15
x equals 15. Cross-multiplying 9 over 12 equals x over 20 gives 12x equals 180, so x is 15. 17 comes from adding and subtracting the three numbers as 9 plus 20 minus 12 instead of keeping the ratio. 13 and 11 are both too small: 13 over 20 and 11 over 20 are well under the 9 over 12 ratio, which is three quarters.
- In a parking lot the ratio of sedans to vans is 7 to 2. If there are 54 vehicles total, how many are sedans?
Correct answer: 42
There are 42 sedans. The ratio 7 to 2 makes 9 parts, so each part is 54 divided by 9, or 6 vehicles, and the 7 sedan parts come to 42. 12 is the number of vans, which is the 2-part share. 24 doubles the van count instead of scaling the sedan share. 30 splits the lot roughly 5 to 4, far nearer even than the 7 to 2 the question gives.
- If 5 workers can stack a load in 18 minutes, how long would 15 workers take at the same rate?
Correct answer: 6 min
Fifteen workers take 6 min. Tripling the crew from 5 to 15 cuts the time to a third, and a third of 18 minutes is 6 minutes. 5 min repeats the original crew size as though it were a time. 4 min divides 18 by 5 rather than by the factor the crew grew by. 2 min cuts the time by a factor of nine, which would need 45 workers.
- A freight train travels 84 miles in 1 hour and 30 minutes. At the same speed, how far will it go in 4 hours?
Correct answer: 224
The train travels 224 miles. Its speed is 84 divided by 1.5 hours, or 56 miles per hour, and 56 times 4 hours is 224. 336 multiplies the 84 miles by the 4 hours and ignores that the 84 miles took an hour and a half. 280 uses 5 hours at 56 miles per hour, and 392 uses 7 hours; both stretch the journey beyond the 4 hours asked for.
- Two boats leave a dock heading in opposite directions, one at 22 mph and the other at 18 mph. How far apart are they after 3 hours?
Correct answer: 120
They are 120 miles apart. Heading in opposite directions, the boats separate at 22 plus 18, or 40 miles per hour, and 40 times 3 hours is 120 miles. 114 uses a combined rate of 38 miles per hour, 126 uses 42, and 132 uses 44; each mis-adds the two speeds, and only 22 plus 18 equals 40.
- A jogger runs for 20 minutes at 9 miles per hour. How far does he run?
Correct answer: 3 mi
The jogger covers 3 mi. Twenty minutes is a third of an hour, and a third of 9 miles per hour is 3 miles. 9 mi repeats the speed as though the run lasted a full hour. 5 mi and 7 mi both assume far more than 20 minutes of running: at 9 miles per hour, 5 miles takes about 33 minutes and 7 miles about 47.
- A helicopter covers 660 miles in 1 hour and 12 minutes. What is its average speed in miles per hour?
Correct answer: 550
The average speed is 550 miles per hour. One hour and 12 minutes is 1.2 hours, and 660 divided by 1.2 gives 550. 440 divides 660 by 1.5, reading the 12 minutes as half an hour. 660 drops the extra 12 minutes and treats the flight as exactly one hour. 770 is higher still and would cover the 660 miles in under 52 minutes.
- A traveler drives the first 90 miles at 45 mph and the next 90 miles at 90 mph. What is the total travel time?
Correct answer: 3 hr
The total time is 3 hr. The first 90 miles at 45 mph take 2 hours and the next 90 miles at 90 mph take 1 hour, so the trip takes 3 hours. 4 hr runs the whole 180 miles at the slower 45 mph. 5 hr and 7 hr are longer still and would need average speeds of 36 and about 26 miles per hour, both below the slower leg.
- A scout hikes at 3 mph for 90 minutes, then rests, then hikes 2 more miles. What total distance does the scout cover?
Correct answer: 6.5 mi
The scout covers 6.5 mi. Ninety minutes is 1.5 hours, so the hiking leg is 3 times 1.5, or 4.5 miles, and the final 2 miles bring the total to 6.5. 5.0 mi treats the 90 minutes as a single hour, giving only 3 miles before the last 2. 8.0 mi treats it as two full hours, giving 6 miles. 3.5 mi adds the 1.5 hours to the 2 miles as though hours and miles were the same unit.
- A car uses 4 gallons of fuel to travel 120 miles. How many miles can it travel on 7 gallons at the same rate?
Correct answer: 210
The car travels 210 miles. It covers 120 divided by 4, or 30 miles per gallon, and 30 times 7 gallons is 210 miles. 180 is the distance on 6 gallons, 240 the distance on 8, and 270 the distance on 9; each uses a fuel figure other than the 7 gallons asked for.
- A standard deck has 52 cards, 13 of which are hearts. If one card is drawn at random, what is the probability it is a heart?
Correct answer: 1/4
The probability is 1/4. There are 13 hearts among the 52 cards, and 13 over 52 reduces to 1 over 4. 1/3 compares the 13 hearts with the 39 cards that are not hearts instead of with the whole deck. 3/4 is the chance of drawing something other than a heart. 2/3 would mean about 35 of the 52 cards were hearts, far more than the 13 the deck holds.
- A box has 5 working fuses and 3 defective fuses. If one fuse is picked at random, what is the probability it is defective?
Correct answer: 3/8
The probability is 3/8. Three of the 8 fuses in the box are defective, so the chance of picking one is 3 over 8. 5/8 is the chance of picking a working fuse. 3/5 compares the 3 defective fuses with the 5 working ones rather than with all 8. 1/8 is the chance of picking one particular fuse, not any of the three defective ones.
- A single die is rolled once. What is the probability of rolling an even number?
Correct answer: 1/2
The probability is 1/2. The even faces are 2, 4 and 6, so 3 of the 6 faces succeed and 3 over 6 reduces to 1 over 2. 1/6 is the chance of one particular face rather than of any even face. 5/6 would make five of the six faces even, which counts the odd faces as successes too. 1/4 would mean only one and a half of the six faces were even.
- A prize wheel pays $50 with probability 0.1 and pays nothing otherwise. What is the expected payout per spin?
Correct answer: $5.00
The expected payout is $5.00. The $50 prize is paid with probability 0.1, so the long-run average is 50 times 0.1, and the empty outcome contributes nothing. $4.50 works from a $45 prize, $4.00 from a probability of 0.08, and $5.50 from a probability of 0.11; the wheel pays $50 at a probability of 0.1, so none of the three matches the figures given.
- Two dice are rolled. What is the probability that the sum of the two dice equals 12?
Correct answer: 1/36
The probability is 1/36. Two dice fall in 36 equally likely ways and only 6 with 6 totals 12, so the chance is 1 over 36. 1/11 treats the eleven possible totals from 2 to 12 as equally likely, which they are not. 1/12 reads the 12 in the question as the size of the sample space. 2/36 counts 6 with 6 twice, but the two dice showing 6 is a single outcome.
- A jar holds 6 white, 4 black, and 10 gray buttons. If one button is drawn at random, what is the probability it is white?
Correct answer: 3/10
The probability is 3/10. The jar holds 6 plus 4 plus 10, or 20 buttons, and 6 of them are white, so 6 over 20 reduces to 3 over 10. 6/14 compares the 6 white buttons with the 14 that are not white rather than with all 20. 7/10 is the chance of drawing a button that is not white. 1/20 is the chance of drawing one particular button.
- A carnival game costs $5 and offers a 1 in 50 chance to win $150. What is the expected value of the winnings before subtracting the cost?
Correct answer: $3.00
The expected value of the winnings is $3.00. A $150 prize won once in every 50 plays averages 150 divided by 50, or $3.00 a play, before the $5 cost is taken into account. $4.50 reads the 1 in 50 chance as 0.03 rather than 0.02. $3.75 works from a 1 in 40 chance. $2.50 works from a $125 prize, but the prize on offer is $150.
- What is the average of 16, 20, 24, and 28?
Correct answer: 22
The average is 22. The four values add to 88, and 88 divided by 4 is 22. 18 comes from a mis-addition of 72 rather than 88. 20 is simply one of the listed values rather than the mean of all four. 25 would require the four values to total 100, but 16, 20, 24 and 28 total 88.
- A trainee records times of 30, 36, and 33 seconds on three sprints. What is the average time?
Correct answer: 33
The average is 33. The three sprints total 99 seconds and 99 divided by 3 is 33. The figure 36 is simply the slowest single sprint and 30 the fastest, neither of which is an average, and 39 follows from mis-totaling the three times as 117 before dividing.
- The average of six numbers is 14. What is their total sum?
Correct answer: 84
The total sum is 84, because an average of 14 across 6 numbers means the total is 14 multiplied by 6. Multiplying 14 by a count misread as 5 gives 70, by 7 gives 98, and by 4 gives 56, so each of those totals belongs to a different quantity of numbers than the six stated.
- The average of seven readings is 40. If a new reading of 56 is added, what is the new average?
Correct answer: 42
The new average is 42. Seven readings averaging 40 total 280, adding 56 gives 336, and 336 divided by 8 readings is 42. Dividing 336 by the original seven readings gives 48, treating the old total as 8 times 40, or 320, gives 376 over 8, which is 47, and mis-multiplying 7 by 40 as 296 gives 352 over 8, which is 44.
- A team's scores average 76 over 5 games. Four of the scores are 70, 80, 74, and 82. What is the fifth score?
Correct answer: 74
The fifth score is 74. An average of 76 across 5 games means a total of 380, the four known scores sum to 306, and 380 minus 306 leaves 74. The figures 70 and 82 are only scores already listed rather than the missing one, and mis-summing the four known scores as 302 leaves 78.
- The average age of 8 players is 24 years. If a 33-year-old coach is included, what is the new average age of the group?
Correct answer: 25
The new average is 25. The 8 players total 192 years, the coach's 33 years bring the total to 225, and 225 divided by 9 people is 25. The coach sits 9 years above the old average, so spreading those 9 years over 3 people rather than 9 gives 27, subtracting the share instead of adding it gives 23, and averaging the two figures 24 and 33 gives 28.5, which rounds to 29.
- Mentally compute 36 plus 29 plus 64 plus 21 by pairing numbers into round sums.
Correct answer: 150
The sum is 150. Pairing 36 with 64 gives 100 and pairing 29 with 21 gives 50, so the total is 150. Dropping the carry so that 36 and 64 pair to 90 gives 140, mis-adding 29 and 21 as 55 gives 155, and reading the 64 as 59 gives 145.
- Mentally multiply 14 by 50.
Correct answer: 700
The product is 700. Since 50 is half of 100, multiply 14 by 100 to get 1,400 and halve it. Multiplying 14 by a second factor misread as 40 gives 560, as 60 gives 840, and as 70 gives 980, so each of those answers a different question from the one asked.
- What is 20 percent of 350 computed mentally?
Correct answer: 70
The result is 70. Ten percent of 350 is 35, and doubling that for 20 percent gives 70. Stopping at the ten percent step leaves 35, and taking 20 percent of a base misread as 300 or 250 gives 60 and 50 respectively.
- Mentally divide 360 by 6.
Correct answer: 60
The quotient is 60. Splitting 360 into 300 plus 60 and dividing each part by 6 gives 50 plus 10. Dividing 360 by 5 instead of 6 gives 72, dividing it by 4 gives 90, and dividing 480 by 6 gives 80, which misreads the dividend.
- Mentally compute 8 times 102.
Correct answer: 816
The product is 816. Treating 102 as 100 plus 2 turns the work into 800 plus 16. Multiplying 8 by a second factor misread as 100 drops the 2 entirely and gives 800, misreading it as 104 gives 832, and misreading it as 98 gives 784.
- Mentally find 24 times 25.
Correct answer: 600
The product is 600. Treating 25 as 100 divided by 4, multiply 24 by 100 to get 2,400 and then divide by 4. Multiplying 24 by 20 instead of 25 gives 480, while 525 and 575 are 25 multiplied by 21 and by 23, so both misread the first factor.
- Mentally compute 500 minus 264.
Correct answer: 236
The difference is 236, which checks out because 264 plus 236 equals 500. Subtracting 200 and then 60 brings 500 down to 240, and adding the final 4 instead of subtracting it gives 244, transposing the subtrahend to 246 gives 254, and 264 is only the amount taken away restated as the answer.
- If 6y equals 54, what is the value of y?
Correct answer: 9
The value of y is 9. Dividing both sides of 6y equals 54 by 6 leaves y as 54 divided by 6. The figure 6 is only the coefficient copied out of the equation, while 5 and 4 come from dividing 30 and 24 by 6, each of which misreads the 54.
- A number decreased by 7 equals 19. What is the number?
Correct answer: 26
The number is 26. Writing the statement as n minus 7 equals 19 and adding 7 to both sides gives n equals 26. Subtracting the 7 instead of adding it gives 12, the figure 19 is only the stated result restated as though it were the number, and adding 7 a second time gives 33.
- Four times a number plus 6 equals 34. What is the number?
Correct answer: 7
The number is 7. The equation 4n plus 6 equals 34 becomes 4n equals 28 once 6 is subtracted, and dividing by 4 leaves 7. The figure 6 is only the constant copied out of the equation and 4 only the coefficient, while dividing 34 by 4 before subtracting the 6 leaves 2.5, which reads as 2.
- Anna is 3 years younger than Ben. Together their ages total 27. How old is Anna?
Correct answer: 12
Anna is 12. Letting Anna's age be a makes Ben a plus 3, so a plus a plus 3 equals 27, giving 2a equals 24 and a equals 12. Adding the 3-year gap instead of subtracting it gives 15, which is Ben's age; halving 27 while ignoring the gap gives 13.5, read as 13; and halving 27 and then adding the gap gives 16.5, read as 17.
- The sum of three consecutive integers is 60. What is the largest of them?
Correct answer: 21
The largest is 21. Three consecutive integers n, n plus 1 and n plus 2 sum to 3n plus 3, so 3n equals 57, n equals 19 and the largest is 21. The figure 19 is the smallest of the three rather than the largest, subtracting 3 from the middle value of 20 instead of adding gives 17, and dividing 60 by 4 terms instead of 3 gives 15.
- The perimeter of a rectangle is 54 inches, and the length is twice the width. What is the length?
