- A statement reads 'eight more than the product of a number and 5.' Which expression correctly translates it?
- 5n plus 8
- 5(n plus 8)
- 8n plus 5
- 5n minus 8
Correct answer: 5n plus 8
The expression is 5n plus 8. 'The product of a number and 5' becomes 5n, and 'eight more than' that product means adding 8, giving 5n plus 8.
- A delivery company charges a $6 base fee plus $2 for each package p. Which equation gives the total charge T?
- T equals 6p plus 2
- T equals 6 plus 2p
- T equals 8p
- T equals 2 plus 6p
Correct answer: T equals 6 plus 2p
The equation is T equals 6 plus 2p. The $6 base fee stays fixed, while the $2 per-package cost multiplies by the number of packages p, producing 6 plus 2p.
- A statement says 'three times a number is the same as the number plus 14.' Which equation models it?
- 3n equals 14n
- 3(n plus 14) equals n
- 3n equals n plus 14
- N plus 3 equals 14
Correct answer: 3n equals n plus 14
The model is 3n equals n plus 14. 'Three times a number' is 3n and 'the number plus 14' is n plus 14, so the two expressions are set equal to each other.
- A clerk earns a $40 daily allowance plus $9 per completed report r. Which expression shows total daily earnings?
- 9 plus 40r
- 49r
- 40r plus 9
- 40 plus 9r
Correct answer: 40 plus 9r
The expression is 40 plus 9r. The $40 allowance is constant, and $9 multiplies by the number of reports r, so total earnings are 40 plus 9r.
- 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 x equals 7.
- A phrase reads 'the difference between a number and 9, divided by 2.' Which expression matches it?
- N minus (9 divided by 2)
- (n minus 9) divided by 2
- (9 minus n) divided by 2
- N divided by (9 minus 2)
Correct answer: (n minus 9) divided by 2
The expression is (n minus 9) divided by 2. 'The difference between a number and 9' is n minus 9, and dividing that whole quantity by 2 places it over 2.
- Two parts together number 75. The first part is 4 times the second part, which is x. Which equation finds x?
- 4x equals 75
- X plus 4 equals 75
- X plus 4x equals 75
- X(4x) equals 75
Correct answer: X plus 4x equals 75
The equation is x plus 4x equals 75. The second part is x and the first part is 4x, so their combined total of 75 means adding x and 4x and setting the sum equal to 75.
- 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 x equals 5.
- A statement reads 'the height of a pole is 7 feet less than twice its shadow length s.' Which expression gives the height?
- 2s minus 7
- 7 minus 2s
- 2(s minus 7)
- 7s minus 2
Correct answer: 2s minus 7
The height is 2s minus 7. 'Twice the shadow length' is 2s, and '7 feet less than' that means subtracting 7, producing 2s minus 7.
- 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 n equals 27.
- 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 had 675 units. This quarter's 540 represents 80 percent of last quarter's total, so dividing 540 by 0.80 gives 675 units.
- 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?
- $250.00
- $246.00
- $252.00
- $248.40
Correct answer: $248.40
The price is $248.40. An 8 percent rise brings $250 to $270, and an 8 percent drop of $270 is $21.60, leaving $270 minus $21.60 equals $248.40.
- Membership in a club grew from 320 members to 400 members. What was the percent increase?
- 25 percent
- 30 percent
- 20 percent
- 15 percent
Correct answer: 25 percent
The increase was 25 percent. Membership grew by 80 members, and 80 divided by the original 320 equals 0.25, or 25 percent.
- After a 40 percent markdown, a coat costs $54. What was its original price?
Correct answer: $90
The original price was $90. After a 40 percent markdown, the $54 price represents 60 percent of the original, so dividing 54 by 0.60 gives $90.
- 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 adding that increase to 350 gives 413.
- 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 that increase to the original $900 gives $945.
- 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 cutting $600 by 20 percent removes $120, producing $480.
- A tire loses 6 percent of its tread, which equals 0.9 millimeters. What was the original tread depth?
- 14 millimeters
- 15 millimeters
- 16 millimeters
- 13.5 millimeters
Correct answer: 15 millimeters
The original tread depth was 15 millimeters. Since the 0.9-millimeter loss equals 6 percent of the original, dividing 0.9 by 0.06 gives 15 millimeters.
- 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 has 5 total parts, so antifreeze is 2 parts out of 5, and two-fifths of 25 quarts is 10 quarts.
- If 6 identical bolts weigh 18 ounces, how much do 15 of the same bolts weigh?
- 40 ounces
- 42 ounces
- 48 ounces
- 45 ounces
Correct answer: 45 ounces
Fifteen bolts weigh 45 ounces. Each bolt weighs 18 divided by 6, which is 3 ounces, so 15 bolts weigh 15 times 3, equal to 45 ounces.
- 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 equals 8 meters, so multiplying 8 meters by the 7 centimeters gives 56 meters.
- 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 has 1 plus 4 plus 5, or 10 parts, so each part is $5,000 divided by 10, equal to $500; the largest share is 5 parts, which is $2,500.
- A proportion states that 9 is to 12 as x is to 20. What is x?
Correct answer: 15
x equals 15. Cross-multiplying the proportion 9/12 equals x/20 gives 12x equals 180, and dividing by 12 yields x equals 15.
- 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 has 7 plus 2, or 9 parts, so each part equals 54 divided by 9, which is 6; the sedans are 7 parts, giving 42.
- If 5 workers can stack a load in 18 minutes, how long would 15 workers take at the same rate?
- 6 minutes
- 9 minutes
- 5 minutes
- 8 minutes
Correct answer: 6 minutes
Fifteen workers take 6 minutes. Tripling the workers from 5 to 15 divides the time by 3, so 18 minutes becomes 6 minutes.
- A freight train travels 84 miles in 1 hour and 30 minutes. At the same speed, how far will it go in 4 hours?
- 240 miles
- 224 miles
- 210 miles
- 252 miles
Correct answer: 224 miles
It will travel 224 miles. The speed is 84 divided by 1.5 hours, or 56 miles per hour, and 56 times 4 hours equals 224 miles.
- 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?
- 100 miles
- 110 miles
- 120 miles
- 130 miles
Correct answer: 120 miles
They are 120 miles apart. Their combined separation rate is 22 plus 18, or 40 miles per hour, and over 3 hours that gives 120 miles.
- A jogger runs for 20 minutes at 9 miles per hour. How far does he run?
- 4 miles
- 2 miles
- 2.5 miles
- 3 miles
Correct answer: 3 miles
He runs 3 miles. Twenty minutes is one-third of an hour, and multiplying 9 miles per hour by one-third gives 3 miles.
- A helicopter covers 660 miles in 1 hour and 12 minutes. What is its average speed in miles per hour?
- 550 mph
- 560 mph
- 540 mph
- 520 mph
Correct answer: 550 mph
The average speed is 550 mph. One hour and 12 minutes equals 1.2 hours, and dividing 660 miles by 1.2 hours gives 550 miles per hour.
- A traveler drives the first 90 miles at 45 mph and the next 90 miles at 90 mph. What is the total travel time?
- 2 hours
- 3 hours
- 4 hours
- 2.5 hours
Correct answer: 3 hours
The total time is 3 hours. The first leg takes 90 divided by 45, or 2 hours, and the second takes 90 divided by 90, or 1 hour, summing to 3 hours.
- A scout hikes at 3 mph for 90 minutes, then rests, then hikes 2 more miles. What total distance does the scout cover?
- 5.5 miles
- 7 miles
- 6.5 miles
- 6 miles
Correct answer: 6.5 miles
The total distance is 6.5 miles. Hiking 3 mph for 90 minutes (1.5 hours) covers 4.5 miles, and adding the final 2 miles gives 6.5 miles.
- A car uses 4 gallons of fuel to travel 120 miles. How many miles can it travel on 7 gallons at the same rate?
- 200 miles
- 220 miles
- 180 miles
- 210 miles
Correct answer: 210 miles
It can travel 210 miles. The car gets 120 divided by 4, or 30 miles per gallon, and 30 times 7 gallons equals 210 miles.
- 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 out of 52 cards, and 13/52 reduces to 1/4.
- 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. There are 3 defective fuses out of 8 total, giving a probability of 3/8.
- 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 results are 2, 4, and 6, giving 3 favorable outcomes out of 6, and 3/6 reduces to 1/2.
- 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. Multiplying the $50 prize by its 0.1 probability gives $5.00, and the no-payout outcome adds nothing.
- 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. A sum of 12 requires both dice to show 6, which is 1 outcome out of the 36 equally likely combinations.
- 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. There are 6 white buttons out of 20 total, and 6/20 reduces to 3/10.
- 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 is $3.00. Multiplying the $150 prize by its 1/50 probability gives $3.00 as the average return before the $5 cost is considered.
- What is the average of 16, 20, 24, and 28?
Correct answer: 22
The average is 22. Adding 16, 20, 24, and 28 gives 88, and dividing that sum by the 4 values equals 22.
- A trainee records times of 30, 36, and 33 seconds on three sprints. What is the average time?
- 33 seconds
- 32 seconds
- 34 seconds
- 35 seconds
Correct answer: 33 seconds
The average time is 33 seconds. The three times total 99 seconds, and dividing 99 by 3 sprints equals 33 seconds.