Correct answer: 18
The length is 18 inches. Writing the length as 2w makes the perimeter 2(2w) plus 2w equal 54, which simplifies to 6w equals 54, so the width is 9 and the length 2w is 18. Halving the perimeter gives 27, treating the perimeter as length plus width makes 3w equal 54 and the length 36, and subtracting the 9-inch width from the perimeter gives 45.
- A piggy bank holds nickels and dimes worth $2.30 total. There are 10 nickels. The equation 0.05(10) plus 0.10d equals 2.30 finds the number of dimes d. What is d?
Correct answer: 18
There are 18 dimes. The 10 nickels are worth $0.50, so 0.10d equals 1.80 and d equals 18. Dividing the whole $2.30 by 0.10 and forgetting the nickels gives 23, adding the $0.50 instead of subtracting it gives 28, and removing the 10 nickels as coins rather than as 50 cents gives 13.
- Carl is three times as old as Dana. In 4 years, the sum of their ages will be 32. How old is Dana now?
Correct answer: 6
Dana is 6 now. Letting Dana be d makes Carl 3d, and in 4 years their ages d plus 4 and 3d plus 4 sum to 4d plus 8 equals 32, so 4d equals 24 and d equals 6. Dividing 32 by 4 without first removing the 8 added years gives 8, removing those 8 years twice gives 4, and dividing the 8 added years by 4 rather than the 24 that remains gives 2.
- Five times a number, decreased by 4, equals twice the number plus 11. What is the number?
Correct answer: 5
The number is 5. The equation 5n minus 4 equals 2n plus 11 rearranges to 3n equals 15, so n equals 5. Dividing 15 by the 5 on the left rather than by 3 gives 3, combining the two terms as 7n equals 15 gives roughly 2, and 4 is only a constant lifted straight out of the equation.
- A worker's total pay P is given by P equals 18h plus 25, where h is hours worked. If she earned $313, how many hours did she work?
Correct answer: 16
She worked 16 hours. Subtracting the $25 flat amount from $313 leaves 18h equals 288, and 288 divided by 18 is 16. Dividing the whole $313 by 25 gives about 12, dividing the 288 remainder by 20 gives about 14, and dividing it by 16 gives 18, so each of those uses a figure the formula never supplies.
- A toolbox holds wrenches and pliers totaling 24 tools. There are twice as many wrenches as pliers. How many pliers are there?
Correct answer: 8
There are 8 pliers. Letting pliers be p makes wrenches 2p, so p plus 2p equals 24, giving 3p equals 24 and p equals 8. Splitting the 24 tools into 4 equal shares gives 6, into 6 shares gives 4 and into 8 shares gives 3, but a two-to-one ratio makes 3 shares the only correct count.
- A warehouse receives 9 pallets of 45 boxes each, then ships out 130 boxes. How many boxes remain?
Correct answer: 275
There are 275 boxes remaining. Nine pallets of 45 boxes total 405, and subtracting the 130 shipped leaves 275. The figure 405 is the delivery with nothing shipped out, counting 10 pallets instead of 9 leaves 320, and mis-multiplying 9 by 45 as 495 leaves 365.
- A squad orders meals for 6 members at $11 each, plus a $9 delivery fee, then splits the total evenly among the 6 members. How much does each member pay?
Correct answer: $12.50
Each member pays $12.50. Six meals at $11 come to $66, the $9 delivery fee brings the total to $75, and $75 split among 6 members is $12.50. Leaving the delivery fee out gives $11.00, mis-adding the total as $81 gives $13.50, and dividing $75 among 5 members gives $15.00.
- A printer produces 24 pages per minute. After running for 5 minutes, 18 pages are discarded as misprints. How many usable pages remain?
Correct answer: 102
There are 102 usable pages. Twenty-four pages a minute over 5 minutes is 120 pages, and removing the 18 misprints leaves 102. The figure 120 is the run before any misprints are removed, adding the 18 instead of subtracting gives 138, and running the printer for 7 minutes gives 168 less 18, or 150.
- A depot stores 1,500 sandbags and uses 175 per day for 7 days. How many sandbags are left?
Correct answer: 275
There are 275 sandbags left. Seven days at 175 a day uses 1,225, and 1,500 minus 1,225 leaves 275. Counting 8 days uses 1,400 and leaves 100, counting 6 days uses 1,050 and leaves 450, and counting 5 days uses 875 and leaves 625.
- A purchase includes 4 helmets at $32 each and 3 vests at $48 each. What is the total cost?
Correct answer: $272
The total cost is $272. Four helmets at $32 come to $128 and three vests at $48 come to $144, and $128 plus $144 is $272. Pricing all seven items at the $32 helmet rate gives $224, buying four vests rather than three gives $320, and counting only one vest gives $176.
- A reservoir gains 14 gallons per hour from rain but loses 9 gallons per hour to a drain. Starting with 200 gallons, how many gallons are present after 8 hours?
Correct answer: 240
There are 240 gallons present. The net change is 14 minus 9, or 5 gallons an hour, so 8 hours add 40 gallons to the starting 200. Counting only the 9 gallons lost each hour leaves 128, subtracting the 40-gallon net gain instead of adding it leaves 160, and 200 is the starting volume with the two rates wrongly treated as cancelling.
- A courier makes 5 trips carrying 34 letters each, then 4 trips carrying 22 letters each. How many letters are delivered in all?
Correct answer: 258
A total of 258 letters are delivered. Five trips of 34 letters is 170, four trips of 22 letters is 88, and 170 plus 88 is 258. Counting five trips at each load gives 170 plus 110, or 280; counting six trips of 34 gives 204 plus 88, or 292; and treating all nine trips as 34 letters each gives 306.
- CAR is to GARAGE as PLANE is to ___.
Correct answer: HANGAR
HANGAR completes the analogy. A garage is the structure that stores and shelters a car, so the matching word for a plane is the building that stores and shelters it. A runway is the strip a plane takes off from, a tarmac is the open apron it parks on briefly, and an engine is a part of the plane rather than any kind of building.
- HAPPY is to SAD as HOT is to ___.
Correct answer: COLD
COLD completes the analogy. Happy and sad are opposites, so the word opposite in meaning to hot is needed, and cold is its direct antonym. Warm names a lesser degree of the same heat, heat names the quality itself rather than its opposite, and an oven is a thing that produces heat.
- PETAL is to FLOWER as ___ is to BOOK.
Correct answer: PAGE
PAGE completes the analogy. A petal is one part of a whole flower, so the missing word must be one part of a whole book, and a page is exactly that. A tome is another name for a whole book rather than a part of one, read is the action performed on a book, and a desk is furniture a book rests on.
- FIRE is to SMOKE as ___ is to FLOOD.
Correct answer: RAIN
RAIN completes the analogy. Fire is the cause and smoke the effect it produces, so the missing word must be the cause that produces a flood, and heavy rain is that cause. Silt is what a flood leaves behind and so sits in the effect position, a lake is a standing body of water rather than a cause, and a boat is used to escape flooding.
- KNIFE is to CUT as PEN is to ___.
Correct answer: WRITE
WRITE completes the analogy. A knife is a tool and cutting is the action it performs, so the missing word is the action a pen performs, which is to write. Carve names a second action of a knife rather than of a pen, erase is the action of a rubber, and speak is a way of communicating that needs no tool at all.
- WARM is to HOT as COOL is to ___.
Correct answer: COLD
COLD completes the analogy. Warm names a lesser degree and hot a greater degree of the same quality, so the missing word is the greater degree matching cool, which is cold. Mild names another lesser degree rather than a greater one, wind is a thing that cools rather than a point on the scale, and damp describes moisture instead of temperature.
- BIG is to LARGE as TINY is to ___.
Correct answer: SMALL
SMALL completes the analogy. Big and large mean nearly the same thing, so the analogy pairs synonyms, and the synonym of tiny is small. Bulky means the opposite of tiny, broad describes width alone, and thick describes depth or density rather than overall size.
- WHEEL is to BICYCLE as ___ is to HOUSE.
Correct answer: ROOM
ROOM completes the analogy. A wheel is a component part of a whole bicycle, so the missing word must be a part of a whole house, and a room is exactly that. A town is the larger whole a house belongs to, which reverses the direction of the relationship; rent is a payment made for a house; and a deed is the document recording who owns it.
- SCISSORS is to CLOTH as ___ is to WOOD.
Correct answer: SAW
SAW completes the analogy. Scissors are the tool that cuts cloth, so the missing word is the tool that cuts wood, and a saw does that. An awl bores small holes rather than cutting, a peg fastens two pieces of wood together, and wax polishes a finished surface.
- DOCTOR is to HOSPITAL as TEACHER is to ___.
Correct answer: SCHOOL
SCHOOL completes the analogy. A doctor works in a hospital, so the analogy links a worker to the place of work, and a teacher works in a school. A course is the program of study rather than a place, a parent is someone the teacher deals with, and a salary is the pay the work earns.
- OVEREAT is to OBESITY as ___ is to DROWSINESS.
- FATIGUE
- CRAVING
- STRETCH
- SUNRISE
Correct answer: FATIGUE
FATIGUE completes the analogy. Overeating is the cause and obesity the result, so the blank must name a cause that brings on drowsiness, and fatigue does exactly that. A craving is a wish for food and belongs to the eating half of the pair, a stretch is something people do to shake drowsiness off, and a sunrise wakes a sleeper rather than making one drowsy.
- In the analogy SAPLING is to TREE as KITTEN is to ___, the relationship between the first pair is one of:
- Whole to parts
- Maker to goods
- Plant to fruit
- Young to adult
Correct answer: Young to adult
Young to adult is the relationship. A sapling is a young tree that grows into a mature one, and a kitten grows into a cat in the same way, so the bridge is the step from an immature form to its grown form. The pair is not a whole set beside its parts, not a maker set beside the goods produced, and not a plant set beside the fruit it bears.
- WHISPER is to SHOUT as SIP is to ___.
Correct answer: GULP
GULP completes the analogy. A whisper is a small amount of sound and a shout is a large amount, so the blank must be the large version of a sip, and a gulp is a large swallow. To pour is to move liquid from one container into another rather than to drink it, to spit is to force liquid back out, and to lick takes even less than a sip does.
- BIRD is to FLOCK as ___ is to HERD.
Correct answer: COW
COW completes the analogy. A bird is one member of the group called a flock, so the blank must name one member of a herd, and a cow is one animal within a herd. A hen belongs to a flock rather than to a herd, hay is what the animals eat, and a pen is the enclosure that holds them.
- BRAVE is to COWARDLY as GENEROUS is to ___.
Correct answer: STINGY
STINGY completes the analogy. Brave and cowardly are opposites, so the blank must be the opposite of generous, and a stingy person is unwilling to give. Giving and lavish both describe the same open-handedness that generous describes, and humble names modesty about oneself rather than any refusal to share.
- BROOM is to SWEEP as SHOVEL is to ___.
Correct answer: DIG
DIG completes the analogy. A broom is the tool and sweeping is what it does, so the blank must be what a shovel does, and a shovel is used to dig. A pit is the hole left behind once the work is done, sod is the material lifted, and a hoe is a different garden tool rather than an action.
- GLOVE is to HAND as SHOE is to ___.
Correct answer: FOOT
FOOT completes the analogy. A glove is worn on a hand, so the blank must name the body part a shoe is worn on, and a shoe is worn on a foot. A shin is a body part that a shoe leaves bare, a boot is another covering rather than the part being covered, and a step is something the wearer takes.
- DRIZZLE is to DOWNPOUR as ___ is to BLIZZARD.
Correct answer: FLURRY
FLURRY completes the analogy. A drizzle is light rain and a downpour is heavy rain, so the blank must be the light form of the snowfall that grows into a blizzard, and a flurry is light snow. A shower is light rain and merely repeats the first pair, winter is the season blizzards fall in, and an icicle is formed by freezing water rather than by falling snow.
- DRUMMER is to BAND as ___ is to TEAM.
Correct answer: PLAYER
PLAYER completes the analogy. A drummer is one member of the group called a band, so the blank must name one member of a team, and a player is one member of a team. An umpire oversees the game without belonging to either side, a jersey is what a member wears, and a league is the wider body that whole teams belong to.
- VIRUS is to ILLNESS as SPARK is to ___.
Correct answer: FIRE
FIRE completes the analogy. A virus is the cause and illness is the result, so the blank must be what a spark brings about, and a spark starts a fire. Fuel is the material that burns rather than the event itself, a wire is a source a spark can jump from and so comes before it, and a lamp is a manufactured light that no spark produces.
- PENCIL is to WRITE as SCISSORS is to ___.
Correct answer: CUT
CUT completes the analogy. A pencil is the tool and writing is what it does, so the blank must be what scissors do, and scissors are used to cut. To sew is the action of a needle rather than of scissors, a pin is another sewing tool rather than an action, and the tip is a part of the scissors themselves.
- ANCIENT is to OLD as ___ is to BIG.
Correct answer: HUGE
HUGE completes the analogy. Ancient means extremely old, the intensified form of old, so the blank must be the intensified form of big, and huge means extremely big. Tiny runs the other way and means extremely small, while tall and long each describe one dimension rather than an extreme of overall size.
- FISH is to SCHOOL as WOLF is to ___.
Correct answer: PACK
PACK completes the analogy. A group of fish is called a school, so the blank must name the group that wolves form, and wolves travel in a pack. A herd is the collective term for grazing animals rather than for wolves, a lair is the place a wolf shelters in, and a howl is the sound it makes.
- In the analogy KEY is to LOCK as SOLUTION is to ___, the bridge between the first pair is best described as:
- Tool to the problem it solves
- Piece to the pattern it forms
- Cause to the result it brings
- Word to the opposite it names
Correct answer: Tool to the problem it solves
Tool to the problem it solves is the relationship. A key is the implement that opens a lock, and a solution is the implement that settles a problem, so the bridge joins a device to the difficulty it clears. The pair is not a piece set within a pattern, not a cause set before a result, and not a word set against its opposite.
- THERMOMETER is to TEMPERATURE as CLOCK is to ___.
Correct answer: TIME
TIME completes the analogy. A thermometer is the instrument and temperature is the quantity it measures, so the blank must be the quantity a clock measures, and a clock measures time. A dial is the face that gets read, a gear is a part inside the case, and a tick is the sound the mechanism makes.
- ENEMY is to FRIEND as FAILURE is to ___.
- SETBACK
- SUCCESS
- ATTEMPT
- FORTUNE
Correct answer: SUCCESS
SUCCESS completes the analogy. Enemy and friend are opposites, so the blank must be the opposite of failure, and success is the opposite of failure. A setback means much the same as a failure, an attempt is the try that may end either way, and fortune names luck or riches rather than the reverse of failing.
- BAKER is to BREAD as ___ is to FURNITURE.
- ARMCHAIRS
- CARPENTER
- SANDPAPER
- APARTMENT
Correct answer: CARPENTER
CARPENTER completes the analogy. A baker is the worker who makes bread, so the blank must name the worker who makes furniture, and a carpenter makes furniture. Armchairs are examples of the furniture itself, sandpaper is a supply used in the workshop, and an apartment is where the finished pieces end up.
- PUDDLE is to LAKE as HILL is to ___.
- MOUNTAIN
- FOOTHILL
- LOWLANDS
- RIVERBED
Correct answer: MOUNTAIN
MOUNTAIN completes the analogy. A puddle is a small body of water and a lake is a far larger one of the same kind, so the blank must be a far larger landform than a hill, and a mountain is exactly that. A foothill is a small rise at the base of a range and so is no larger, lowlands lie below the surrounding ground rather than above it, and a riverbed is a channel rather than a raised landform.
- NEEDLE is to SEW as KEY is to ___.
Correct answer: UNLOCK
UNLOCK completes the analogy. A needle is the tool and sewing is what it does, so the blank must be what a key does, and a key is used to unlock. Thread is what a needle carries and belongs to the first pair, a pocket is where a key is kept, and hinges are parts of the door rather than anything the key does.
- DARK is to LIGHT as ROUGH is to ___.
Correct answer: SMOOTH
SMOOTH completes the analogy. Dark and light are opposites, so the blank must be the opposite of rough, and smooth is the opposite of rough. Gritty and jagged both describe the same coarse texture that rough describes, and sturdy describes strength rather than the feel of a surface.
- DROUGHT is to FAMINE as ___ is to FLOOD.
- SANDBAGS
- DRAINAGE
- RAINFALL
- HUMIDITY
Correct answer: RAINFALL
RAINFALL completes the analogy. A drought is the cause that ends in famine, so the first blank must be the cause that ends in a flood, and heavy rainfall is what makes a river burst its banks. Sandbags are piled up to hold floodwater back, drainage carries water away and so prevents the flood, and humidity is moisture held in the air rather than water falling in quantity.
- STRING is to GUITAR as ___ is to PIANO.
Correct answer: KEY
KEY completes the analogy. A string is the part of a guitar that is set moving to sound a note, so the blank must be the matching part of a piano, and a key is the part pressed to sound a note. A peg is a tuning part of the guitar and belongs to the first pair, a bow is drawn across a violin and not a piano, and an ear hears the note instead of producing it.
- PAINTER is to BRUSH as WRITER is to ___.
Correct answer: PEN
PEN completes the analogy. A painter works with a brush, so the blank must name the tool a writer works with, and a writer works with a pen. Ink is the substance the pen lays down rather than the tool itself, oil is the painter's medium and belongs to the first pair, and an ode is something a writer produces.
- GLAD is to ECSTATIC as ___ is to FURIOUS.
- ENRAGED
- RELAXED
- CONTENT
- ANNOYED
Correct answer: ANNOYED
ANNOYED completes the analogy. Glad is a mild feeling and ecstatic is that feeling at full strength, so the blank must be the mild form of furious, and being annoyed is mild anger. Enraged is anger at full strength and so matches furious itself, relaxed describes calm rather than any degree of anger, and content describes quiet pleasure in the way glad does.
- In the analogy DICTIONARY is to WORDS as ATLAS is to ___, the relationship between the first pair is one of:
- A reference and its contents
- A statement and its opposite
- A component and its assembly
- A landslide and its injuries
Correct answer: A reference and its contents
A reference and its contents is the relationship. A dictionary is a reference book holding words and an atlas is a reference book holding maps, so the bridge joins a work of reference to the material gathered inside it. The pair is not a statement set beside its opposite, not a component set beside the assembly it belongs to, and not a landslide set beside the injuries it causes.
- FIN is to FISH as WING is to ___.
Correct answer: BIRD
BIRD completes the analogy. A fin is a body part of the whole fish, so the blank must name the creature a wing belongs to, and a wing is part of a bird. A gill is a part of the fish and belongs to the first pair, a beak is another part rather than the whole creature, and a flap is the motion a wing makes.
- FAUCET is to WATER as SWITCH is to ___.
- TEMPERATURE
- ELECTRICITY
- VENTILATION
- TRANSFORMER
Correct answer: ELECTRICITY
ELECTRICITY completes the analogy. A faucet is the device that governs the flow of water, so the blank must be the flow a switch governs, and a switch governs the flow of electricity. Temperature is what a thermostat governs, ventilation is the airflow a vent governs, and a transformer is another piece of equipment rather than a flow at all.
- BEGIN is to START as CONCLUDE is to ___.
Correct answer: FINISH
FINISH completes the analogy. Begin and start mean the same thing, so the blank must mean the same as conclude, and to finish is to conclude. To launch is to begin and runs the other way, to linger is to stay on past the proper end, and to resume is to take something up again after a break.
- BACTERIA is to INFECTION as POLLUTION is to ___.
Correct answer: SMOG
SMOG completes the analogy. Bacteria are the cause and infection is the result, so the blank must be the result that pollution produces, and heavy pollution settles over a city as smog. Cars give off pollution and sit on the cause side, soot is itself a pollutant rather than an outcome of pollution, and a mask is worn to keep the dirty air out.
- SCALE is to WEIGHT as RULER is to ___.
Correct answer: LENGTH
LENGTH completes the analogy. A scale is the instrument and weight is the quantity it reports, so the blank must be the quantity a ruler reports, and a ruler measures length. Volume is read from a measuring jug, a degree is an angle read from a protractor, and a marker is a separate drawing tool.
- COMPOSER is to SYMPHONY as ___ is to NOVEL.
Correct answer: AUTHOR
AUTHOR completes the analogy. A composer is the person who creates a symphony, so the blank must name the person who creates a novel, and an author writes the novel. An editor works over a manuscript someone else has written, a critic judges the finished book, and a sequel is another book rather than a person.
- COTTAGE is to MANSION as PEBBLE is to ___.
- SHINGLE
- GRANITE
- MINERAL
- BOULDER
Correct answer: BOULDER
BOULDER completes the analogy. A cottage is a small dwelling and a mansion is a far larger one of the same kind, so the blank must be a far larger rock than a pebble, and a boulder is exactly that. Shingle is the small loose stone of a beach and is no bigger than a pebble, granite names the kind of rock rather than its size, and mineral is a broad category that says nothing about size.
- ROOF is to BUILDING as LID is to ___.
Correct answer: JAR
JAR completes the analogy. A roof is the top that closes a building, so the blank must name the container a lid closes, and a lid closes a jar. Jam is what sits inside the jar, the rim is the edge the lid rests on rather than the whole vessel, and a cap is another closure and so matches the lid instead of the container.
- INHALE is to EXHALE as IMPORT is to ___.
Correct answer: EXPORT
EXPORT completes the analogy. Inhale and exhale are opposite movements, so the blank must be the opposite of import, and to export is to send goods out where importing brings them in. A tariff is a tax charged on trade, a market is the place where trade happens, and supply is the quantity available rather than a movement in either direction.
- SUN is to SHADOW as ___ is to ECHO.
Correct answer: SOUND
SOUND completes the analogy. The sun is the cause that throws a shadow, so the first blank must be the cause that throws an echo, and a sound returning off a surface is an echo. Light is what the sun gives out and belongs to the first pair, a cliff is the surface an echo bounces from, and quiet is the absence that leaves nothing to return.
- AX is to CHOP as HAMMER is to ___.
Correct answer: POUND
POUND completes the analogy. An ax is the tool and chopping is what it does, so the blank must be what a hammer does, and a hammer is used to pound. Nails are what the hammer strikes rather than the action, a plank is the material worked on, and a shelf is the thing built once the work is done.
- In the analogy CALM is to PANIC as TRICKLE is to FLOOD, the relationship in BOTH pairs is best described as:
- A near match against a twin phrase
- A mild degree against a severe one
- A known cause against a grave harm
- A handy tool against a daily chore
Correct answer: A mild degree against a severe one
A mild degree against a severe one is the relationship. Calm is the quiet state and panic is the same feeling at full pitch, while a trickle is a thin flow and a flood is that flow at full pitch, so both pairs set a low degree beside an intense one. The pairs are not a near match beside a twin phrase, not a known cause beside the grave harm it works, and not a handy tool beside the daily chore it performs.
- BLADE is to KNIFE as ___ is to AX.
Correct answer: HANDLE
HANDLE completes the analogy. A blade is one of the parts a knife is built from, so the blank must name a part an ax is built from, and an ax has a handle. Timber is the wood an ax cuts, a sheath is a cover that holds a knife and belongs to the first pair, and a strike is the action rather than a part.
- CHEF is to KITCHEN as JUDGE is to ___.
- TESTIMONY
- PRECEDENT
- COURTROOM
- ATTORNEYS
Correct answer: COURTROOM
COURTROOM completes the analogy. A chef does the work in a kitchen, so the blank must name the place a judge does the work, and a judge sits in a courtroom. Testimony is the evidence heard there, precedent is the body of past rulings a judge draws on, and attorneys are the other people in the case.
- WEALTHY is to POOR as EXPAND is to ___.
- INCREASE
- MULTIPLY
- CONTRACT
- SOLIDIFY
Correct answer: CONTRACT
CONTRACT completes the analogy. Wealthy and poor are opposites, so the blank must be the opposite of expand, and to contract is to draw in and grow smaller. To increase and to multiply both mean to grow in the same direction as expanding, and to solidify is to harden into a solid rather than to change in size.
- SEED is to PLANT as EGG is to ___.
Correct answer: BIRD
BIRD completes the analogy. A seed grows into a plant, so the pair links a starting form to the living thing it becomes, and an egg grows into a bird. A coop is only where poultry are housed, a yolk is material inside the egg rather than the creature it turns into, and hens lay eggs instead of developing out of them.
- TELESCOPE is to ASTRONOMER as ___ is to SURGEON.
- ANATOMY
- SCALPEL
- BANDAGE
- THEATER
Correct answer: SCALPEL
SCALPEL completes the analogy. A telescope is the instrument an astronomer works with, so the missing word must be the instrument a surgeon works with, and a surgeon works with a scalpel. Anatomy is the subject studied rather than a tool, a bandage is a dressing applied once the cutting is over, and a theater is the room the operation happens in.
- NIBBLE is to DEVOUR as TAP is to ___.
Correct answer: POUND
POUND completes the analogy. A nibble is a small act and to devour is that same act at full force, so the missing word must be a tap at full force, and to pound is to strike with full force. To brush and to graze are as light as a tap rather than stronger, and to press is steady pressure rather than a heavy blow.
- PILOT is to AIRPLANE as ___ is to SHIP.
- SEAPORT
- CAPTAIN
- COMPASS
- CURRENT
Correct answer: CAPTAIN
CAPTAIN completes the analogy. A pilot is the person who commands an airplane, so the missing word must be the person who commands a ship, and a captain commands a ship. A seaport is where the vessel ties up, a compass is equipment carried aboard, and a current is water the hull moves through, so not one of them is a person.
- LEAF is to TREE as FEATHER is to ___.
Correct answer: BIRD
BIRD completes the analogy. A leaf is a part that grows on the whole tree, so the missing word must be the whole creature a feather grows on, and a feather grows on a bird. A nest is a structure the creature builds, a beak is one more part rather than the whole, and a flap is a motion the wings make.
- PRACTICE is to SKILL as ___ is to MUSCLE.
- SKELETON
- NUTRIENT
- EXERCISE
- RECOVERY
Correct answer: EXERCISE
EXERCISE completes the analogy. Practice is the activity that builds skill, so the first term must be the activity that builds muscle, and exercise builds muscle. A skeleton is a separate body structure, a nutrient supplies fuel without doing any work, and recovery is the rest that follows the work rather than the work itself.
- VACCINE is to DISEASE as ___ is to FIRE.
- FLAMETHROWER
- ASPHYXIATION
- FLAMMABILITY
- EXTINGUISHER
Correct answer: EXTINGUISHER
EXTINGUISHER completes the analogy. A vaccine is what stops a disease from taking hold, so the missing word must be what stops a fire, and an extinguisher puts a fire out. A flamethrower starts one, asphyxiation is harm that a fire causes, and flammability is the property that lets material burn, so not one of them halts a blaze.
- CHAPTER is to BOOK as ACT is to ___.
Correct answer: PLAY
PLAY completes the analogy. A chapter is one major division of a whole book, so the missing word must be the whole work an act divides, and an act is one major division of a play. A cast is the group of performers, a line is a far smaller unit of the script, and a hall is the building the work is staged in.
- In the analogy LION is to CUB as HORSE is to FOAL, the relationship between the first pair is one of:
- Hunter to its prey
- Unit to its system
- Species to its kin
- Adult to its young
Correct answer: Adult to its young
Adult to its young is the relationship. A lion is the grown animal and a cub is its offspring, exactly as a horse is grown and a foal is its offspring, so the bridge runs from a mature animal to an immature one. Neither pair names a hunter and what it eats, neither names a unit sitting inside a larger system, and neither sorts a species into related kinds.
- WET is to DRY as ASLEEP is to ___.
Correct answer: AWAKE
AWAKE completes the analogy. Wet and dry are opposite conditions, so the missing word must be the opposite of asleep, and awake is the opposite of asleep. Weary describes being close to sleep rather than out of it, dozed names a light form of sleep, and a dream is something that occurs while sleep continues.
- BEE is to HIVE as ___ is to COLONY.
Correct answer: ANT
ANT completes the analogy. A bee is the individual creature that lives in a hive, so the missing word must be the individual creature that lives in a colony, and an ant lives in a colony. Wax is material the hive is built from, a den is a different kind of animal home, and an egg is a stage the insect passes through before it forages.