- The average of six numbers is 14. What is their total sum?
Correct answer: 84
The sum is 84. Since the average equals the total divided by the count, multiplying the average of 14 by the 6 numbers gives 84.
- 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. The seven readings total 7 times 40, or 280, and adding 56 gives 336, which divided by 8 readings equals 42.
- 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 over 5 games means a total of 380, and the four known scores sum to 306, so the fifth must be 380 minus 306, equal to 74.
- 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?
- 25 years
- 24.5 years
- 26 years
- 25.5 years
Correct answer: 25 years
The new average is 25 years. The 8 players total 192 years, and adding the coach's 33 years gives 225 years across 9 people, which divided equals 25 years.
- 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, and 100 plus 50 equals 150.
- Mentally multiply 14 by 50.
Correct answer: 700
The product is 700. Since 50 is half of 100, multiplying 14 by 100 gives 1,400, and taking half of that yields 700.
- What is 20 percent of 350 computed mentally?
Correct answer: 70
The result is 70. Ten percent of 350 is 35, and doubling that to get 20 percent gives 70.
- Mentally divide 360 by 6.
Correct answer: 60
The quotient is 60. Splitting 360 into 300 plus 60 and dividing each by 6 gives 50 plus 10, which equals 60.
- Mentally compute 8 times 102.
Correct answer: 816
The product is 816. Treating 102 as 100 plus 2, the calculation is 800 plus 16, which equals 816.
- 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, then divide by 4 to get 600.
- Mentally compute 500 minus 264.
Correct answer: 236
The difference is 236. Subtracting 264 from 500 leaves 236, which can be checked by noting 264 plus 236 equals 500.
- If 6y equals 54, what is the value of y?
Correct answer: 9
y equals 9. Dividing both sides of the equation 6y equals 54 by 6 isolates y as 54 divided by 6, which is 9.
- A number decreased by 7 equals 19. What is the number?
Correct answer: 26
The number is 26. Setting up the equation n minus 7 equals 19 and adding 7 to both sides gives n equals 26.
- 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 after subtracting 6, and dividing by 4 gives 7.
- 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, Ben is a plus 3, and a plus (a plus 3) equals 27 gives 2a equals 24, so a equals 12.
- 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 equals 60, so 3n equals 57 and n equals 19, making the largest 21.
- The perimeter of a rectangle is 54 inches, and the length is twice the width. What is the length?
- 9 inches
- 18 inches
- 21 inches
- 27 inches
Correct answer: 18 inches
The length is 18 inches. With length equal to 2w, the perimeter equation 2(2w) plus 2w equals 54 simplifies to 6w equals 54, so w equals 9 and the length 2w is 18 inches.
- 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 2.30 minus 0.50, or 1.80, and dividing 1.80 by 0.10 gives d equals 18 dimes.
- 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 and Carl be 3d, in 4 years their ages are d plus 4 and 3d plus 4, summing to 4d plus 8 equals 32, so 4d equals 24 and d equals 6.
- 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 becomes 3n equals 15 after moving terms, so n equals 5.
- 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 25 from 313 gives 18h equals 288, and dividing both sides by 18 yields h equals 16 hours.
- 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 and wrenches be 2p, the equation p plus 2p equals 24 gives 3p equals 24, so p equals 8 pliers.
- 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 boxes, and subtracting the 130 shipped leaves 275.
- 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 each is $66, and adding the $9 delivery fee gives $75, which divided by 6 members is $12.50 each.
- 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 per minute over 5 minutes produces 120 pages, and subtracting the 18 misprints leaves 102.
- 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 sandbags per day uses 1,225 sandbags, and subtracting 1,225 from the 1,500 stored leaves 275.
- 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 each is $128 and three vests at $48 each is $144, and adding $128 and $144 gives $272.
- 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 gain is 14 minus 9, or 5 gallons per hour, and over 8 hours that adds 40 gallons to the starting 200, giving 240.
- 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, and four trips of 22 letters is 88, and 170 plus 88 equals 258.
- CAR is to GARAGE as PLANE is to ___.
Correct answer: HANGAR
HANGAR completes the analogy. A garage is the structure where a car is stored and sheltered, so the matching word for a plane is the building that stores and shelters it, which is a hangar. A runway is where a plane takes off, a pilot operates it, and fuel powers it, but none of those is the storage structure.
- HAPPY is to SAD as HOT is to ___.
Correct answer: COLD
COLD completes the analogy. Happy and sad are opposites, so the word that is the opposite of hot is needed. Cold is the direct antonym of hot. Warm is only milder than hot rather than opposite, and sun and fire are merely sources of heat.
- PETAL is to FLOWER as ___ is to BOOK.
Correct answer: PAGE
PAGE completes the analogy. A petal is one part of the whole flower, so the matching word must be a single part of a book. A page is one part of the whole book. Reading is an action, a library stores books, and an author writes them, so none expresses the part-to-whole relationship.
- FIRE is to SMOKE as ___ is to FLOOD.
Correct answer: RAIN
RAIN completes the analogy. Fire is the cause and smoke is the effect it produces, so the first blank must be the cause that produces a flood. Heavy rain causes a flood. Water and a river are merely involved in flooding rather than its cause, and a dam prevents flooding.
- KNIFE is to CUT as PEN is to ___.
Correct answer: WRITE
WRITE completes the analogy. A knife is a tool used to cut, so the analogy pairs a tool with its action; a pen is a tool used to write. Ink and paper are materials involved, and a desk is a surface, but none of them is the action the pen performs.
- WARM is to HOT as COOL is to ___.
- BREEZE
- MILD
- COLD
- TEMPERATURE
Correct answer: COLD
COLD completes the analogy. Warm is a mild degree and hot is a greater degree of the same quality, so the analogy moves from a lesser to a greater intensity; cool is the lesser degree and cold is the greater degree. Mild restates a low intensity, while breeze and temperature are unrelated to the intensity scale.
- 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; the synonym of tiny is small. Huge and giant mean the opposite of tiny, and wide describes breadth 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 the whole bicycle, so the missing word must be a part of the whole house. A room is a part of a house. A garden and a street are outside the house, and a builder constructs it, so none expresses part-to-whole.
- SCISSORS is to CLOTH as ___ is to WOOD.
Correct answer: SAW
SAW completes the analogy. Scissors are the tool used to cut cloth, so the missing word must be the tool used to cut wood. A saw cuts wood. A nail and glue join wood, and a hammer drives nails, so none of them performs the cutting function.
- 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; a teacher works in a school. A student is a person the teacher serves, a lesson is what is taught, and a book is a tool, so none names the workplace.
- OVEREAT is to OBESITY as ___ is to DROWSINESS.
- EXERCISE
- FATIGUE
- REST
- APPETITE
Correct answer: FATIGUE
FATIGUE completes the analogy. Overeating is the cause that leads to the effect of obesity, so the first blank must be a cause that leads to drowsiness; fatigue brings on drowsiness. Exercise and rest tend to relieve drowsiness, and appetite is unrelated to becoming drowsy.
- In the analogy SAPLING is to TREE as KITTEN is to ___, the relationship between the first pair is one of:
- Opposites
- Cause and effect
- Part to whole
- Young to adult
Correct answer: Young to adult
Young to adult is the relationship. A sapling is a young tree that grows into an adult tree, so the bridge linking the words is the progression from a young form to its grown form; a kitten likewise grows into a cat. The pair are not opposites, not a part and its whole, and not a cause and its effect.
- 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 analogy moves from a small action to a large one; a sip is a small drink and a gulp is a large one. Drink and taste are general, and pour describes transferring liquid rather than consuming a large amount.
- BIRD is to FLOCK as ___ is to HERD.
Correct answer: COW
COW completes the analogy. A bird is a single member of the larger group called a flock, so the missing word must be one member of a herd; a cow is one member of a herd. A farm and a barn are places, and grass is food, so none names a single member of the group.
- BRAVE is to COWARDLY as GENEROUS is to ___.
Correct answer: STINGY
STINGY completes the analogy. Brave and cowardly are opposites, so the missing word must be the opposite of generous; stingy is the opposite of generous. Kind and giving are similar in meaning to generous, and wealthy describes having money rather than being unwilling to give.
- BROOM is to SWEEP as SHOVEL is to ___.
Correct answer: DIG
DIG completes the analogy. A broom is the tool used to sweep, so the analogy pairs a tool with its action; a shovel is the tool used to dig. Dirt is the material moved, a garden is a place, and a handle is a part of the shovel, so none is the action the tool performs.
- GLOVE is to HAND as SHOE is to ___.
Correct answer: FOOT
FOOT completes the analogy. A glove is worn on a hand, so the analogy links a covering with the body part it covers; a shoe is worn on a foot. A lace is part of the shoe, a sock is another covering, and walk is an action, so none is the body part the shoe covers.
- DRIZZLE is to DOWNPOUR as ___ is to BLIZZARD.
Correct answer: FLURRY
FLURRY completes the analogy. A drizzle is light rain and a downpour is the same thing in a far heavier degree, so the missing word must be a light version that intensifies into a blizzard; a flurry is light snow that becomes a blizzard. Storm, wind, and ice do not form the light-to-heavy pairing of the same weather.
- DRUMMER is to BAND as ___ is to TEAM.