- BROOM is to JANITOR as WRENCH is to ___.
- FASTENER
- WORKSHOP
- MECHANIC
- HARDWARE
Correct answer: MECHANIC
MECHANIC completes the analogy. A broom is the tool a janitor works with, so the missing word must be the worker who handles a wrench, and a mechanic handles a wrench. A fastener is the part the wrench turns, a workshop is the place the job is done, and hardware is a class of parts rather than a person.
- FREEZE is to BOIL as ___ is to MAXIMUM.
- EXTREME
- OPTIMUM
- MINIMUM
- TYPICAL
Correct answer: MINIMUM
MINIMUM completes the analogy. Freeze and boil sit at opposite ends of one temperature scale, so the missing word must be the end opposite maximum, and minimum is the opposite of maximum. An extreme names either end without choosing one, an optimum is a best value somewhere inside the range, and typical describes an ordinary middling value.
- YAWN is to BOREDOM as SHIVER is to ___.
Correct answer: COLD
COLD completes the analogy. A yawn is the reaction boredom produces, so the missing word must be the condition that produces a shiver, and cold produces a shiver. Heat is the opposite condition, an ache is a separate bodily symptom rather than a cause, and a coat is what relieves the chill instead of bringing it on.
- HUNGRY is to FAMISHED as ___ is to EXHAUSTED.
Correct answer: TIRED
TIRED completes the analogy. Hungry is a mild state and famished is that same state at its extreme, so the first term must be the mild form of exhausted, and tired is the mild form of exhausted. Bored describes interest rather than energy, tense describes strain, and alert is the opposite of being worn out.
- GARDENER is to RAKE as FISHERMAN is to ___.
Correct answer: NET
NET completes the analogy. A gardener works with a rake, so the missing word must be the implement a fisherman works with, and a fisherman works with a net. The sea is where the work is done, a cod is part of the catch, and a tug is a craft that carries the crew rather than an implement.
- In the analogy STITCH is to TAILOR as WELD is to BLACKSMITH, the relationship in BOTH pairs is best described as:
- Outcome to the source producing it
- Action to the worker performing it
- Portion to the whole containing it
- Symbol to the opposite negating it
Correct answer: Action to the worker performing it
Action to the worker performing it is the relationship. A stitch is the act a tailor carries out, exactly as a weld is the act a blacksmith carries out, so each pair runs from a task to the tradesman doing it. Neither pair runs from an outcome back to the source producing it, neither sets a portion against the whole containing it, and neither sets a symbol against the opposite negating it.
- PAGE is to BOOK as ___ is to CHAIN.
Correct answer: LINK
LINK completes the analogy. A page is the single repeated unit a whole book is made of, so the missing word must be the repeated unit a chain is made of, and a link is one unit of a chain. Iron is the material rather than the unit, a cord is a different object entirely, and a hook is an attachment fitted to one end.
- PARCHED is to MOIST as STARVING is to ___.
Correct answer: FULL
FULL completes the analogy. Parched means extremely dry and moist is its opposite, so the missing word must be the opposite of starving, and full is the opposite of starving. Arid describes dryness instead of hunger, thin is a result of going without food rather than its reverse, and dine names an action rather than a condition.
- What number comes next in the series 9, 13, 17, 21, ___ ?
Correct answer: 25
The next number is 25. The terms climb by a fixed difference of 4, so 21 plus 4 gives 25. Adding that difference twice gives 29 and adding it three times gives 33, while 22 comes from treating the step as 1, so none of those advances the series by a single proper step.
- Find the next term in the series 14, 22, 30, 38, ___ .
Correct answer: 46
The next term is 46. Each term rises by a constant 8, so 38 plus 8 equals 46. Taking the step as 2 gives 40, as 5 gives 43 and as 11 gives 49, and none of those matches the difference that actually separates every pair of listed terms.
- What is the missing number in the series 60, 52, 44, 36, ___ ?
Correct answer: 28
The missing number is 28. Each term falls by a constant 8, so 36 minus 8 leaves 28. Subtracting 12 gives 24, subtracting 16 gives 20 and subtracting 20 gives 16, so every one of those removes more than the series takes away at each step.
- Which number continues the series 2, 11, 20, 29, ___ ?
Correct answer: 38
The continuing number is 38. The terms rise by a fixed 9, so 29 plus 9 equals 38. Taking the step as 5 gives 34, as 13 gives 42 and as 17 gives 46, so none of those respects the gap that holds between every pair of listed terms.
- What number is missing in the series 88, 76, 64, 52, ___ ?
Correct answer: 40
The missing number is 40. Each term drops by 12, so 52 minus 12 leaves 40. Subtracting only 6 gives 46, subtracting 18 gives 34 and subtracting 24 gives 28, so each of those applies a step the series never uses.
- Find the next number in the series 15, 26, 37, 48, 59, ___ .
Correct answer: 70
The next number is 70. The series adds 11 at every step, so 59 plus 11 equals 70. Adding 5 gives 64, adding 8 gives 67 and adding 14 gives 73, and none of those matches the difference found between any two listed terms.
- A series begins 8, 16, 24, 32, 40. What is the ninth term?
Correct answer: 72
The ninth term is 72. The common difference is 8 and the ninth term is the first term plus eight steps of 8, so 8 plus 64 gives 72. Taking ten steps instead of eight gives 88, counting 40 as the fourth term gives 80, and multiplying 8 by twelve gives 96, so each of those miscounts how far along the ninth term sits.
- What number completes the series 18, 24, 30, ___, 42 ?
Correct answer: 36
The completing number is 36. The series rises by 6 each step, so the value after 30 is 30 plus 6, which is 36, and 36 plus 6 gives 42. Splitting the gap from 30 to 42 into four parts gives 33, working back from 42 by 3 gives 39 and working back by 1 gives 41, so none of those leaves an equal step on both sides of the blank.
- Find the missing term in the series 5, ___, 17, 23, 29.
Correct answer: 11
The missing term is 11. The series rises by 6 each step, so the value between 5 and 17 is 5 plus 6, which is 11. Working back from 17 by 3 gives 14, by 2 gives 15 and by 1 gives 16, and each of those leaves the first gap far wider than every later one.
- What number comes next in the series 21, 19, 17, 15, ___ ?
Correct answer: 13
The next number is 13. Each term falls by a constant 2, so 15 minus 2 leaves 13. Subtracting 3 gives 12, subtracting 5 gives 10 and subtracting 6 gives 9, so each of those takes away more than the steady step this series uses.
- What number comes next in the series 4, 12, 36, 108, ___ ?
Correct answer: 324
The next number is 324. Each term is multiplied by 3, so 108 times 3 gives 324. Adding the two previous terms gives 144, doubling 108 gives 216, and reading the series as successive powers of 3 gives 243, so none of those applies the constant multiplier the terms actually follow.
- Find the next term in the series 7, 14, 28, 56, ___ .
Correct answer: 112
The next term is 112. Each value doubles, so 56 times 2 gives 112. Tripling 56 gives 168, adding the previous gap of 84 gives 140, and squaring 14 gives 196, so none of those holds the ratio of 2 that links every listed pair.
- What number comes next in the series 1, 6, 36, 216, ___ ?
Correct answer: 1296
The next number is 1296. Each term is multiplied by 6, so 216 times 6 gives 1296. Reading the previous term as 206 gives 1236, as 226 gives 1356 and as 236 gives 1416, so each of those multiplies a misread figure rather than the term the series actually reached.
- Which number continues the series 2, 14, 98, 686, ___ ?
Correct answer: 4802
The continuing number is 4802. Each term is multiplied by 7, so 686 times 7 gives 4802. Multiplying by 3 gives 2058, by 5 gives 3430 and by 6 gives 4116, so each of those uses a ratio smaller than the one the earlier terms establish.
- What is the missing number in the series 96, 48, 24, 12, ___ ?
Correct answer: 6
The missing number is 6. Each term is divided by 2, so 12 divided by 2 leaves 6. The gaps shrink by half each time, and subtracting 3, 4 or 5 from 12 gives 9, 8 or 7, so each of those treats the pattern as a plain subtraction rather than a halving.
- Find the next term in the series 5, 15, 45, 135, ___ .
Correct answer: 405
The next term is 405. Each value is multiplied by 3, so 135 times 3 gives 405. Dropping the middle carry from that product gives 315, while reading 135 as 165 gives 495 and reading it as 195 gives 585, so each of those mishandles the very same multiplication.
- What number completes the series 625, 125, 25, 5, ___ ?
Correct answer: 1
The completing number is 1. Each term is divided by 5, so 5 divided by 5 leaves 1. The gaps between terms run 500, 100 and 20, so the next gap is 4, and mistaking that gap for the term itself gives 4, while adding rather than dividing gives 6 or 9.
- What number completes the series 3, 12, ___, 192, 768 ?
Correct answer: 48
The completing number is 48. Each term is multiplied by 4, so the value after 12 is 12 times 4, which is 48, and 48 times 4 gives 192. Adding 4 to 12 gives 16, doubling that gives 32, and dividing 192 by 3 gives 64, so none of those keeps the same ratio on both sides of the blank.
- Find the next term in the series 1000, 100, 10, 1, ___ .
Correct answer: 0.10
The next term is 0.10. Each value is divided by 10, so 1 divided by 10 leaves 0.10. Slipping the decimal two places instead of one gives 0.01, while treating the pattern as a steady subtraction from 1 gives 0.07 or 0.04, so none of those divides by ten exactly once.
- What number comes next in the series 9, 18, 36, 72, ___ ?
Correct answer: 144
The next number is 144. Each term doubles, so 72 times 2 gives 144. Tripling 72 gives 216, adding 90 gives 162 and adding 126 gives 198, so none of those keeps the ratio of 2 that links every listed pair.
- What number comes next in the series 1, 2, 3, 5, 8, 13, 21, ___ ?
Correct answer: 34
The next number is 34. Each term is the sum of the two before it, so 13 plus 21 gives 34. Adding 5 to 21 skips a term and gives 26, adding 5 and 13 gives 18, and doubling 21 gives 42, so none of those sums the pair that immediately precedes the blank.
- Find the next term in the series 4, 4, 8, 12, 20, ___ .
Correct answer: 32
The next term is 32. Each value equals the sum of the two preceding values, so 12 plus 20 equals 32. 22, 26 and 29 would each require the term sitting before 20 to be 2, 6 or 9 rather than the 12 that is actually there, so none of them completes the additive rule.
- What number continues the series 2, 4, 6, 10, 16, ___ ?
Correct answer: 26
The continuing number is 26. Each term is the sum of the two before it, so 10 plus 16 equals 26. 20 pairs 16 with the 4 instead of the 10 that immediately precedes it, 32 simply doubles 16, and 38 totals every term in the series rather than only the last two.
- What is the missing number in the series 6, 9, 15, 24, 39, ___ ?
Correct answer: 63
The missing number is 63. Each term equals the sum of the two preceding terms, so 24 plus 39 equals 63. 48 adds 39 to the earlier 9 rather than to 24, while 53 and 58 would need the term before 39 to be 14 or 19, neither of which appears in the series.
- Find the next number in the series 1, 3, 4, 7, 11, 18, ___ .
Correct answer: 29
The next number is 29. Each term is the sum of the two before it, so 11 plus 18 equals 29. 23 and 26 add only 5 or 8 to 18, and 33 adds 15; none of those amounts is the 11 that actually precedes 18, so each breaks the additive rule.
- What number comes next in the series 10, 10, 20, 30, 50, ___ ?
Correct answer: 80
The next number is 80. Each term is the sum of the two previous terms, so 30 plus 50 equals 80. 84, 88 and 92 would each require the term before 50 to be 34, 38 or 42 rather than the 30 that is actually there, so none of them follows the rule.
- Find the next term in the series 2, 5, 7, 12, 19, 31, ___ .
Correct answer: 50
The next term is 50. Each value is the sum of the two preceding values, so 19 plus 31 equals 50. 62 doubles 31 instead of adding the term before it, and 56 and 68 would need 31 to be paired with 25 or 37, neither of which is in the series.
- What number is missing in the series 3, 3, 6, 9, 15, 24, ___ ?
Correct answer: 39
The missing number is 39. Each term equals the sum of the two before it, so 15 plus 24 equals 39. 48 doubles 24 rather than adding 15 to it, and 44 and 52 would require a partner of 20 or 28 for 24, so neither matches the additive rule.
- In the series 2, 9, 4, 18, 6, 27, ___, what number comes next?
Correct answer: 8
The next number is 8. The series interleaves two runs: the odd positions count up by two as 2, 4, 6 and the even positions count up by nine as 9, 18, 27. The seventh position belongs to the counting-by-two run, so it advances past 6 to 8. The values 2, 4 and 6 are terms that run has already used and cannot appear again.
- Find the next term in the series 20, 3, 30, 6, 40, 9, ___ .
Correct answer: 50
The next term is 50. Two runs alternate: one rises by ten as 20, 30, 40 and the other rises by three as 3, 6, 9. The seventh position belongs to the tens run, so 40 becomes 50. 43 applies the other run's step of three to 40, while 57 and 64 match the step of neither run.
- What number continues the series 5, 40, 10, 35, 15, 30, ___ ?
Correct answer: 20
The continuing number is 20. The series alternates two runs: 5, 10, 15 rising by five and 40, 35, 30 falling by five. The seventh position returns to the rising run, so 15 plus 5 gives 20. 11, 14 and 17 all fall short of a five-step rise from 15 and belong to neither run.
- What number is missing in the series 4, 1, 8, 5, 12, 9, 16, ___ ?
Correct answer: 13
The missing number is 13. Two runs interleave: the odd positions go 4, 8, 12, 16 adding four, and the even positions go 1, 5, 9 adding four as well. The eighth position belongs to the second run, so 9 plus 4 equals 13. 10 adds only one to 9, 16 repeats the term just given, and 19 adds three to 16 instead of continuing the second run.
- Find the next number in the series 200, 6, 180, 12, 160, 18, ___ .
Correct answer: 140
The next number is 140. One run falls by twenty as 200, 180, 160 and the other rises by six as 6, 12, 18. The seventh position returns to the falling run, so 160 minus 20 equals 140. 150 drops only ten, 160 repeats the term already given, and 180 steps back to a term the run has passed.