Correct answer: PLAYER
PLAYER completes the analogy. A drummer is one member of the whole group called a band, so the missing word must be one member of a team; a player is one member of a team. A coach directs the team, a stadium is the venue, and a trophy is a prize, so none is a single member of the group.
- VIRUS is to ILLNESS as SPARK is to ___.
Correct answer: FIRE
FIRE completes the analogy. A virus is the cause and illness is the resulting effect, so the missing word must be the effect that a spark causes; a spark causes a fire. A match produces a spark, heat is a property of fire, and wood is fuel, so none is the effect the spark produces.
- PENCIL is to WRITE as SCISSORS is to ___.
Correct answer: CUT
CUT completes the analogy. A pencil is the tool used to write, so the analogy pairs a tool with its action; scissors are used to cut. Paper is the material, sharp describes the edge, and a blade is a part of the scissors, so none is the action the tool performs.
- ANCIENT is to OLD as ___ is to BIG.
Correct answer: HUGE
HUGE completes the analogy. Ancient means extremely old, an intensified version of old, so the missing word must be an intensified version of big; huge means extremely big. Tiny is the opposite, while wide and heavy describe other qualities rather than an extreme degree of 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 missing word must be the name for a group of wolves; a group of wolves is a pack. A forest and a den are places, and hunt is an action, so none is the collective group name.
- In the analogy KEY is to LOCK as SOLUTION is to ___, the bridge between the first pair is best described as:
- Object to its action
- Part to whole
- Opposites
- Thing to the problem it opens or solves
Correct answer: Thing to the problem it opens or solves
Thing to the problem it opens or solves is the relationship. A key is the thing that opens a lock, just as a solution is the thing that resolves a problem, so the bridge links a device to the obstacle it overcomes. It is not an object paired with its action, not a part of a whole, and not a pair of opposites.
- THERMOMETER is to TEMPERATURE as CLOCK is to ___.
Correct answer: TIME
TIME completes the analogy. A thermometer is an instrument that measures temperature, so the missing word must be what a clock measures; a clock measures time. A hand and a number are parts of the clock, and a wall is where it hangs, so none is the quantity the instrument measures.
- ENEMY is to FRIEND as FAILURE is to ___.
Correct answer: SUCCESS
SUCCESS completes the analogy. Enemy and friend are opposites, so the missing word must be the opposite of failure; success is the opposite of failure. Defeat and loss are similar in meaning to failure, and effort is the work that may lead to either result, so none is the antonym.
- BAKER is to BREAD as ___ is to FURNITURE.
Correct answer: CARPENTER
CARPENTER completes the analogy. A baker is the worker who makes bread, so the missing word must be the worker who makes furniture; a carpenter makes furniture. Wood is the material, a hammer is a tool, and a house is where furniture is used, so none names the maker.
- PUDDLE is to LAKE as HILL is to ___.
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 missing word must be a much larger version of a hill; a mountain is a much larger landform than a hill. A valley is a low area, a slope is a side, and a rock is unrelated to overall size.
- NEEDLE is to SEW as KEY is to ___.
Correct answer: UNLOCK
UNLOCK completes the analogy. A needle is the tool used to sew, so the analogy pairs a tool with its action; a key is used to unlock. A lock and a door are objects the key acts upon, and metal is the material, so none is the action the key performs.
- DARK is to LIGHT as ROUGH is to ___.
Correct answer: SMOOTH
SMOOTH completes the analogy. Dark and light are opposites, so the missing word must be the opposite of rough; smooth is the opposite of rough. Coarse and bumpy mean nearly the same as rough, and hard describes firmness rather than texture, so none is the antonym.
- DROUGHT is to FAMINE as ___ is to FLOOD.
- SUNSHINE
- RAINFALL
- DESERT
- HEAT
Correct answer: RAINFALL
RAINFALL completes the analogy. A drought is the cause that leads to famine, so the first blank must be the cause that leads to a flood; excessive rainfall causes a flood. Sunshine and heat contribute to drought instead, and a desert is a dry place, so none causes flooding.
- STRING is to GUITAR as ___ is to PIANO.
Correct answer: KEY
KEY completes the analogy. A string is a part of the whole guitar that produces its notes, so the missing word must be the corresponding part of a piano; a key is the part of a piano that is pressed to produce notes. Music and sound are results, and a player operates the instrument, so none is a physical part of the piano.
- PAINTER is to BRUSH as WRITER is to ___.
Correct answer: PEN
PEN completes the analogy. A painter uses a brush as the tool of the craft, so the missing word must be the tool a writer uses; a writer uses a pen. A book and a story are the products of writing, and a desk is furniture, so none is the writing tool.
- GLAD is to ECSTATIC as ___ is to FURIOUS.
Correct answer: ANNOYED
ANNOYED completes the analogy. Glad is a mild feeling and ecstatic is the same feeling at an extreme degree, so the missing word must be the mild form of furious; being annoyed is the mild form of being furious. Calm and tired are unrelated emotions, and happy matches glad rather than the anger scale.
- In the analogy DICTIONARY is to WORDS as ATLAS is to ___, the relationship between the first pair is one of:
- Opposites
- Cause and effect
- A reference and what it contains
- Part to whole that is reversed
Correct answer: A reference and what it contains
A reference and what it contains is the relationship. A dictionary is a reference book that contains words, just as an atlas is a reference book that contains maps, so the bridge links a reference work to its contents. The pair show no cause and effect, no opposition, and no reversed part-to-whole structure.
- FIN is to FISH as WING is to ___.
Correct answer: BIRD
BIRD completes the analogy. A fin is a body part belonging to the whole fish, so the missing word must be the creature to which a wing belongs; a wing is a part of a bird. A feather is a smaller part of the wing, the sky is where the bird flies, and a nest is its home, so none is the whole creature.
- FAUCET is to WATER as SWITCH is to ___.
Correct answer: ELECTRICITY
ELECTRICITY completes the analogy. A faucet is the device that controls the flow of water, so the missing word must be what a switch controls; a switch controls the flow of electricity. A wall holds the switch, a wire carries current, and a light is one device the current powers, so none is the flow being controlled.
- BEGIN is to START as CONCLUDE is to ___.
Correct answer: FINISH
FINISH completes the analogy. Begin and start mean the same thing, so the missing word must mean the same as conclude; finish is a synonym of conclude. Open means to begin, continue means to keep going, and pause means to stop temporarily, so none matches the meaning of conclude.
- BACTERIA is to INFECTION as POLLUTION is to ___.
Correct answer: SMOG
SMOG completes the analogy. Bacteria are the cause and infection is the resulting effect, so the missing word must be the effect that pollution causes; pollution causes smog. A factory produces pollution, cleanup removes it, and air is what gets polluted, so none is the effect that results from pollution.
- SCALE is to WEIGHT as RULER is to ___.
Correct answer: LENGTH
LENGTH completes the analogy. A scale is an instrument that measures weight, so the missing word must be what a ruler measures; a ruler measures length. A line is what is drawn, a pencil is a separate tool, and a number is a marking, so none is the quantity the ruler measures.
- COMPOSER is to SYMPHONY as ___ is to NOVEL.
- READER
- AUTHOR
- LIBRARY
- CHAPTER
Correct answer: AUTHOR
AUTHOR completes the analogy. A composer is the person who creates a symphony, so the missing word must be the person who creates a novel; an author creates a novel. A reader consumes the novel, a library stores it, and a chapter is a part of it, so none names the creator.
- COTTAGE is to MANSION as PEBBLE is to ___.
Correct answer: BOULDER
BOULDER completes the analogy. A cottage is a small dwelling and a mansion is a much larger one of the same kind, so the missing word must be a much larger version of a pebble; a boulder is a much larger rock than a pebble. Sand and gravel are smaller, and stone is a general term, so none expresses the large degree.
- ROOF is to BUILDING as LID is to ___.
Correct answer: JAR
JAR completes the analogy. A roof is the top part that covers a building, so the missing word must be the object that a lid covers; a lid covers a jar. A kitchen is where a jar is kept, a handle is a separate part, and open is an action, so none is the whole object the lid tops.
- INHALE is to EXHALE as IMPORT is to ___.
Correct answer: EXPORT
EXPORT completes the analogy. Inhale and exhale are opposite actions, so the missing word must be the opposite of import; export is the opposite of import. Trade is the general activity, a ship carries the goods, and goods are the items moved, so none is the opposing action.
- SUN is to SHADOW as ___ is to ECHO.
Correct answer: SOUND
SOUND completes the analogy. The sun is the cause that produces a shadow, so the first blank must be the cause that produces an echo; a sound produces an echo. A cave is where an echo occurs, silence is the absence of sound, and noise is unwanted sound, so none is the cause of the echo.
- AX is to CHOP as HAMMER is to ___.
Correct answer: POUND
POUND completes the analogy. An ax is the tool used to chop, so the analogy pairs a tool with its action; a hammer is used to pound. A nail is what the hammer strikes, wood is the material, and tool is a general category, so none is the action the hammer performs.
- In the analogy CALM is to PANIC as TRICKLE is to FLOOD, the relationship in BOTH pairs is best described as:
- A small or mild form versus an extreme form
- Synonyms
- Cause and effect
- Object to its action
Correct answer: A small or mild form versus an extreme form
A small or mild form versus an extreme form is the relationship. Calm is the mild state and panic is its extreme, just as a trickle is a small flow and a flood is its extreme, so both pairs contrast a low degree with an intense one. The pairs are not synonyms, not cause and effect, and not an object with its action.