- What number comes next in the series 3, 64, 6, 32, 12, 16, ___ ?
Correct answer: 24
The next number is 24. Two runs interleave: 3, 6, 12 doubles each time and 64, 32, 16 halves each time. The seventh position returns to the doubling run, so 12 times 2 equals 24. 18, 21 and 27 each add a fixed amount to 12 instead of doubling it, which is the wrong operation for this run.
- Find the next term in the series 1, 100, 2, 90, 3, 80, ___ .
Correct answer: 4
The next term is 4. Two runs alternate: 1, 2, 3 counting up by one and 100, 90, 80 dropping by ten. The seventh position returns to the counting run, so 3 plus 1 equals 4. 5, 6 and 7 carry that count too far ahead, skipping the positions the falling run still occupies.
- What number comes next in the series 6, 50, 12, 45, 24, 40, ___ ?
Correct answer: 48
The next number is 48. Two runs interleave: 6, 12, 24 doubles each time and 50, 45, 40 drops by five. The seventh position returns to the doubling run, so 24 times 2 equals 48. 64 adds 24 to 40 across the two runs, 72 triples 24, and 56 belongs to neither run.
- What number comes next in the series 3, 4, 7, 12, 19, ___ ?
Correct answer: 28
The next number is 28. The gaps between terms are 1, 3, 5 and 7, growing by two each step, so the next gap is 9 and 19 plus 9 equals 28. 22 and 25 use gaps of only 3 or 6, which break the steady growth, and 19 merely repeats the term just given.
- Find the next term in the series 2, 4, 9, 17, 28, ___ .
Correct answer: 42
The next term is 42. The gaps are 2, 5, 8 and 11, rising by three each time, so the next gap is 14 and 28 plus 14 equals 42. 39 repeats the previous gap of 11 instead of advancing it, while 36 and 33 fall back to gaps of only 8 or 5.
- What number continues the series 5, 6, 9, 14, 21, ___ ?
Correct answer: 30
The continuing number is 30. The gaps are 1, 3, 5 and 7, growing by two each step, so the next gap is 9 and 21 plus 9 equals 30. 24 and 27 use gaps of only 3 or 6, and 33 overshoots with a gap of 12, so none of them keeps the gaps rising by exactly two.
- What number is missing in the series 10, 12, 16, 22, 30, ___ ?
Correct answer: 40
The missing number is 40. The gaps are 2, 4, 6 and 8, rising by two each time, so the next gap is 10 and 30 plus 10 equals 40. 34 and 37 add only 4 or 7 to 30, and 43 adds 13; none of those continues the even, steadily growing gaps.
- Find the next number in the series 1, 3, 8, 16, 27, ___ .
Correct answer: 41
The next number is 41. The gaps are 2, 5, 8 and 11, climbing by three each time, so the next gap is 14 and 27 plus 14 equals 41. 38 repeats the previous gap of 11 rather than advancing it, while 44 and 47 would need gaps of 17 or 20.
- What number comes next in the series 4, 7, 13, 22, 34, ___ ?
Correct answer: 49
The next number is 49. The gaps are 3, 6, 9 and 12, increasing by three each step, so the next gap is 15 and 34 plus 15 equals 49. 46 repeats the previous gap of 12, while 52 and 55 would require gaps of 18 or 21, which skip past the next value in the pattern.
- Find the next term in the series 3, 4, 8, 17, 33, ___ .
Correct answer: 58
The next term is 58. The gaps are 1, 4, 9 and 16, which are the perfect squares, so the next gap is 25 and 33 plus 25 equals 58. 46 and 52 add only 13 or 19 to 33, and 64 adds 31; none of those three amounts is a square, so none fits the pattern of gaps.
- What number is missing in the series 0, 1, 4, 9, 16, ___ ?
Correct answer: 25
The missing number is 25. The gaps are 1, 3, 5 and 7, rising by two each step, so the next gap is 9 and 16 plus 9 equals 25. 19 and 22 add only 3 or 6 to 16, and 16 simply repeats the term just given rather than advancing the series.
- Which rule governs the series 8, 15, 22, 29, 36?
- Each term climbs by a fixed seven
- Each term is double the one prior
- Each term sums the two just above
- Each term shrinks by a flat eight
Correct answer: Each term climbs by a fixed seven
Each term climbs by a fixed seven: 8 goes to 15, then 22, then 29, then 36, so the governing rule is repeated addition of seven. Shrinking by a flat eight reverses the direction the values actually move. Doubling the one prior would put 16 after 8 rather than 15. Summing the two just above would put 23 after 8 and 15 rather than 22.
- What rule explains the series 2, 8, 32, 128, 512?
- Each term climbs by a steady nine
- Each term is the last one squared
- Each term adds the two seen above
- Each term is four times the value
Correct answer: Each term is four times the value
Each term is four times the value before it: 2 becomes 8, then 32, then 128, then 512, so a constant multiplier rather than a constant addition explains the series. Climbing by a steady nine would put 11 after 2 rather than 8. The last one squared would put 64 after 8 rather than 32. Adding the two seen above would put 10 after 2 and 8 rather than 32.
- Which rule produces the series 1, 2, 4, 7, 11, 16?
- Each term is twice the one just past
- Each term adds a gap one unit bigger
- Each term sums the two seen above it
- Each term gains a set three per turn
Correct answer: Each term adds a gap one unit bigger
Each term adds a gap one unit bigger than the gap before it: the increases run 1, 2, 3, 4 and 5, so the amount added grows by one every step. Twice the one just past would put 8 after 4 rather than 7. Summing the two seen above it would put 6 after 2 and 4 rather than 7. Gaining a set three per turn would put 5 after 2 rather than 4.
- What rule governs the series 64, 32, 16, 8, 4?
- Each term drops by a flat sixteen
- Each term totals the two above it
- Each term is half the value shown
- Each term is one quarter the last
Correct answer: Each term is half the value shown
Each term is half the value shown before it: 64 becomes 32, then 16, then 8, then 4, so a constant division and not a fixed subtraction governs the series. Dropping by a flat sixteen would put 48 after 64 rather than 32. One quarter the last would put 16 after 64 rather than 32. Totalling the two above it would make the values grow instead of fall.
- Find the next term in the series 1, 4, 9, 16, 25, 36, ___ .
Correct answer: 49
The next term is 49. These are the perfect squares, since 1 through 36 are 1 through 6 squared, so the seventh term is 7 times 7. 42, 56 and 63 are 7 times 6, 8 and 9, which pair the seventh index with the wrong factor instead of squaring it.
- What number comes next in the series 8, 27, 64, 125, ___ ?
Correct answer: 216
The next number is 216. These are the perfect cubes, since 8, 27, 64 and 125 are 2, 3, 4 and 5 cubed, so the next term is 6 times 6 times 6. 189 adds the two previous terms, 64 and 125, instead of cubing; 162 and 243 are not cubes of any whole number.
- Which rule governs the series 3, 9, 27, 81?
- Each term is twice the earlier one
- Each term sums the two shown items
- Each term climbs by a steady eight
- Each term is three times the value
Correct answer: Each term is three times the value
Each term is three times the value before it: 3 becomes 9, then 27, then 81, so a constant multiplier of three and not a fixed addition governs the series. Twice the earlier one would put 6 after 3 rather than 9. Climbing by a steady eight would put 11 after 3 rather than 9. Summing the two shown items would put 12 after 3 and 9 rather than 27.
- What rule governs the series 90, 81, 72, 63, 54?
- Each term is triple the one prior
- Each term falls by a wider margin
- Each term is simply half the last
- Each term slides by a steady nine
Correct answer: Each term slides by a steady nine
Each term slides by a steady nine: 90 drops to 81, then 72, then 63, then 54, so a constant downward difference of nine governs the series. Triple the one prior would drive the values upward instead. A wider margin would make the gaps grow, yet every gap here is exactly nine. Simply half the last would put 45 after 90 rather than 81.
- Which rule governs the series 2, 5, 11, 23, 47?
- Each term is three past the total seen
- Each term is the two values added here
- Each term is double the value plus one
- Each term is the last one times itself
Correct answer: Each term is double the value plus one
Each term is double the value plus one: 2 gives 5, then 11, then 23, then 47, since twice a value with one added reproduces the next. Three past the total seen would put 8 after 5 rather than 11. The two values added here would put 7 after 2 and 5 rather than 11. The last one times itself would put 25 after 5 rather than 11.
- What rule governs the series 1, 4, 13, 40, 121?
- Each term is triple the value plus one
- Each term climbs by a flat three units
- Each term adds the two values above it
- Each term is just the last one squared
Correct answer: Each term is triple the value plus one
Each term is triple the value plus one: 1 gives 4, then 13, then 40, then 121, since three times a value plus one reproduces the next. Climbing by a flat three units would put 7 after 4 rather than 13. Adding the two values above it would put 5 after 1 and 4 rather than 13. The last one squared would put 16 after 4 rather than 13.
- What number comes next in the series 19, 25, 31, 37, ___ ?
Correct answer: 43
The next number is 43. Each term increases by a constant 6, so 37 plus 6 equals 43. 39 advances by only 2, while 47 and 51 advance by 10 or 14; none of those keeps the fixed step that makes this an arithmetic sequence.
- Find the next term in the series 47, 41, 35, 29, ___ .
Correct answer: 23
The next term is 23. Each value falls by 6, so 29 minus 6 equals 23. 26 subtracts only 3, 20 subtracts 9, and 17 subtracts 12; none of those keeps the constant decrease that defines this arithmetic sequence.
- What number comes next in the series 2, 20, 200, 2000, ___ ?
Correct answer: 20000
The next number is 20000. Each term is multiplied by 10, so 2000 times 10 equals 20000. 25000, 30000 and 35000 each add a fixed amount to 2000 rather than applying the constant ratio of 10, so none of them continues the geometric sequence.
- Find the next term in the series 11, 22, 44, 88, ___ .
Correct answer: 176
The next term is 176. Each value doubles, so 88 times 2 equals 176. 132 adds the two previous terms, 44 and 88, instead of doubling; 220 adds 132 to 88; and 264 triples 88. Only doubling gives a constant ratio of 2.
- What number is missing in the series 5, 8, 13, 21, 34, ___ ?
Correct answer: 55
The missing number is 55. Each term equals the sum of the two before it, so 21 plus 34 equals 55. 68 doubles 34 rather than adding 21 to it, and 61 and 74 would require 34 to be paired with 27 or 40, neither of which appears in the series.
- Find the next term in the series 1, 5, 6, 11, 17, 28, ___ .
Correct answer: 45
The next term is 45. Each value is the sum of the two preceding values, so 17 plus 28 equals 45. 39 adds 11 to 28 instead of 17, 34 adds only 6, and 42 adds 14; none of those is the term that actually precedes 28.
- What comes next in the series 70, 2, 65, 5, 60, 8, ___ ?
Correct answer: 55
The next number is 55. One run decreases by 5 as 70, 65, 60 and the other increases by 3 as 2, 5, 8. The seventh position returns to the decreasing run, so 60 minus 5 equals 55. 59 takes only 1 off 60, 51 takes 9, and 47 takes 13.
- Find the next term in the series 2, 7, 6, 10, 18, 13, ___ .
Correct answer: 54
The next term is 54. Two sequences interleave: 2, 6, 18 triples each time while 7, 10, 13 rises by 3, and after 13 the tripling sequence takes its turn, so 18 times 3 gives 54. The value 16 continues the adding sequence instead, which is not the one due; 39 applies the tripling to 13, a term that belongs to the adding sequence; 31 adds the last two printed terms, a rule the earlier terms do not obey.
- What number comes next in the series 5, 8, 14, 23, 35, ___ ?
Correct answer: 50
The next number is 50. The gaps run 3, 6, 9, 12 and grow by 3 each time, so the gap due is 15 and 35 plus 15 gives 50. The value 47 repeats the previous gap of 12 instead of growing it; 44 reuses the earlier gap of 9; 53 lets the gap grow one extra step to 18 before it is due.
- Find the next term in the series 20, 21, 24, 29, 36, ___ .
Correct answer: 45
The next term is 45. The gaps run 1, 3, 5, 7 and rise by 2 each time, so the gap due is 9 and 36 plus 9 gives 45. The value 43 repeats the previous gap of 7 rather than advancing it; 41 reaches back to the gap of 5; 47 skips 9 and jumps to the following odd number, 11.
- Which rule governs the series 4, 12, 36, 108?
- Doubled totals at each step
- Regular eights at each step
- Squared inputs at each step
- Tripled values at each step
Correct answer: Tripled values at each step
The rule is tripled values at each step: every term is three times the one before it, since 4 gives 12, 12 gives 36 and 36 gives 108. Doubled totals would produce 8 then 16, never 12 then 36. Regular eights, a constant addition of 8, fits only the first pair and fails at 12 to 36. Squared inputs would turn 4 into 16 rather than 12.
- What rule governs the series 75, 70, 65, 60, 55?
- Rising values at each step
- Bigger losses at each step
- Subtract five at each step
- Halved totals at each step
Correct answer: Subtract five at each step
The rule is subtract five at each step: every term is five less than the one before, since 75 gives 70, then 65, then 60, then 55. Rising values reverses the direction of a sequence that plainly descends. Bigger losses would mean the gap widens, yet the gap holds at exactly five throughout. Halved totals would send 75 to 37.5, not to 70.
- What number comes next in the series 1, 3, 6, 10, 15, ___ ?
Correct answer: 21
The next number is 21. These are the triangular numbers, and the gaps run 2, 3, 4, 5, so the gap due is 6 and 15 plus 6 gives 21. The value 19 reuses the earlier gap of 4; 23 lets the gap leap to 8 instead of the orderly 6; 25 adds the two printed terms 10 and 15, a rule the earlier terms do not follow.
- Find the next term in the series 13, 16, 19, 22, ___ .
Correct answer: 25
The next term is 25. Every term rises by a constant 3, so 22 plus 3 gives 25. The value 28 doubles the step to 6; 41 adds the two printed terms 19 and 22, treating a constant-step sequence as a summing one; 35 adds the first and last printed terms, 13 and 22, which no step rule supports.