- BLADE is to KNIFE as ___ is to AX.
Correct answer: HANDLE
HANDLE completes the analogy. A blade is a part of the whole knife, so the missing word must be a part of an ax; a handle is a part of an ax. A tree is what an ax cuts, cut is the action, and sharp describes the edge, so none is a physical part of the ax in the way a blade is part of a knife.
- CHEF is to KITCHEN as JUDGE is to ___.
Correct answer: COURTROOM
COURTROOM completes the analogy. A chef works in a kitchen, so the analogy links a worker to the place of work; a judge works in a courtroom. Law is the subject, a verdict is the outcome, and a robe is clothing, so none names the workplace.
- WEALTHY is to POOR as EXPAND is to ___.
Correct answer: CONTRACT
CONTRACT completes the analogy. Wealthy and poor are opposites, so the missing word must be the opposite of expand; contract means the opposite of expand. Grow, stretch, and widen all mean to get larger, like expand, so none is the antonym.
- SEED is to PLANT as EGG is to ___.
Correct answer: BIRD
BIRD completes the analogy. A seed grows into a plant, so the analogy links a beginning form to what it becomes; an egg grows into a bird. A nest holds the egg, a shell surrounds it, and a hen lays it, so none is the grown form that develops from the egg.
- TELESCOPE is to ASTRONOMER as ___ is to SURGEON.
- SCALPEL
- PATIENT
- HOSPITAL
- NURSE
Correct answer: SCALPEL
SCALPEL completes the analogy. A telescope is the instrument an astronomer uses, so the missing word must be the instrument a surgeon uses; a surgeon uses a scalpel. A patient is the person treated, a hospital is the place, and a nurse is a coworker, so none is the surgeon's instrument.
- NIBBLE is to DEVOUR as TAP is to ___.
Correct answer: POUND
POUND completes the analogy. A nibble is a small action and to devour is the same idea taken to an extreme, so the missing word must be the forceful extreme of a tap; to pound is the forceful extreme of a tap. Touch and knock are similar in mildness to a tap, and push is a steady action rather than a forceful blow, so none is the intense degree.
- PILOT is to AIRPLANE as ___ is to SHIP.
Correct answer: CAPTAIN
CAPTAIN completes the analogy. A pilot is the person who operates an airplane, so the missing word must be the person who operates a ship; a captain operates a ship. An ocean is where the ship travels, a harbor is where it docks, and a sail is a part, so none is the operator.
- 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 creature a feather grows on; a feather is a part of a bird. Flight is an action, a wing is a larger part of the bird, and the sky is where it flies, so none is the whole creature.
- PRACTICE is to SKILL as ___ is to MUSCLE.
Correct answer: EXERCISE
EXERCISE completes the analogy. Practice is the cause that builds skill, so the first blank must be the cause that builds muscle; exercise builds muscle. Rest allows recovery, food provides fuel, and a bone is a separate body structure, so none is the cause that develops muscle.
- VACCINE is to DISEASE as ___ is to FIRE.
- MATCH
- EXTINGUISHER
- SMOKE
- FUEL
Correct answer: EXTINGUISHER
EXTINGUISHER completes the analogy. A vaccine is used to prevent or stop a disease, so the missing word must be the thing used to stop a fire; an extinguisher is used to put out a fire. A match starts a fire, smoke is a byproduct, and fuel feeds it, so none is the thing that stops it.
- CHAPTER is to BOOK as ACT is to ___.
Correct answer: PLAY
PLAY completes the analogy. A chapter is a major division of the whole book, so the missing word must be the whole that an act divides; an act is a major division of a play. A stage is where it is performed, an actor performs in it, and a scene is a smaller part, so none is the whole work.
- In the analogy LION is to CUB as HORSE is to FOAL, the relationship between the first pair is one of:
- Predator to prey
- Adult to its young
- Part to whole
- Opposites
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, just as a horse is the grown animal and a foal is its offspring, so the bridge links an adult to its young. The pair are not predator and prey, not a part and its whole, and not opposites.
- WET is to DRY as ASLEEP is to ___.
- TIRED
- RESTING
- AWAKE
- DREAMING
Correct answer: AWAKE
AWAKE completes the analogy. Wet and dry are opposites, so the missing word must be the opposite of asleep; awake is the opposite of asleep. Tired and resting describe states near sleep, and dreaming happens during sleep, so none is the antonym of asleep.
- BEE is to HIVE as ___ is to COLONY.
Correct answer: ANT
ANT completes the analogy. A bee is a single member of the group that lives in a hive, so the missing word must be the single member of a colony; an ant is a member of a colony. A queen is one specific role, a tunnel is part of the home, and soil is the surrounding ground, so none is the ordinary single member.
- BROOM is to JANITOR as WRENCH is to ___.
- BOLT
- GARAGE
- MECHANIC
- TOOLBOX
Correct answer: MECHANIC
MECHANIC completes the analogy. A broom is the tool used by a janitor, so the missing word must be the worker who uses a wrench; a mechanic uses a wrench. A bolt is what the wrench turns, a garage is the workplace, and a toolbox stores it, so none is the worker.
- FREEZE is to BOIL as ___ is to MAXIMUM.
- AVERAGE
- LIMIT
- MIDDLE
- MINIMUM
Correct answer: MINIMUM
MINIMUM completes the analogy. Freeze and boil mark opposite ends of a temperature scale, so the missing word must be the opposite end from maximum; minimum is the opposite of maximum. Average and middle fall between the extremes, and a limit is a general boundary, so none is the opposite extreme.
- YAWN is to BOREDOM as SHIVER is to ___.
Correct answer: COLD
COLD completes the analogy. A yawn is an effect produced by boredom, so the missing word must be the cause that produces a shiver; cold causes a shiver. A blanket relieves cold, warmth is the opposite condition, and a fever is a separate illness, so none is the cause of shivering.
- HUNGRY is to FAMISHED as ___ is to EXHAUSTED.
Correct answer: TIRED
TIRED completes the analogy. Hungry is a mild state and famished is its extreme form, so the missing word must be the mild form of exhausted; being tired is the mild form of being exhausted. Asleep is a different state, lazy describes attitude, and strong is the opposite of tired, so none is the milder degree.
- GARDENER is to RAKE as FISHERMAN is to ___.
Correct answer: NET
NET completes the analogy. A gardener uses a rake as a tool of the trade, so the missing word must be the tool a fisherman uses; a fisherman uses a net. Water is where the work happens, a boat is transportation, and fish are what is caught, so none is the handheld tool of the trade.
- In the analogy STITCH is to TAILOR as WELD is to BLACKSMITH, the relationship in BOTH pairs is best described as:
- Cause and effect
- Part to whole
- Opposites
- Action to the worker who performs it
Correct answer: Action to the worker who performs it
Action to the worker who performs it is the relationship. A stitch is the action a tailor performs, just as a weld is the action a blacksmith performs, so the bridge links a task to the worker who carries it out. The pairs are not a part and its whole, not opposites, and not a cause and its effect.
- PAGE is to BOOK as ___ is to CHAIN.
Correct answer: LINK
LINK completes the analogy. A page is a single repeated part that makes up the whole book, so the missing word must be the single repeated part that makes up a chain; a link is one piece of a chain. Metal is the material, a lock may secure a chain, and rope is a different object, so none is the repeating part.
- PARCHED is to MOIST as STARVING is to ___.
Correct answer: FULL
FULL completes the analogy. Parched means extremely dry, the opposite of moist, so the missing word must be the opposite of starving; being full is the opposite of starving. Hungry is a milder version of starving, thirsty refers to dryness, and eating is an action, so none is the antonym.
- What number comes next in the series 9, 13, 17, 21, ___ ?
Correct answer: 25
The next number is 25. Each term increases by a constant difference of 4, so adding 4 to 21 gives 25. A fixed step between every pair of terms is the signature of an arithmetic sequence.
- Find the next term in the series 14, 22, 30, 38, ___ .
Correct answer: 46
The next term is 46. The series climbs by a fixed amount of 8 each step, so 38 plus 8 equals 46. A constant difference between terms confirms an arithmetic sequence.
- What is the missing number in the series 60, 52, 44, 36, ___ ?
Correct answer: 28
The missing number is 28. Each term drops by a constant 8, so subtracting 8 from 36 gives 28. A steady decrease is still an arithmetic sequence, only with a negative common difference.
- Which number continues the series 2, 11, 20, 29, ___ ?
Correct answer: 38
The continuing number is 38. The terms rise by a constant difference of 9, so 29 plus 9 equals 38. Recognizing the fixed gap of 9 identifies this as an arithmetic sequence.
- What number is missing in the series 88, 76, 64, 52, ___ ?
Correct answer: 40
The missing number is 40. Each term decreases by 12, so 52 minus 12 equals 40. The constant step down marks this as an arithmetic sequence with a difference of negative 12.
- Find the next number in the series 15, 26, 37, 48, 59, ___ .
Correct answer: 70
The next number is 70. The series adds 11 each time, so 59 plus 11 equals 70. A consistent difference between every pair of terms confirms an arithmetic sequence.