- What number comes next in the series 4, 8, 16, 32, 64, ___ ?
Correct answer: 128
The next number is 128. Every term is twice the one before, so 64 doubled gives 128. The value 192 triples 64 rather than doubling it; 256 doubles twice over and lands on the term after the one asked for; 384 multiplies 64 by the term's position rather than by the constant ratio of 2.
- In a figure analogy, the first figure shows an arrow pointing UP and the second shows the same arrow pointing RIGHT. If the third figure shows an arrow pointing LEFT, which way should the fourth arrow point to keep the same relationship?
- It sinks to the base
- It leans to the left
- It tips to the right
- It swings to the top
Correct answer: It swings to the top
The fourth arrow is the one where it swings to the top. A quarter turn clockwise carries an up arrow round to a right arrow, so the identical quarter turn clockwise carries a left arrow round to an up arrow. It tips to the right merely repeats the second figure instead of transforming the third. It sinks to the base turns the arrow the opposite way round. It leans to the left applies no turn at all.
- A figure analogy reads: a triangle pointing up is to a triangle pointing down, as a triangle pointing left is to ___.
- It points to the base
- It turns to the right
- It returns to the top
- It swells to the left
Correct answer: It turns to the right
The answer is the one where it turns to the right. The up triangle becomes the down triangle through a half turn, and a half turn carries a left-pointing triangle to a right-pointing one. It points to the base copies the second figure rather than transforming the third. It returns to the top is a quarter turn, not the half turn the first pair shows. It swells to the left alters size while leaving the direction alone.
- In a figure analogy, the letter L is to the same letter L rotated one quarter turn clockwise, as the letter T is to ___.
- The letter T spun backward instead
- The letter T copied without change
- The letter T turned clockwise once
- The letter T rotated head downward
Correct answer: The letter T turned clockwise once
The answer is the letter T turned clockwise once, that is, given the same single quarter turn clockwise the L receives in the first pair. The letter T spun backward instead reverses the direction of the turn. The letter T copied without change applies no transformation whatever. The letter T rotated head downward is a half turn, twice the rotation the first pair shows.
- A clock hand points to 12. After the transformation in the first pair, it points to 3. If a second hand starts pointing to 9, where will it point after the same transformation?
- The bottom mark
- The twelve mark
- The former mark
- The eleven mark
Correct answer: The twelve mark
The hand finishes on the twelve mark. Moving from 12 to 3 is a quarter turn clockwise, and the same quarter turn clockwise carries a hand resting on 9 round to 12. The former mark is 9 itself, where the hand began, so no turn has been applied. The bottom mark is 6, which is a quarter turn the wrong way round. The eleven mark falls an hour short, counting two marks instead of three.
- In a figure analogy, a comma shape (like a curved tail) is shown, then the same shape rotated a half turn. If the third figure is a small flag pointing up on its pole, the fourth figure should be:
- The flag spun to point downward
- The flag mirrored left to right
- The flag shown at double height
- The flag kept alone and upright
Correct answer: The flag spun to point downward
The answer is the flag spun to point downward. The comma shape is given a half turn, a rotation of 180 degrees, so the flag must be given that same half turn and end up pointing down. The flag mirrored left to right is a reflection, which reverses the figure rather than spinning it round. The flag shown at double height alters scale, a rule the first pair never uses. The flag kept alone and upright applies no transformation.
- A figure analogy shows an upper-case letter R, then the letter R rotated 90 degrees counterclockwise. Which transformation links these two figures?
- A flipping of the letter
- A turning of the outline
- A swelling of the figure
- A deepening of the tints
Correct answer: A turning of the outline
The link between the two figures is a turning of the outline. The second R has been swung 90 degrees counterclockwise while keeping the same handedness, which is exactly what a rotation does. A flipping of the letter describes a reflection, and that would leave the R reversed as a mirror image. A swelling of the figure would alter size, which has not changed. A deepening of the tints would alter fill, which has not changed either.
- In a figure analogy, an arrow pointing northeast (up and to the right) becomes an arrow pointing southeast (down and to the right) after a quarter turn clockwise. If a new arrow points northwest (up and to the left), where will it point after the same quarter turn clockwise?
- Northwest
- Southwest
- Northeast
- Southeast
Correct answer: Northeast
The arrow will point Northeast. A quarter turn clockwise carries a northeast arrow to southeast, so the same quarter turn carries a northwest arrow one step clockwise, which lands it on northeast. Northwest is where the new arrow starts, so choosing it applies no turn. Southwest swings the arrow anticlockwise, the wrong way round the compass. Southeast repeats the second figure rather than transforming the third.
- A figure analogy reads: the number 6 is to the number 9, as a single shape rotated a half turn. Which pair of shapes follows the SAME rule as 6 becoming 9?
- A small square growing to twice size
- A rising arrow turning to point down
- A white circle filling up with black
- A letter mirroring into its own copy
Correct answer: A rising arrow turning to point down
The matching pair is a rising arrow turning to point down. Spinning a 6 through a half turn produces a 9, and spinning an upward arrow through that same half turn produces a downward arrow. A small square growing to twice size changes scale rather than orientation. A white circle filling up with black changes shading only. A letter mirroring into its own copy is a reflection, which is not a rotation.
- In a figure analogy, the first figure is the letter b and the second is the letter d. If the third figure is the letter p, the fourth figure should be:
- The letter d, half rotate
- The letter b, upside flip
- The letter q, side mirror
- The letter p, kept intact
Correct answer: The letter q, side mirror
The answer is the letter q, the side mirror of p. Turning b into d reflects it across a vertical line, and reflecting p that same way, side to side, produces q. The letter d, half rotate, is what a 180-degree turn of p would give instead. The letter b, upside flip, comes from a reflection across a horizontal line. The letter p, kept intact, applies no transformation at all.
- A figure analogy shows the letter p, then the letter b. Which transformation links these two figures?
- A flip from left to right
- A tilt of one quarter arc
- A fold from top to bottom
- A rise in the whole scale
Correct answer: A fold from top to bottom
The transformation is a fold from top to bottom. Reflecting p across a horizontal line lifts its descending loop into an ascending one, which is exactly what turns it into b. A flip from left to right would turn p into q instead. A tilt of one quarter arc is a rotation, which would carry the loop round rather than lift it. A rise in the whole scale changes size, and neither letter has changed size.
- In a figure analogy, an arrow pointing right is to an arrow pointing left, as a hand with the thumb pointing right is to ___.
- The thumb remains aimed right
- The thumb swivels one quarter
- The thumb stretches far wider
- The thumb now points leftward
Correct answer: The thumb now points leftward
The answer is the thumb now points leftward. The right arrow becomes a left arrow through a reflection across a vertical line, so the same reflection reverses the hand and carries its thumb to the left. The thumb remains aimed right applies no transformation. The thumb swivels one quarter is a rotation rather than a mirror reversal. The thumb stretches far wider changes size, which the first pair never does.
- A figure analogy reads: a triangle sitting flat on its base (point up) is to a triangle balancing on its point (point down), as a capital letter E facing right is to ___, when the rule is a top-to-bottom flip.
- A capital E flipped top down once
- A capital E mirrored to face left
- A capital E kept top side unmoved
- A capital E drawn at twice height
Correct answer: A capital E flipped top down once
The answer is a capital E flipped top down once. The triangle is reflected across a horizontal line so its point swaps from top to bottom, and the E must be reflected on that same horizontal axis. A capital E mirrored to face left uses a vertical axis instead, which is the other flip. A capital E kept top side unmoved applies no transformation. A capital E drawn at twice height changes size rather than orientation.
- In a figure analogy, a shape that looks like a checkmark in the bottom-left corner is to the same checkmark mirrored into the bottom-right corner. Which transformation has occurred?
- A reversing of the two sides
- A swapping of the upper half
- A dimming of the shaded area
- A widening of the whole mark
Correct answer: A reversing of the two sides
The transformation is a reversing of the two sides: a reflection across a vertical line carries the mark from the left of the frame to the right as a mirror image. A swapping of the upper half describes a top-to-bottom reflection, which would move the mark up rather than across. A dimming of the shaded area changes tone, and no tone has changed. A widening of the whole mark changes size, and no size has changed.
- A figure analogy shows a sailboat with its sail leaning to the right, then the same boat with its sail leaning to the left. If a third figure shows a person leaning to the right, the fourth figure should show:
- The person drawn rather larger
- The person hauled back upright
- The person flipped heels aloft
- The person tipped leftward now
Correct answer: The person tipped leftward now
The answer is the person tipped leftward now. The sail is mirrored from the right side to the left by a reflection across a vertical line, so the same reflection carries the figure's lean over to the left. The person drawn rather larger changes size, which the two boats do not. The person hauled back upright removes the lean altogether. The person flipped heels aloft inverts the figure top to bottom rather than reversing it side to side.
- In a figure analogy, the letter N is to its mirror image (a backward N), as the letter S is to ___.
- The letter S redrawn unchanged
- The letter S rotated crosswise
- The letter S mirrored sideways
- The letter S printed oversized
Correct answer: The letter S mirrored sideways
The answer is the letter S mirrored sideways, which gives a backward S. The N is reflected across a vertical line into its mirror image, so the same side-to-side reflection turns the S into a backward S. The letter S redrawn unchanged applies no transformation. The letter S rotated crosswise turns the figure instead of mirroring it, and a turn is not a reflection. The letter S printed oversized changes size, which the first pair leaves alone.
- A figure analogy reads: a small circle is to a large circle, as a small square is to ___.
- A circle made large
- A square made small
- A square made large
- A square made round
Correct answer: A square made large
The answer is a square made large. The first pair keeps the shape exactly and only enlarges it, so the square must be enlarged and stay a square. A circle made large carries over the first pair's shape rather than its rule. A square made small reverses the direction of the change. A square made round alters the shape, which the rule holds constant.
- In a figure analogy, a large star is to a small star, as a large hexagon is to ___.
- A hexagon drawn large
- A star pulled smaller
- A hexagon shrunk down
- A hexagon spun around
Correct answer: A hexagon shrunk down
The answer is a hexagon shrunk down. The first pair reduces the star while it stays a star, so the hexagon must be reduced and stay a hexagon. A hexagon drawn large applies no reduction and points the wrong way entirely. A star pulled smaller carries over the first pair's shape rather than its rule. A hexagon spun around turns the figure instead of resizing it.
- A figure analogy shows a small triangle, then a triangle twice as large. Which transformation links the two figures?
- A growth of the whole shape
- A change of the inner shade
- A turning of the flat plane
- A mirror of the outer edges
Correct answer: A growth of the whole shape
The link between the two figures is a growth of the whole shape. The second triangle is the same triangle made larger, with orientation and shading untouched, which is what scaling means. A change of the inner shade would alter the fill, and the fill is identical. A turning of the flat plane would alter orientation, and the triangle sits the same way up. A mirror of the outer edges would reverse the figure, and it is not reversed.
- In a figure analogy, the first figure is a tiny dot inside a box and the second is a large dot inside the same box. If the third figure is a tiny heart inside a box, the fourth figure should be:
- A large heart inside the box
- A small heart inside the box
- A large point inside the box
- A huge heart outside the box
Correct answer: A large heart inside the box
The answer is a large heart inside the box. The dot grows while its box stays exactly as it was, so the heart must grow while its box stays exactly as it was. A small heart inside the box applies no growth. A large point inside the box carries over the first pair's inner figure rather than its rule. A huge heart outside the box moves the figure out of its frame, which the first pair never does.
- A figure analogy reads: two small circles side by side is to two large circles side by side, as three small squares in a row is to ___.
- Three small squares in a row
- Three large squares in a row
- Three large circles in a row
- Two great squares in one row
Correct answer: Three large squares in a row
The answer is three large squares in a row. The circles are enlarged while their count of two is held, so the squares must be enlarged while their count of three is held. Three small squares in a row applies no enlargement. Three large circles in a row swaps the shape for the one used in the first pair. Two great squares in one row copies the first pair's count instead of keeping three.
- In a figure analogy, a large arrow pointing right is to a small arrow pointing right, as a large arrow pointing up is to ___.
- A large arrow aimed aloft
- A small arrow held upward
- A small arrow facing down
- A small arrow aimed right
Correct answer: A small arrow held upward
The answer is a small arrow held upward. The first pair shrinks the arrow and leaves its direction alone, so the second arrow must shrink and go on aiming up. A large arrow aimed aloft applies no shrinking. A small arrow facing down shrinks the arrow but also turns it, adding a change the rule does not make. A small arrow aimed right shrinks the arrow but borrows its direction from the first pair.
- A figure analogy shows a hollow (unshaded) circle, then a solid (fully shaded) circle. If the third figure is a hollow square, the fourth figure should be:
- A square filled jet dark
- A square left quite bare
- A circle made pure black
- A square drawn far wider
Correct answer: A square filled jet dark
The answer is a square filled jet dark, that is, colored in solid. The circle goes from hollow to fully shaded, so the same change must fill the square completely. A square left quite bare applies no change of fill at all. A circle made pure black carries over the first pair's shape rather than its rule. A square drawn far wider changes size, and the rule alters fill alone.
- In a figure analogy, a black (filled) triangle is to a white (unfilled) triangle, as a black star is to ___.
- A star left plain white
- A star shown still dark
- A triangle set to white
- A star drawn far bigger
Correct answer: A star left plain white
The answer is a star left plain white. The filled triangle becomes unfilled, so the same change empties the star to white while its shape is held constant. A star shown still dark applies no change of fill. A triangle set to white carries over the first pair's shape instead of its rule. A star drawn far bigger changes size, which the first pair leaves untouched.
- A figure analogy shows a circle that is white on the left half and black on the right half, then the same circle with the shading reversed (black on the left, white on the right). Which transformation has occurred?
- A turn of the whole disc
- A gain of the total size
- A loss of the inner half
- A trade of the two tones
Correct answer: A trade of the two tones
The transformation is a trade of the two tones: the black and white halves change places while the circle keeps its shape and its size. A turn of the whole disc would move the boundary round rather than swap the fills, and a plain circle turned about its center looks the same. A gain of the total size is ruled out because the circle is the same size. A loss of the inner half is ruled out because both halves are still present.