- 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 equals the first term plus eight steps of 8, which is 8 plus 64. Multiplying the difference by the number of steps lets you jump ahead without listing every term.
- What number completes the series 18, 24, 30, ___, 42 ?
Correct answer: 36
The completing number is 36. The series increases by 6 each step, so the value between 30 and 42 must be 30 plus 6, which is 36, and 36 plus 6 is 42. A constant difference lets you fill any gap in an arithmetic sequence.
- Find the missing term in the series 5, ___, 17, 23, 29.
Correct answer: 11
The missing term is 11. The series increases by 6 each step, so the term between 5 and 17 must be 5 plus 6, which is 11. A constant difference lets you fill any gap in an arithmetic sequence.
- What number comes next in the series 21, 19, 17, 15, ___ ?
Correct answer: 13
The next number is 13. Each term decreases by a constant 2, so 15 minus 2 equals 13. A steady step down is an arithmetic sequence with a difference of negative 2.
- 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 to get the next, so 108 times 3 equals 324. A constant multiplier between terms is the mark of a geometric sequence.
- 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 equals 112. Doubling is a geometric sequence with a common ratio of 2.
- 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 equals 1296. A constant ratio of 6 between consecutive terms defines this geometric sequence.
- 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 equals 4802. A fixed ratio of 7 makes this a geometric sequence.
- 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 equals 6. Dividing by a constant is the same as a geometric sequence with a ratio of one-half.
- 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 equals 405. A consistent ratio of 3 identifies this geometric sequence.
- 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 equals 1. Dividing by a fixed value is a geometric sequence with a ratio of one-fifth.
- 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 between 12 and 192 must be 12 times 4, which is 48, and 48 times 4 is 192. A constant ratio lets you fill any missing term in a geometric sequence.
- Find the next term in the series 1000, 100, 10, 1, ___ .
Correct answer: 0.1
The next term is 0.1. Each value is divided by 10, so 1 divided by 10 equals 0.1. Dividing by a constant of 10 is a geometric sequence with a ratio of one-tenth.
- 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 equals 144. Doubling each value is a geometric sequence with a ratio of 2.
- 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 equals 34. Adding the two previous terms is the rule of a Fibonacci-style additive sequence.
- 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. Summing the two prior terms is the hallmark of a Fibonacci-style additive sequence seeded with two fours.
- 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. This additive build is a Fibonacci-style sequence even with different seed numbers.
- 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. Adding the two prior values is the defining rule of a Fibonacci-style additive sequence.
- 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. This Fibonacci-style additive pattern is seeded with 1 and 3.
- 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. Summing consecutive pairs is the Fibonacci-style additive rule, here seeded with two tens.
- 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. Summing the two prior terms is the Fibonacci-style additive pattern.
- 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. Adding consecutive pairs is the Fibonacci-style additive rule, seeded with two threes.
- 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 patterns: the odd positions count by 2 as 2, 4, 6 and the even positions count by 9 as 9, 18, 27. After 27, the next term belongs to the counting-by-two pattern, which continues to 8.
- Find the next term in the series 20, 3, 30, 6, 40, 9, ___ .
Correct answer: 50
The next term is 50. Two patterns alternate: one rises by tens as 20, 30, 40 and the other rises by threes as 3, 6, 9. After the 9, the tens pattern resumes, giving 50.
- What number continues the series 5, 40, 10, 35, 15, 30, ___ ?
Correct answer: 20
The continuing number is 20. The series alternates between two sequences: 5, 10, 15 increasing by 5 and 40, 35, 30 decreasing by 5. After 30, the rising sequence continues, so 15 plus 5 gives 20.
- What number is missing in the series 4, 1, 8, 5, 12, 9, 16, ___ ?
Correct answer: 13
The missing number is 13. Two interleaved patterns run together: the odd positions go 4, 8, 12, 16 adding 4, and the even positions go 1, 5, 9 adding 4. After 16 the next term belongs to the second pattern, so 9 plus 4 equals 13.
- Find the next number in the series 200, 6, 180, 12, 160, 18, ___ .
Correct answer: 140
The next number is 140. One pattern falls by 20 as 200, 180, 160 and the other rises by 6 as 6, 12, 18. After 18, the falling pattern returns, so 160 minus 20 equals 140.
- What number comes next in the series 3, 64, 6, 32, 12, 16, ___ ?
Correct answer: 24
The next number is 24. Two patterns interleave: 3, 6, 12 doubles each time and 64, 32, 16 halves each time. After 16, the doubling pattern resumes, so 12 times 2 equals 24.
- Find the next term in the series 1, 100, 2, 90, 3, 80, ___ .
Correct answer: 4
The next term is 4. Two patterns alternate: 1, 2, 3 counts up by one and 100, 90, 80 drops by ten. After 80, the counting pattern returns, so 3 plus 1 equals 4.
- What number comes next in the series 6, 50, 12, 45, 24, 40, ___ ?
Correct answer: 48
The next number is 48. Two patterns interleave: 6, 12, 24 doubles each time and 50, 45, 40 drops by 5. After 40, the doubling pattern resumes, so 24 times 2 equals 48.
- What number comes next in the series 3, 4, 7, 12, 19, ___ ?
Correct answer: 28
The next number is 28. The differences between terms are 1, 3, 5, 7, which themselves grow by 2, so the next gap is 9 and 19 plus 9 equals 28. Looking at the differences of the differences uncovers this second-order pattern.
- Find the next term in the series 2, 4, 9, 17, 28, ___ .
Correct answer: 42
The next term is 42. The differences are 2, 5, 8, 11, increasing by 3 each time, so the next difference is 14 and 28 plus 14 equals 42. The constant change in the differences is a second-order pattern.
- What number continues the series 5, 6, 9, 14, 21, ___ ?
Correct answer: 30
The continuing number is 30. The differences are 1, 3, 5, 7, growing by 2 each step, so the next difference is 9 and 21 plus 9 equals 30. Examining how the gaps change reveals this second-order pattern.
- What number is missing in the series 10, 12, 16, 22, 30, ___ ?
Correct answer: 40
The missing number is 40. The differences are 2, 4, 6, 8, increasing by 2, so the next gap is 10 and 30 plus 10 equals 40. The steady rise in the differences themselves is a second-order pattern.
- Find the next number in the series 1, 3, 8, 16, 27, ___ .
Correct answer: 41
The next number is 41. The differences are 2, 5, 8, 11, climbing by 3 each time, so the next difference is 14 and 27 plus 14 equals 41. Tracking the differences of the differences exposes the rule.
- What number comes next in the series 4, 7, 13, 22, 34, ___ ?
Correct answer: 49
The next number is 49. The differences are 3, 6, 9, 12, increasing by 3 each step, so the next gap is 15 and 34 plus 15 equals 49. The steady growth in the gaps is a second-order pattern.
- Find the next term in the series 3, 4, 8, 17, 33, ___ .
Correct answer: 58
The next term is 58. The differences are 1, 4, 9, 16, which are the perfect squares, so the next difference is 25 and 33 plus 25 equals 58. Looking at how the gaps change exposes this second-order pattern.
- What number is missing in the series 0, 1, 4, 9, 16, ___ ?
Correct answer: 25
The missing number is 25. The differences are 1, 3, 5, 7, rising by 2 each step, so the next difference is 9 and 16 plus 9 equals 25. The increasing gaps form a second-order pattern.
- Which rule governs the series 8, 15, 22, 29, 36?
- Add 7 to each term
- Add the two previous terms
- Multiply each term by 2
- Subtract 7 from each term
Correct answer: Add 7 to each term
The rule is to add 7 to each term. Moving from 8 to 15 to 22 and beyond shows a constant increase of 7, so the governing rule is repeated addition of 7. Identifying the consistent difference is how you deduce the rule.
- What rule explains the series 2, 8, 32, 128, 512?
- Add 6 each time
- Multiply each term by 4
- Square each term
- Add the two previous terms
Correct answer: Multiply each term by 4
The rule is to multiply each term by 4. Each value is four times the one before it, since 2 becomes 8, then 32, then 128, then 512. A constant multiplier rather than a constant addition is what defines this rule.
- Which rule produces the series 1, 2, 4, 7, 11, 16?
- Multiply each term by 2
- Subtract 1 each time
- Add a number that increases by 1 each step
- Add the two previous terms
Correct answer: Add a number that increases by 1 each step
The rule adds an amount that grows by 1 each step. The increases are 1, then 2, then 3, then 4, then 5, so each added value rises by one. Recognizing that the differences themselves climb steadily reveals the governing rule.
- What rule governs the series 64, 32, 16, 8, 4?
- Subtract 16 each time
- Add the two previous terms
- Divide each term by 4
- Divide each term by 2
Correct answer: Divide each term by 2
The rule is to divide each term by 2. Each value is half of the previous one, since 64 becomes 32, then 16, then 8, then 4. A constant division identifies the rule rather than a fixed subtraction.
- 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 equal 1 through 6 squared, so the seventh term is 7 squared, which is 49. Spotting that each term is a square number reveals the pattern.
- 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 equal 2, 3, 4, and 5 cubed, so the next term is 6 cubed, which is 216. Recognizing the cubes uncovers the governing rule.
- Which rule governs the series 3, 9, 27, 81?
- Multiply each term by 2
- Add the two previous terms
- Multiply each term by 3
- Add 6 each time
Correct answer: Multiply each term by 3
The rule is to multiply each term by 3. Each value is triple the one before, since 3 becomes 9, then 27, then 81. A constant multiplier of 3 rather than a fixed addition defines the rule.
- What rule governs the series 90, 81, 72, 63, 54?
- Add 9 each time
- Subtract a growing amount
- Divide each term by 2
- Subtract 9 each time
Correct answer: Subtract 9 each time
The rule is to subtract 9 each time. Every term is 9 less than the previous one, since 90 drops to 81, then 72, then 63, then 54. A constant difference of negative 9 defines this descending arithmetic rule.
- Which rule governs the series 2, 5, 11, 23, 47?
- Double each term and add 1
- Add 3 each time
- Add the two previous terms
- Square each term
Correct answer: Double each term and add 1
The rule is to double each term and add 1. Each value is twice the previous one plus 1, since 2 becomes 5, then 11, then 23, then 47. Testing a multiply-then-add rule uncovers the governing pattern.
- What rule governs the series 1, 4, 13, 40, 121?
- Add 3 each time
- Multiply each term by 3 and add 1
- Add the two previous terms
- Square each term
Correct answer: Multiply each term by 3 and add 1
The rule is to multiply each term by 3 and add 1. Each value is triple the previous one plus 1, since 1 becomes 4, then 13, then 40, then 121. Testing a multiply-then-adjust rule reveals the governing pattern.
- 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. The fixed step of 6 marks 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. A constant decrease of 6 makes this an 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. A constant ratio of 10 identifies this 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. Doubling is a geometric sequence with a 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. Adding consecutive pairs is the Fibonacci-style additive rule.
- 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. Summing the two prior terms is the Fibonacci-style additive pattern.
- What comes next in the series 70, 2, 65, 5, 60, 8, ___ ?
Correct answer: 55
The next number is 55. One pattern decreases by 5 as 70, 65, 60 and the other increases by 3 as 2, 5, 8. After the 8, the decreasing pattern resumes, so 60 minus 5 equals 55.
- Find the next term in the series 2, 7, 6, 10, 18, 13, ___ .
Correct answer: 54
The next term is 54. Two patterns interleave: 2, 6, 18 multiplies by 3 and 7, 10, 13 adds 3. After 13, the multiplying pattern returns, so 18 times 3 equals 54.
- What number comes next in the series 5, 8, 14, 23, 35, ___ ?
Correct answer: 50
The next number is 50. The differences are 3, 6, 9, 12, increasing by 3 each step, so the next gap is 15 and 35 plus 15 equals 50. Examining how the differences grow reveals the second-order pattern.
- Find the next term in the series 20, 21, 24, 29, 36, ___ .
Correct answer: 45
The next term is 45. The differences are 1, 3, 5, 7, rising by 2 each step, so the next difference is 9 and 36 plus 9 equals 45. The steady growth in the gaps is a second-order pattern.
- Which rule governs the series 4, 12, 36, 108?
- Multiply each term by 3
- Multiply each term by 2
- Add the two previous terms
- Add 8 each time
Correct answer: Multiply each term by 3
The rule is to multiply each term by 3. Each value is triple the one before, since 4 becomes 12, then 36, then 108. A constant multiplier of 3 rather than a fixed addition defines the rule.
- What rule governs the series 75, 70, 65, 60, 55?
- Add 5 each time
- Subtract 5 each time
- Subtract a growing amount
- Divide each term by 2
Correct answer: Subtract 5 each time
The rule is to subtract 5 each time. Every term is 5 less than the previous one, since 75 drops to 70, then 65, then 60, then 55. A constant difference of negative 5 defines this descending arithmetic rule.
- What number comes next in the series 1, 3, 6, 10, 15, ___ ?
Correct answer: 21
The next number is 21. The differences are 1, 2, 3, 4, increasing by 1 each step, so the next gap is 6 and 15 plus 6 equals 21. These are the triangular numbers, and tracking how the gaps grow reveals the second-order pattern.
- Find the next term in the series 13, 16, 19, 22, ___ .
Correct answer: 25
The next term is 25. Each term increases by a constant 3, so 22 plus 3 equals 25. A fixed step of 3 marks an arithmetic sequence.
- What number comes next in the series 4, 8, 16, 32, 64, ___ ?
Correct answer: 128
The next number is 128. Each term doubles, so 64 times 2 equals 128. Doubling each value is a geometric sequence with a 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?
Correct answer: UP
The fourth arrow should point UP. Turning the up arrow a quarter turn clockwise produces the right arrow, so the same quarter turn clockwise must be applied to the third figure; a left arrow rotated a quarter turn clockwise points up. The rule is a consistent 90-degree clockwise rotation, not a flip or a reversal.
- A figure analogy reads: a triangle pointing up is to a triangle pointing down, as a triangle pointing left is to ___.
- A triangle pointing down
- A triangle pointing up
- A triangle pointing right
- A larger triangle pointing left
Correct answer: A triangle pointing right
The answer is a triangle pointing right. The up triangle is rotated a half turn to become the down triangle, so the same half-turn rotation must be applied to the left-pointing triangle; a half turn sends a left-pointing triangle to a right-pointing triangle. The transformation is a 180-degree rotation, which reverses the direction the point faces.
- 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 rotated a quarter turn clockwise
- The letter T made larger
- The letter T unchanged
- The letter T flipped upside down
Correct answer: The letter T rotated a quarter turn clockwise
The answer is the letter T rotated a quarter turn clockwise. The first pair shows the L turned one quarter clockwise, so the identical quarter turn clockwise must be applied to the T. Matching the exact rotation, not leaving it unchanged or flipping it, preserves the analogy.
- 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?
Correct answer: To 12
The hand will point to 12. Moving from 12 to 3 is a quarter turn clockwise, so the same quarter turn clockwise applied to a hand at 9 advances it to 12. A consistent 90-degree clockwise rotation governs both figures.
- 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 mirrored left to right
- The flag enlarged
- The flag pointing up, unchanged
- The flag pointing down (rotated a half turn)
Correct answer: The flag pointing down (rotated a half turn)
The answer is the flag pointing down. The first pair applies a half turn, a 180-degree rotation, to the comma shape, so the same half turn must rotate the flag so it points down. A half turn spins the whole figure around, which is different from a mirror flip.
- A figure analogy shows an upper-case letter R, then the letter R rotated 90 degrees counterclockwise. Which transformation links these two figures?
- A change in shading
- A reflection across a vertical line
- A change in size
- A rotation
Correct answer: A rotation
The transformation is a rotation. The second R has been turned 90 degrees counterclockwise while keeping the same handedness, which is the defining feature of a rotation. A reflection would reverse the letter into a mirror image, and neither the size nor the shading has changed.
- 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 (unchanged)
- Northeast (up and to the right)
- Southwest (down and to the left)
- Southeast (down and to the right)
Correct answer: Northeast (up and to the right)
The arrow will point northeast. A quarter turn clockwise moves northeast to southeast, so the same quarter turn clockwise moves a northwest arrow to northeast. Tracking the diagonal one step clockwise around the compass gives the new direction.
- 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?
- An upward arrow becoming a downward arrow
- A square becoming a larger square
- A triangle becoming its mirror image
- A hollow circle becoming a filled circle
Correct answer: An upward arrow becoming a downward arrow
The matching pair is an upward arrow becoming a downward arrow. Turning a 6 a half turn produces a 9, a 180-degree rotation, and rotating an up arrow a half turn produces a down arrow. The other pairs change size, shading, or apply a mirror flip rather than 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
- The letter p turned upside down
- The letter b
- The letter q
Correct answer: The letter q
The answer is the letter q. The letter b becomes d by a left-to-right mirror flip, so the same horizontal flip must be applied to p; flipping p left to right produces q. A horizontal reflection reverses the figure across a vertical line.
- A figure analogy shows the letter p, then the letter b. Which transformation links these two figures?
- A vertical flip (top to bottom)
- A horizontal flip (left to right)
- A 90-degree rotation
- An increase in size
Correct answer: A vertical flip (top to bottom)
The transformation is a vertical flip. Reflecting p across a horizontal line moves its descending loop to an ascending position, turning it into b. A horizontal flip (left to right) would turn p into q instead. A vertical flip mirrors the figure from top to bottom.
- In a figure analogy, an arrow pointing right is to an arrow pointing left, as a hand with the thumb pointing right is to ___.
- A hand with the thumb pointing left
- A hand rotated a quarter turn
- A hand with the thumb pointing right
- A larger hand with the thumb pointing right
Correct answer: A hand with the thumb pointing left
The answer is a hand with the thumb pointing left. The right arrow becomes a left arrow by a horizontal flip across a vertical line, so the same flip reverses the hand so its thumb points left. A horizontal reflection mirrors the figure from side to side.
- 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 facing right, unchanged
- A capital E facing left
- A larger capital E
- A capital E reflected so its arms point up and down differently (flipped top to bottom)
Correct answer: A capital E reflected so its arms point up and down differently (flipped top to bottom)
The answer is the E flipped top to bottom. The triangle is reflected across a horizontal line so its point swaps from top to bottom, so the same horizontal-axis flip must be applied to the E. A vertical flip mirrors a figure across a horizontal line, swapping its top and bottom.