- In a figure analogy, a square split into four small squares with the top-left one shaded is to the same square with the top-RIGHT one shaded. If a third figure is a square with the bottom-left small square shaded, the fourth should have which square shaded, following the same left-to-right pairing?
- The box back in bottom left
- The box in the topmost left
- The box in four outer cells
- The box in the bottom right
Correct answer: The box in the bottom right
The shaded cell should be the box in the bottom right. The shading moves from a cell on the left side to the matching cell on the right side, so shading at bottom left must move to bottom right. The box back in bottom left leaves the shading where it started. The box in the topmost left moves the shading up the grid instead of across it. The box in four outer cells shades the whole grid, which the first pair never does.
- A figure analogy reads: a light gray circle is to a dark gray circle, as a light gray pentagon is to ___.
- A pentagon gone darker
- A pentagon drawn light
- A circle turned darker
- A pentagon shown empty
Correct answer: A pentagon gone darker
The answer is a pentagon gone darker. The circle deepens from light gray to dark gray, so the pentagon must deepen in the same way while its shape and size stay put. A pentagon drawn light applies no deepening. A circle turned darker carries over the first pair's shape rather than its rule. A pentagon shown empty removes the fill altogether rather than deepening it.
- In a figure analogy, a triangle with horizontal stripes is to a triangle with vertical stripes, as a circle with horizontal stripes is to ___.
- A circle banded sideways
- A circle striped upright
- A triangle ruled upright
- A circle shaded entirely
Correct answer: A circle striped upright
The answer is a circle striped upright, that is, ruled with vertical stripes. The triangle's stripes turn from horizontal to vertical, so the circle's stripes must turn the same way while the outer shape holds. A circle banded sideways keeps the stripes horizontal and applies no change. A triangle ruled upright carries over the first pair's shape. A circle shaded entirely replaces the stripes with a solid fill.
- A figure analogy shows a square with one dot inside it, then a square with two dots inside it. If the third figure is a circle with one dot inside it, the fourth figure should be:
- A circle round still one speck
- A circle round three dark dots
- A square round twin small dots
- A circle round two little dots
Correct answer: A circle round two little dots
The answer is a circle round two little dots. The square gains exactly one dot, going from one to two, so the circle must gain one dot and hold two. A circle round still one speck adds nothing. A circle round three dark dots adds two dots instead of one. A square round twin small dots gets the count right but swaps the outer shape, which the rule keeps unchanged.
- In a figure analogy, a triangle with three small circles inside is to a triangle with two small circles inside, as a square with three small circles inside is to ___.
- A square holding three tiny discs
- A square holding one lonely speck
- A triangle holding just two blobs
- A square holding merely two rings
Correct answer: A square holding merely two rings
The answer is a square holding merely two rings, the rings being the inner circles. The triangle loses exactly one inner circle, dropping from three to two, so the square must lose one and be left with two. A square holding three tiny discs removes nothing. A square holding one lonely speck removes two rather than one. A triangle holding just two blobs gets the count right but swaps the outer shape, which the rule holds constant.
- A figure analogy shows a plain circle, then a circle with an X drawn through it. Which transformation links the two figures?
- A flip of one flat plane
- A gain of one fresh part
- A twist of one half turn
- A drop of one size grade
Correct answer: A gain of one fresh part
The transformation is a gain of one fresh part. The second figure is the same circle with an X newly drawn inside it, so a feature has been put in. A flip of one flat plane would reverse the figure, and a plain circle would look no different. A twist of one half turn would leave the circle exactly as it was. A drop of one size grade would shrink it, and the circle is the same size in both figures.
- In a figure analogy, the first figure is a stack of four horizontal bars and the second is a stack of three horizontal bars. If the third figure is a stack of five horizontal bars, the fourth figure should be:
- A pile of six narrow bars
- A pile of three gray bars
- A stack of five thin bars
- A pile of four solid bars
Correct answer: A pile of four solid bars
The answer is a pile of four solid bars. The first pair takes away exactly one bar, going from four to three, so taking one bar from a stack of five leaves four. A stack of five thin bars takes nothing away. A pile of three gray bars copies the count shown in the second figure instead of applying the rule to the third. A pile of six narrow bars adds a bar rather than removing one.
- A figure analogy reads: a face with two eyes and no mouth is to a face with two eyes and a mouth, as a clock face with hour marks but no hands is to ___.
- A clock stripped of those marks
- A clock given narrow hour hands
- A clock drawn large minus hands
- A clock emptied of hour details
Correct answer: A clock given narrow hour hands
The answer is a clock given narrow hour hands. The face gains the mouth it was missing while everything already drawn stays, so the clock must gain the hands it was missing while its hour marks stay. A clock stripped of those marks takes a feature away instead of adding one. A clock emptied of hour details strips the figure bare rather than completing it. A clock drawn large minus hands changes size and still supplies nothing that was missing.
- In a figure analogy, a square containing a single triangle is to a square containing a triangle and a circle, as a pentagon containing a single triangle is to ___.
- A pentagon revealing one triangle plus a square
- A pentagon enclosing a triangle plus one circle
- A pentagon wrapping one triangle with no circle
- A pentagon covering two triangles but no circle
Correct answer: A pentagon enclosing a triangle plus one circle
The answer is a pentagon enclosing a triangle plus one circle. The first pair keeps the outer shape and the original inner triangle while adding a circle beside it, so the pentagon must gain a circle in the same way. A pentagon revealing one triangle plus a square adds the wrong new element. A pentagon wrapping one triangle with no circle leaves the figure untouched, so no rule has been applied. A pentagon covering two triangles but no circle repeats an element instead of adding the new one.
- A figure analogy reads: a small black square is to a large white square, as a small black circle is to ___. Two changes happen at once: the figure grows AND its shading flips.
- A bigger white circle
- A bigger black circle
- A little white circle
- A little black circle
Correct answer: A bigger white circle
The answer is a bigger white circle. The first pair enlarges the figure and at the same time switches its shading from black to white, so the circle must end up both larger and unfilled. A bigger black circle applies the growth but keeps the original shading. A little white circle flips the shading but ignores the growth. A little black circle applies neither change at all.
- Figure analogies are usually written in the form A is to B as C is to D. In this structure, what must the fourth figure (D) do?
- Be bigger than each of the other three figures on the sheet
- Stand to the third figure as the second stands to the first
- Remain exactly true to the first figure in shape as in size
- Be the mirror image of that third figure in each case shown
Correct answer: Stand to the third figure as the second stands to the first
The fourth figure must stand to the third figure as the second stands to the first. A figure analogy names a transformation in the opening pair and then demands the very same transformation in the closing pair. Being bigger than each of the other three figures is a size claim the form never makes. Remaining exactly true to the first figure ignores the transformation instead of repeating it. Being the mirror image of the third figure fixes on one particular rule that only some analogies use.
- In the figure analogy A is to B as C is to D, the first figure is a circle and the second is a square. The third figure is a star. To solve for D, what is the FIRST thing you should determine?
- The exact shade of color brushed into the star
- How the first shape turned into the second one
- How much space each of the four answers covers
- The size that the missing figure ought to show
Correct answer: How the first shape turned into the second one
You should first work out how the first shape turned into the second one, because that transformation is exactly what gets applied to the third shape. The exact shade of color brushed into the star is a surface detail that plays no part in the rule. How much space each of the four answers covers judges the options before the rule is known. The size that the missing figure ought to show is an outcome, not a starting point.
- A figure analogy shows: an up arrow is to a down arrow, as a right arrow is to a left arrow. What single rule explains BOTH pairs?
- The arrow is colored in with solid black ink
- The arrow spins half a turn to face backward
- The arrow gains a second thin line beside it
- The arrow is grown to a clearly bigger scale
Correct answer: The arrow spins half a turn to face backward
The single rule is that the arrow spins half a turn to face backward. Up becomes down and right becomes left, and both of those are reversals through one hundred and eighty degrees, so one rule covers each pair. The arrow being colored in with solid black ink describes a fill that never appears. The arrow gaining a second thin line beside it adds an element that neither pair shows. The arrow being grown to a clearly bigger scale changes size, which stays constant throughout.
- In a figure analogy, a white circle is to a black circle, as a white triangle is to a black triangle. Which best describes the relationship that holds the analogy together?
- The figure rotates right around in both of the pairs
- The outline shape switches from one pair to the next
- The figure shrinks into a smaller size in both pairs
- The empty figure becomes filled in both of the pairs
Correct answer: The empty figure becomes filled in both of the pairs
The relationship is that the empty figure becomes filled in both of the pairs. The circle goes from unfilled to solid and the triangle does exactly the same, so one shading rule governs the whole analogy. The figure rotating right around names a turn that never happens. The outline shape switching from one pair to the next describes a difference between pairs, not the link inside each pair. The figure shrinking into a smaller size invents a change of scale that neither pair contains.
- A figure analogy reads: a square with a dot in the center is to a square with no dot, as a circle with a dot in the center is to ___.
- A circle with one center dot
- A square missing its own dot
- A circle with two inner dots
- A circle with its dot erased
Correct answer: A circle with its dot erased
The answer is a circle with its dot erased. The opening pair takes the center dot away from the square and alters nothing else, so the same deletion empties the circle. A circle with one center dot is the third figure left untouched. A square missing its own dot copies the second figure instead of transforming the third. A circle with two inner dots adds elements when the rule removes them.
- In a figure analogy, a large outlined pentagon is to a small filled pentagon, as a large outlined hexagon is to ___. Two things change at once.
- A shrunken hollow hexagon
- A large blackened hexagon
- A large unfilled pentagon
- A miniature solid hexagon
Correct answer: A miniature solid hexagon
The answer is a miniature solid hexagon. The opening pair does two things at once, cutting the figure down in size and filling its interior, so the hexagon must end up both smaller and solid. A shrunken hollow hexagon carries out the size change but leaves the interior empty. A large blackened hexagon fills the interior but skips the size change. A large unfilled pentagon repeats the opening figure and applies neither change.
- A figure analogy shows: a single circle is to two circles, as a single square is to ___.
- A trio of squares
- A lone square now
- A pair of circles
- A pair of squares
Correct answer: A pair of squares
The answer is a pair of squares. The opening pair turns one circle into two circles, so the count doubles while the shape itself is left alone, and one square must therefore become two squares. A trio of squares raises the count past the doubling the rule calls for. A lone square now leaves the count where it started. A pair of circles doubles correctly but abandons the square for the wrong shape.
- In a figure analogy, an arrow pointing right is to the same arrow pointing down after a quarter turn clockwise. If a third arrow points down, the fourth arrow, after the same quarter turn clockwise, points:
- The arrow now points right
- The arrow stays aimed down
- The arrow now turns upward
- The arrow ends angled left
Correct answer: The arrow ends angled left
The arrow ends angled left. A quarter turn clockwise carries right to down, so the very same quarter turn carries down on to left. The arrow now points right would be the result of going one step counterclockwise instead. The arrow stays aimed down applies no turn at all. The arrow now turns upward skips two positions rather than one.
- A figure analogy reads: the letter F is to its mirror image (a backward F facing left), as the letter G is to ___.
- The letter G toppled through a half spin
- The letter G stretched to a greater size
- The letter G flipped to a reversed shape
- The letter G resting exactly as it began
Correct answer: The letter G flipped to a reversed shape
The answer is the letter G flipped to a reversed shape. The F is reflected across a vertical line so that it reads the wrong way round, and the same reflection produces a backward G. The letter G toppled through a half spin rotates the shape instead of reflecting it, which gives a different result. The letter G stretched to a greater size changes scale, a property the opening pair never touches. The letter G resting exactly as it began applies no rule whatever.
- In a figure analogy, a small filled circle is to a large filled circle, as a small filled diamond is to ___.
- A tiny darkened diamond
- A bigger filled diamond
- A widened blank diamond
- A swollen filled circle
Correct answer: A bigger filled diamond
The answer is a bigger filled diamond. The opening pair scales the figure up and alters nothing else, so the diamond must grow while its solid interior stays exactly as it was. A tiny darkened diamond keeps the interior but leaves the size untouched. A widened blank diamond scales up correctly yet empties an interior the rule never empties. A swollen filled circle scales up but swaps the diamond for the wrong outline.
- A figure analogy shows a square containing two stars, then the same square containing four stars. If the third figure is a circle containing two stars, the fourth figure should be:
- A circle drawn around three stars
- A circle enclosing just two stars
- A circle wrapped round four stars
- The square itself with four stars
Correct answer: A circle wrapped round four stars
The answer is a circle wrapped round four stars. The opening pair takes the count from two up to four, a doubling, and leaves the outer boundary alone, so the circle must end up round four stars. A circle drawn around three stars adds only one star instead of doubling the count. A circle enclosing just two stars leaves the count untouched. The square itself with four stars doubles the count but keeps the boundary from the opening pair rather than the third figure.
- In a figure analogy, a clock hand at 12 turns to point at 6. If a second hand starts at 3, where does it point after the same transformation?
- At the twelve point
- At the three marker
- At the nine numeral
- At the six position
Correct answer: At the nine numeral
The hand ends up at the nine numeral. Moving from twelve round to six is a half turn through one hundred and eighty degrees, and a half turn from three lands on nine. At the twelve point is where the first hand began, not where the second hand finishes. At the three marker leaves the second hand exactly where it started. At the six position copies the destination of the first hand instead of applying its rule.
- A figure analogy reads: a hollow square is to a square shaded on its lower half only, as a hollow circle is to ___.
- A circle filled along its bottom half
- A circle blackened along its top half
- A circle given a wholly black surface
- A circle left entirely clear of paint
Correct answer: A circle filled along its bottom half
The answer is a circle filled along its bottom half. The opening pair darkens the lower portion of the square and leaves the upper portion alone, so the circle takes the same partial treatment low down. A circle blackened along its top half treats the wrong region. A circle given a wholly black surface goes past a partial region and covers everything. A circle left entirely clear of paint applies no change whatever.
- In a figure analogy, a figure of three dots arranged in a triangle is to the same three dots with a line connecting them into a triangle outline, as three dots arranged in a row is to ___.
- Three dots in a row and a single line drawn
- Three dots in a row and no mark joins these
- Three dots back in a triangle and not a row
- Two dots in a row and a straight line added
Correct answer: Three dots in a row and a single line drawn
The answer is three dots in a row and a single line drawn. The opening pair keeps every dot exactly where it sat and adds the stroke that joins them, so the row of dots gains a stroke running through it. Three dots in a row and no mark joins these leaves out the addition that defines the rule. Three dots back in a triangle and not a row moves the dots, which the rule never does. Two dots in a row and a straight line added quietly loses one of the dots.
- A figure analogy shows the capital letter A, then the letter A rotated a half turn (upside down). If the third figure is the capital letter V, the fourth figure should be:
- The V with its earlier look intact
- The V with its size greatly raised
- The V with its sides swapped about
- The V with its point turned upward
Correct answer: The V with its point turned upward
The answer is the V with its point turned upward. A half turn carries the letter A onto its head, and the same half turn carries the V so that the apex which normally sits at the bottom now sits at the top. The V with its earlier look intact leaves the letter alone. The V with its size greatly raised changes scale rather than orientation. The V with its sides swapped about reflects the letter instead of rotating it, and a symmetrical V would look no different.
- In a figure analogy, a large gray triangle is to a small gray triangle, as a large gray arrow is to ___.
- A sizable ashen arrow
- A broad gray triangle
- A shrunken gray arrow
- A smaller black arrow
Correct answer: A shrunken gray arrow
The answer is a shrunken gray arrow. The opening pair cuts the figure down in size and alters nothing else, so the arrow must lose size while its shade stays exactly as it was. A sizable ashen arrow leaves the size untouched. A broad gray triangle abandons the arrow for the earlier outline and applies no size change. A smaller black arrow shrinks correctly but darkens a shade the rule never darkens.
- A figure analogy reads: a square with an arrow pointing up inside it is to a square with an arrow pointing right inside it, as a square with an arrow pointing left inside it is to ___, when the rule is a quarter turn clockwise of the inner arrow.
- The square holds its arrow pointed left
- The square sets its arrow aimed skyward
- The square shows one arrow facing right
- The square keeps this arrow turned down
Correct answer: The square sets its arrow aimed skyward
The answer is the square sets its arrow aimed skyward. A quarter turn clockwise carries an upward arrow round to the right, so the same quarter turn carries a leftward arrow round to point up, while the outer boundary never moves. The square holds its arrow pointed left applies no turn at all. The square shows one arrow facing right skips two quarter turns instead of one. The square keeps this arrow turned down runs the quarter turn the wrong way.
- In a figure analogy, a hollow heart is to a filled (black) heart, as a hollow moon (crescent) is to ___.
- A completely black heart
- A bigger hollow crescent
- A blank outline crescent
- A totally black crescent
Correct answer: A totally black crescent
The answer is a totally black crescent. The opening pair takes the figure from empty to solid and touches neither its size nor its outline, so the crescent simply becomes solid. A completely black heart fills correctly but keeps the earlier outline instead of the crescent. A bigger hollow crescent changes scale, which the rule never changes, and leaves the interior empty. A blank outline crescent applies no change of any kind.
- A figure analogy shows a square with rounded corners, then a square with sharp corners but otherwise the same, as a triangle with rounded corners is to ___.
- A square wearing sharp corners
- A triangle with curved corners
- A triangle given sharp corners
- A circle without sharp corners
Correct answer: A triangle given sharp corners
The answer is a triangle given sharp corners. The opening pair swaps soft corners for crisp ones and leaves the outline itself alone, so the triangle keeps three sides and gains points at its vertices. A square wearing sharp corners applies the corner change to the wrong outline. A triangle with curved corners leaves the corners as they were. A circle without sharp corners has no corners to change and abandons the triangle as well.
- In a figure analogy, an upward arrow is to a leftward arrow (a quarter turn counterclockwise). If a third arrow points right, where does the fourth arrow point after the same quarter turn counterclockwise?
- Toward the lower edge
- Toward the right side
- Toward the top border
- Toward the left flank
Correct answer: Toward the top border
The fourth arrow ends up pointing toward the top border. A quarter turn counterclockwise carries an upward arrow round to the left, so the same quarter turn carries a rightward arrow round to point up. Toward the lower edge is the result of two quarter turns rather than one. Toward the right side leaves the third arrow where it began. Toward the left flank runs the quarter turn clockwise instead of counterclockwise.
- A figure analogy reads: a tall thin rectangle standing upright is to a wide flat rectangle lying down, as a tall thin oval standing upright is to ___.
- A squat wide rectangle laid down
- A simple round circle drawn here
- A tall narrow oval still upright
- A broad flattened oval laid down
Correct answer: A broad flattened oval laid down
The answer is a broad flattened oval laid down. The opening pair swings the upright figure through a quarter turn so that its long axis runs sideways, and the oval takes the same quarter turn. A squat wide rectangle laid down turns correctly but keeps the outline from the opening pair. A simple round circle drawn here swaps the oval for an outline the analogy never mentions. A tall narrow oval still upright leaves the figure standing and applies no turn.
- In a figure analogy, a circle divided into four equal quarters with the top-left quarter shaded is to the same circle with the top-left quarter UNshaded and the top-right quarter shaded. Following a quarter-turn-clockwise rule, where is the shaded quarter in the second figure relative to the first?
- One quarter round the clockwise route
- Right where the shading first settled
- Directly opposite where it began from
- One quarter against the clockwise way
Correct answer: One quarter round the clockwise route
The shaded region ends up one quarter round the clockwise route. Shading that sat in the top-left now sits in the top-right, which is a single step in the direction the hands of a clock travel. Right where the shading first settled would mean no movement at all. Directly opposite where it began from describes a half turn, which would carry the shading to the bottom-right. One quarter against the clockwise way reverses the direction the rule names.
- A figure analogy shows a square with a small circle in its top-left corner, then the same square with the small circle in its top-RIGHT corner. If a third figure is a square with a small circle in its bottom-left corner, the fourth figure should have the small circle in the:
- The corner planted low at right
- The corner resting down at left
- The corner raised high at right
- The corner sitting high at left
Correct answer: The corner planted low at right
The small circle belongs in the corner planted low at right. The opening pair mirrors the inner element across a vertical line, moving it from one side to the other while keeping the row it sits in, so a low-left circle becomes a low-right circle. The corner resting down at left leaves the circle where it started. The corner raised high at right mirrors sideways but also jumps rows. The corner sitting high at left moves up a row without mirroring at all.
- In a figure analogy, a figure with one dot is to a figure with three dots, as a figure with two dots is to ___. Each step adds the same number of dots.
- A figure with merely two dots
- A figure with three dots here
- A figure with four dots total
- A figure with five whole dots
Correct answer: A figure with four dots total
The answer is a figure with four dots total. The opening pair moves from one dot to three, an increase of two, so the same increase of two carries a figure of two dots up to four. A figure with merely two dots adds nothing at all. A figure with three dots here adds only a single dot. A figure with five whole dots treats the rule as tripling rather than as a fixed addition.
- A figure analogy reads: a black square is to a black square with a white circle in the middle, as a black triangle is to ___.
- A white triangle with no circle inside
- A dark square enclosing a white circle
- A black triangle around a white circle
- A black triangle and nothing added yet
Correct answer: A black triangle around a white circle
The answer is a black triangle around a white circle. The opening pair keeps the outer shape and its solid shading and drops an unfilled circle into the middle, so the triangle keeps its own shading and gains the same inner circle. A white triangle with no circle inside changes the shading and skips the addition. A dark square enclosing a white circle adds the circle to the shape from the opening pair rather than to the triangle. A black triangle and nothing added yet applies no change at all.
- In a figure analogy, a small white star is to a large black star, as a small white square is to ___. Track BOTH changes.
- A broader black square
- A stretched black star
- A huge unshaded square
- A shrunken dark square
Correct answer: A broader black square
The answer is a broader black square. The opening pair enlarges the figure and darkens it from unfilled to solid, so both changes must reach the square. A stretched black star carries out both changes but keeps the outline from the opening pair. A huge unshaded square grows correctly and leaves the shading untouched. A shrunken dark square darkens correctly but reverses the change of size.
- A figure analogy shows a triangle pointing up, then a triangle pointing up sitting inside a circle. If the third figure is a square, the fourth figure should be:
- A square standing inside a circle
- A lone circle with nothing inside
- A square placed plainly by itself
- A triangle tucked inside a circle
Correct answer: A square standing inside a circle
The answer is a square standing inside a circle. The opening pair keeps the original shape untouched and draws a circle around it, so the square keeps its own form and gains the surrounding circle. A lone circle with nothing inside adds the surround but loses the shape it was meant to surround. A square placed plainly by itself applies no change at all. A triangle tucked inside a circle simply repeats the second figure rather than transforming the third.
- In a figure analogy, the first figure is a square with a diagonal line from the bottom-left to the top-right corner, and the second is the same square with the diagonal from the top-left to the bottom-right corner. Which transformation links them?
- A simple rise in overall size
- A second line drawn across it
- A mirror flip across one axis
- A change of shading inside it
Correct answer: A mirror flip across one axis
The transformation is a mirror flip across one axis. Reflecting the square about a vertical line, or equally about a horizontal one, swings the single diagonal so that it runs the opposite way while everything else stays put. A simple rise in overall size is ruled out because both figures are the same size. A second line drawn across it is ruled out because the count of lines never changes. A change of shading inside it is ruled out because neither square is filled.
- A figure analogy reads: two stacked squares (a small one on top of a large one) is to two stacked squares with their sizes swapped (large on top of small), as a small circle on top of a large circle is to ___.
- The square pair with small one high
- The bigger circle set riding on top
- The tiny circle sitting high on top
- A pair of circles at identical size
Correct answer: The bigger circle set riding on top
The answer is the bigger circle set riding on top. The opening pair exchanges the two sizes so that the larger figure ends up above the smaller one, and the same exchange lifts the large circle above the small circle. The square pair with small one high keeps the outline from the opening pair and applies no exchange. The tiny circle sitting high on top leaves the stack exactly as it was. A pair of circles at identical size makes the two match in size, which the rule never does.
- In a figure analogy, a single horizontal line is to a plus sign (a horizontal line crossed by a vertical line). If the third figure is a single forward slash (a line tilted like /), the fourth figure should be:
- A line drawn quite upright here
- The slash kept wholly as before
- A line stretched dead level now
- A slash plus a second backslash
Correct answer: A slash plus a second backslash
The answer is a slash plus a second backslash, the two strokes together forming an X. The opening pair keeps the original stroke and lays a second stroke across it at right angles, so the tilted stroke gains a stroke tilted the opposite way. A line drawn quite upright here supplies only the crossing stroke and loses the original. The slash kept wholly as before adds nothing at all. A line stretched dead level now replaces the tilted stroke instead of crossing it.
- A figure analogy shows a circle with a vertical line dividing it in half, then the same circle with no dividing line. If the third figure is a square with a vertical line dividing it, the fourth figure should be:
- A square with its middle line gone
- A square with a flat dividing line
- A square with an upright line kept
- A circle emptied of its inner mark
Correct answer: A square with its middle line gone
The answer is a square with its middle line gone. The opening pair deletes the internal stroke and leaves the outer shape exactly as it was, so the square keeps its four sides and loses the stroke. A square with a flat dividing line swaps the stroke for another rather than deleting it. A square with an upright line kept applies no deletion at all. A circle emptied of its inner mark deletes correctly but keeps the outline from the opening pair.
- In a figure analogy, a right-pointing triangle is to a left-pointing triangle, and the rule is a horizontal flip. Which other pair follows the SAME flipping rule?
- A lowercase b gives a mirrored d
- A tiny circle gives a bigger one
- An upward arrow gives a down one
- A white square gives a black one
Correct answer: A lowercase b gives a mirrored d
The pair that follows the same rule is a lowercase b gives a mirrored d, because reflecting a b across a vertical line produces exactly the shape of a d, just as reflecting a rightward triangle produces a leftward one. A tiny circle gives a bigger one changes scale rather than reflecting. An upward arrow gives a down one is a half turn, not a sideways reflection. A white square gives a black one changes only the shading.
- A figure analogy reads: a square with four dots (one in each corner) is to a square with two dots (only the top corners), as a square with four dots is also compared elsewhere; here the rule removes the bottom dots. If a triangle has dots at all three corners and the rule removes the bottom two, the result is:
- A triangle still dotted at each corner
- A triangle retaining just its apex dot
- A triangle showing just its lower dots
- A triangle left with no dots remaining
Correct answer: A triangle retaining just its apex dot
The answer is a triangle retaining just its apex dot. The rule strips away the two dots that sit along the base and leaves whatever sits above them, and a triangle dotted at all three corners has exactly one dot above the base. A triangle still dotted at each corner removes nothing. A triangle showing just its lower dots keeps the very dots the rule takes away. A triangle left with no dots remaining removes all three rather than the lower two.
- In a figure analogy, a medium circle is to a circle four times its area, as a medium square is to ___.
- A square four times its former area
- A square merely two times that area
- A circle held at its unchanged size
- A square left at its unchanged size
Correct answer: A square four times its former area
The answer is a square four times its former area. The opening pair multiplies the area of the figure by four and leaves the outline alone, so the square is scaled by the very same factor. A square merely two times that area applies the wrong multiplier. A circle held at its unchanged size scales nothing and abandons the square as well. A square left at its unchanged size ignores the scaling entirely.
- A figure analogy shows an L-shape, then the L-shape rotated a quarter turn clockwise so its foot points a new direction. If the third figure is a J-shape, the fourth figure should be:
- The J reversed into its reflected counterpart
- The J pivoted precisely one quarter clockwise
- The J rolled a quarter turn counter-clockwise
- The J unchanged since its original appearance
Correct answer: The J pivoted precisely one quarter clockwise
The answer is the J pivoted precisely one quarter clockwise. The opening pair turns the figure through ninety degrees in the direction the hands of a clock travel, and the closing pair must repeat that exact motion on the J. The J reversed into its reflected counterpart mirrors the figure, which is a different operation and would not match the opening pair. The J rolled a quarter turn counter-clockwise turns through the right angle but in the wrong direction. The J unchanged since its original appearance applies no motion at all.