- 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 vertical flip (top to bottom)
- A horizontal flip (left to right)
- A shading change
- A size increase
Correct answer: A horizontal flip (left to right)
The transformation is a horizontal flip. Reflecting the checkmark across a vertical line carries it from the left side to the right side as a mirror image. A vertical flip would move it from bottom to top instead, and no size or shading change has taken place.
- 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:
- A larger person leaning to the right
- The person upside down
- The person leaning to the left
- The person standing straight
Correct answer: The person leaning to the left
The answer is the person leaning to the left. The boat's sail is mirrored from right to left by a horizontal flip, so the same flip reverses the person to lean left. A left-to-right reflection swaps the figure's left and right sides.
- In a figure analogy, the letter N is to its mirror image (a backward N), as the letter S is to ___.
- The letter S unchanged
- A backward S (its mirror image)
- The letter S rotated a quarter turn
- A larger letter S
Correct answer: A backward S (its mirror image)
The answer is a backward S. The N is reflected left to right into its mirror image, so the same horizontal flip turns the S into a backward S. A mirror reflection reverses the orientation of the letter across a vertical line.
- A figure analogy reads: a small circle is to a large circle, as a small square is to ___.
- A large circle
- A small square
- A small triangle
- A large square
Correct answer: A large square
The answer is a large square. The first pair keeps the shape the same and only makes it bigger, so the same enlargement must be applied to the square while keeping it a square. The rule changes size alone, leaving the shape and everything else unchanged.
- In a figure analogy, a large star is to a small star, as a large hexagon is to ___.
- A large circle
- A large hexagon
- A small star
- A small hexagon
Correct answer: A small hexagon
The answer is a small hexagon. The first pair shrinks the star while keeping it a star, so the same shrinking must be applied to the hexagon. The transformation reduces size only, holding the shape constant.
- A figure analogy shows a small triangle, then a triangle twice as large. Which transformation links the two figures?
- A change in size (scaling)
- A change in shading
- A reflection
- A rotation
Correct answer: A change in size (scaling)
The transformation is a change in size. The second triangle is the same triangle made larger, with its shape, orientation, and shading all unchanged, which is the defining feature of scaling. No turning, flipping, or shading change is present.
- 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 box with a tiny heart
- A tiny heart inside a box, unchanged
- A tiny dot inside the box
- A large heart inside the same box
Correct answer: A large heart inside the same box
The answer is a large heart inside the same box. The dot grows larger while the box stays the same, so the heart must grow larger while its box stays the same. Only the inner figure is scaled up; the surrounding box is unchanged.
- 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 ___.
- Two large squares in a row
- Three large squares in a row
- Three small squares in a row
- Three small circles in a row
Correct answer: Three large squares in a row
The answer is three large squares in a row. The pair of circles is enlarged while keeping the count of two, so the squares must be enlarged while keeping the count of three. The rule scales each shape up without adding, removing, or changing the shapes.
- 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 pointing right
- A small arrow pointing down
- A large arrow pointing up
- A small arrow pointing up
Correct answer: A small arrow pointing up
The answer is a small arrow pointing up. The first pair shrinks the arrow while keeping it pointing right, so the same shrinking keeps the second arrow pointing up but smaller. The direction is preserved and only the size changes.
- 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 hollow square, unchanged
- A solid (fully shaded) square
- A larger hollow square
- A solid circle
Correct answer: A solid (fully shaded) square
The answer is a solid square. The circle changes from hollow to fully shaded, so the same shading change must fill in the square. The rule alters shading only, leaving the shape and size the same.
- In a figure analogy, a black (filled) triangle is to a white (unfilled) triangle, as a black star is to ___.
- A larger black star
- A white triangle
- A black star
- A white (unfilled) star
Correct answer: A white (unfilled) star
The answer is a white star. The filled triangle becomes unfilled, so the same change empties the star to white. Only the shading is reversed from filled to unfilled, with the shape held constant.
- 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 rotation
- A size increase
- The removal of an element
- A swap of the shaded and unshaded regions
Correct answer: A swap of the shaded and unshaded regions
The transformation is a swap of the shaded and unshaded regions. The black and white halves trade places while the circle itself stays the same shape and size. This is a shading change, not a rotation, resize, or removal of any part.
- 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 bottom-right small square
- All four small squares
- The bottom-left small square
- The top-left small square
Correct answer: The bottom-right small square
The answer is the bottom-right small square. The shaded cell moves from the left side to the matching cell on the right side, so the bottom-left shading must move to the bottom-right. The shading slides horizontally across the grid while the grid stays fixed.
- A figure analogy reads: a light gray circle is to a dark gray circle, as a light gray pentagon is to ___.
- A light gray pentagon
- A black pentagon outline only
- A dark gray circle
- A dark gray pentagon
Correct answer: A dark gray pentagon
The answer is a dark gray pentagon. The circle becomes darker in shade, so the same darkening must be applied to the pentagon. The rule deepens the shading while keeping the shape and size unchanged.
- 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 with horizontal stripes
- A circle with vertical stripes
- A triangle with vertical stripes
- A solid black circle
Correct answer: A circle with vertical stripes
The answer is a circle with vertical stripes. The triangle's stripes turn from horizontal to vertical, so the same change reorients the circle's stripes to vertical. The fill pattern changes while the outer shape stays the same.
- 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 with one dot inside it
- A circle with two dots inside it
- A circle with three dots inside it
- A square with two dots inside it
Correct answer: A circle with two dots inside it
The answer is a circle with two dots inside it. The square gains one extra dot, going from one to two, so the same addition gives the circle a second dot. The rule adds exactly one element while keeping the outer shape the same.
- 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 triangle with two small circles inside
- A square with one small circle inside
- A square with two small circles inside
- A square with three small circles inside
Correct answer: A square with two small circles inside
The answer is a square with two small circles inside. The triangle loses one circle, dropping from three to two, so the same removal leaves the square with two circles. The rule removes exactly one inner element while the outer shape is unchanged.
- A figure analogy shows a plain circle, then a circle with an X drawn through it. Which transformation links the two figures?
- A reflection
- A rotation
- The addition of an element
- A size decrease
Correct answer: The addition of an element
The transformation is the addition of an element. The second figure is the same circle with a new X added inside it, so an element has been put in. Nothing is turned, resized, or flipped; a feature is simply added.
- 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 stack of six horizontal bars
- A stack of three horizontal bars
- A stack of four horizontal bars
- A stack of five horizontal bars
Correct answer: A stack of four horizontal bars
The answer is a stack of four horizontal bars. The first pair removes one bar, going from four to three, so removing one bar from a stack of five leaves four. The rule takes away exactly one element from the group.
- 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 larger clock face with no hands
- A clock face with the marks removed
- A clock face with hour marks and hands added
- A clock face with no marks and no hands
Correct answer: A clock face with hour marks and hands added
The answer is a clock face with hour marks and hands added. The face gains a mouth that was missing, so the clock should gain the hands that were missing. The rule adds the one missing feature while keeping everything already present.
- 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 containing a single triangle
- A square containing a triangle and a circle
- A pentagon containing two triangles
- A pentagon containing a triangle and a circle
Correct answer: A pentagon containing a triangle and a circle
The answer is a pentagon containing a triangle and a circle. The square gains an added circle alongside the triangle, so the pentagon should likewise gain a circle alongside its triangle. The rule adds one new inner shape while keeping the outer shape and the original inner shape.
- 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 small white circle
- A large white circle
- A small black circle
- A large black circle
Correct answer: A large white circle
The answer is a large white circle. The first pair both enlarges the figure and flips it from black to white, so both changes must apply to the circle, making it large and white. Tracking every transformation that occurs in the first pair and applying all of them is how A is to B becomes C is to D.
- 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?
- Relate to C the same way B relates to A
- Be larger than all the other figures
- Match the first figure A exactly
- Be the mirror image of C in every case
Correct answer: Relate to C the same way B relates to A
The fourth figure must relate to C the same way B relates to A. A figure analogy first establishes a transformation from A to B, then requires the same transformation from C to D. Identifying that shared rule, rather than copying a single figure or always flipping, is the core skill of figure analogies.
- 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 color of the star
- How the circle was changed to become the square
- Which answer choice is largest
- The size of the missing figure
Correct answer: How the circle was changed to become the square
You should first determine how the circle was changed to become the square. The rule that turns A into B is the same rule you apply to C to find D, so naming that change is the starting point. Solving a figure analogy always begins by reading the relationship in the first pair before touching the second pair.
- 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 figure is shaded in
- The figure is rotated a half turn so it points the opposite way
- An extra arrow is added
- The figure is made larger
Correct answer: The figure is rotated a half turn so it points the opposite way
The single rule is a half-turn rotation so each arrow points the opposite way. Up to down and right to left are both reversals by a 180-degree turn, so one rule covers both pairs. A correct figure analogy uses the same relationship in the A-to-B pair and the C-to-D pair.
- 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 in both pairs
- The shape changes from one pair to the next
- The figure shrinks in both pairs
- The shading changes from white to black in both pairs
Correct answer: The shading changes from white to black in both pairs
The relationship is that the shading changes from white to black in both pairs. The circle and the triangle each go from unfilled to filled, so the same shading rule governs both pairs of the analogy. A figure analogy holds together because the C-to-D change mirrors the A-to-B change.
- 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 a dot in the center
- A square with no dot
- A circle with two dots
- A circle with no dot
Correct answer: A circle with no dot
The answer is a circle with no dot. The first pair removes the center dot from the square, so the same removal takes the dot out of the circle. Applying the identical change from the first pair to the third figure produces the matching fourth figure.
- 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 large outlined pentagon
- A small outlined hexagon
- A small filled hexagon
- A large filled hexagon
Correct answer: A small filled hexagon
The answer is a small filled hexagon. The first pair both shrinks the figure and fills it in, so both changes apply to the hexagon, making it small and filled. When two transformations occur together in the A-to-B pair, both must carry over to the C-to-D pair.
- A figure analogy shows: a single circle is to two circles, as a single square is to ___.
- Three squares
- Two squares
- One square
- Two circles
Correct answer: Two squares
The answer is two squares. The first pair doubles the figure from one to two, so the same doubling turns one square into two squares. The relationship that doubles the count of the shape carries from the first pair to the second.
- 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:
Correct answer: Left
The fourth arrow points left. A quarter turn clockwise sends right to down, so the same quarter turn clockwise sends down to left. Continuing one step clockwise around the directions gives the answer.
- 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 upside down
- A backward G (mirror image facing left)
- A larger letter G
- The letter G unchanged
Correct answer: A backward G (mirror image facing left)
The answer is a backward G. The F is reflected left to right into a mirror image, so the same horizontal flip turns the G into a backward G. A mirror reflection reverses the letter across a vertical line.
- In a figure analogy, a small filled circle is to a large filled circle, as a small filled diamond is to ___.
- A large filled diamond
- A small filled diamond
- A large hollow diamond
- A large filled circle
Correct answer: A large filled diamond
The answer is a large filled diamond. The first pair enlarges the circle while keeping it filled, so the diamond must grow larger and stay filled. The rule changes only the size, leaving shape and shading the same.
- 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 containing three stars
- A circle containing two stars
- A circle containing four stars
- A square containing four stars
Correct answer: A circle containing four stars
The answer is a circle containing four stars. The square doubles its stars from two to four, so the circle should double its stars to four as well. The rule adds elements until the count doubles, while the outer shape stays the same.
- 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?
Correct answer: At 9
The hand will point at 9. Moving from 12 to 6 is a half turn, a 180-degree rotation, so a half turn from 3 lands on 9. The same half-turn rotation applies to both hands.
- 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 shaded on its upper half only
- A circle shaded on its lower half only
- A fully shaded circle
- A hollow circle
Correct answer: A circle shaded on its lower half only
The answer is a circle shaded on its lower half only. The square gains shading on its bottom half, so the same partial shading fills only the bottom half of the circle. The rule shades a specific region rather than the whole figure.
- 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 ___.
- Two dots in a row
- The three dots in a row connected by a single line
- Three dots arranged in a triangle
- Three dots in a row, unchanged
Correct answer: The three dots in a row connected by a single line
The answer is the three dots in a row connected by a single line. The first pair adds connecting lines to join the dots, so the same addition draws a line through the row of dots. The rule adds the connecting element while keeping the dots in place.
- 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:
- An upside-down V (rotated a half turn)
- The letter V unchanged
- A larger letter V
- The letter V mirrored left to right
Correct answer: An upside-down V (rotated a half turn)
The answer is an upside-down V. The first pair rotates the A a half turn, so the same 180-degree rotation flips the V to point upward at the bottom. A half turn spins the whole figure around its center.
- In a figure analogy, a large gray triangle is to a small gray triangle, as a large gray arrow is to ___.
- A small gray arrow
- A large gray arrow
- A large gray triangle
- A small black arrow
Correct answer: A small gray arrow
The answer is a small gray arrow. The triangle shrinks while staying gray, so the arrow must shrink while staying gray. Only the size changes; the shape and shading remain the same.
- 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.
- A square with an arrow pointing up inside it
- A square with an arrow pointing left inside it
- A square with an arrow pointing right inside it
- A square with an arrow pointing down inside it
Correct answer: A square with an arrow pointing up inside it
The answer is a square with an arrow pointing up inside it. The inner arrow turns a quarter turn clockwise from up to right, so the same quarter turn clockwise sends the left arrow to up. Only the inner arrow rotates while the surrounding square stays fixed.
- In a figure analogy, a hollow heart is to a filled (black) heart, as a hollow moon (crescent) is to ___.
- A filled heart
- A larger hollow crescent
- A hollow crescent
- A filled (black) crescent
Correct answer: A filled (black) crescent
The answer is a filled crescent. The heart changes from hollow to filled, so the same shading fills in the crescent. The rule alters shading only, keeping the shape and size the same.
- 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 triangle with sharp corners
- A square with sharp corners
- A triangle with rounded corners
- A circle
Correct answer: A triangle with sharp corners
The answer is a triangle with sharp corners. The first pair changes the rounded corners to sharp ones, so the same change sharpens the triangle's corners. The detail of the corners changes while the overall shape stays a triangle.
- 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?
Correct answer: Up
The fourth arrow points up. A quarter turn counterclockwise sends up to left, so the same quarter turn counterclockwise sends right to up. Moving one step counterclockwise around the directions gives the answer.
- 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 wide flat rectangle
- A wide flat oval lying down
- A circle
- A tall thin oval upright
Correct answer: A wide flat oval lying down
The answer is a wide flat oval lying down. The upright rectangle is rotated a quarter turn to lie flat, so the same quarter turn lays the oval on its side. The rule turns the figure so its long direction becomes horizontal.
- 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?
- In the same place
- Directly across from where it started
- One quarter clockwise from where it started
- One quarter counterclockwise from where it started
Correct answer: One quarter clockwise from where it started
The shaded quarter is one quarter clockwise from where it started. The shading moves from the top-left to the top-right, which is one position clockwise around the circle. The rule rotates the shaded region a quarter turn clockwise while the circle's outline stays fixed.
- 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:
- Bottom-left corner
- Center
- Bottom-right corner
- Top-left corner
Correct answer: Bottom-right corner
The answer is the bottom-right corner. The circle is flipped from the left side to the right side across a vertical line, so the bottom-left circle moves to the bottom-right. A horizontal flip mirrors the inner element from one side to the other.
- 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 four dots
- A figure with two dots
- A figure with three dots
- A figure with five dots
Correct answer: A figure with four dots
The answer is a figure with four dots. The first pair adds two dots, going from one to three, so adding two dots to a figure with two dots gives four. The rule adds the same fixed number of elements in both pairs.
- 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 black triangle with a white circle in the middle
- A white triangle
- A black square with a white circle
- A black triangle, unchanged
Correct answer: A black triangle with a white circle in the middle
The answer is a black triangle with a white circle in the middle. The first pair adds a white circle inside the black shape, so the same addition places a white circle inside the triangle. The rule adds the inner element while keeping the outer shape and its shading.
- 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 large black star
- A large white square
- A small black square
- A large black square
Correct answer: A large black square
The answer is a large black square. The first pair both enlarges the figure and fills it from white to black, so both changes apply to the square, making it large and black. Every transformation in the A-to-B pair must be repeated in the C-to-D pair.
- 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 circle alone
- A square, unchanged
- A triangle inside a circle
- A square inside a circle
Correct answer: A square inside a circle
The answer is a square inside a circle. The first pair adds a surrounding circle around the shape, so the same addition places the square inside a circle. The rule adds the enclosing element while keeping the original shape.
- 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 size increase
- A vertical or horizontal flip that mirrors the diagonal
- An added line
- A shading change
Correct answer: A vertical or horizontal flip that mirrors the diagonal
The transformation is a flip that mirrors the diagonal. Reflecting the square across a vertical line swaps the diagonal so it runs the other way. No line is added and no shading or size changes; only the orientation is mirrored.
- 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 ___.
- A small square on top of a large square
- A small circle on top of a large circle
- A large circle on top of a small circle
- Two equal circles
Correct answer: A large circle on top of a small circle
The answer is a large circle on top of a small circle. The first pair swaps which figure is bigger, putting the large one on top, so the same swap places the large circle on top. The rule exchanges the sizes of the two stacked elements.
- 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:
- An X (the slash crossed by a backslash)
- A vertical line
- The slash unchanged
- A horizontal line
Correct answer: An X (the slash crossed by a backslash)
The answer is an X. The first pair adds a perpendicular crossing line to the horizontal line, making a plus, so adding a crossing line to the slash makes an X. The rule adds a second line crossing the first.
- 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 no dividing line
- A square with a horizontal line
- A square with a vertical line
- A circle with no dividing line
Correct answer: A square with no dividing line
The answer is a square with no dividing line. The first pair removes the dividing line from the circle, so the same removal takes the line out of the square. The rule deletes the internal element while keeping the outer shape.
- 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 small circle becoming a large circle
- An up arrow becoming a down arrow
- The letter b becoming the letter d
- A white square becoming a black square
Correct answer: The letter b becoming the letter d
The matching pair is the letter b becoming the letter d. Flipping a right triangle across a vertical line makes it point left, and flipping b the same way makes d. The other pairs change size, shading, or rotate the figure rather than mirror it left to right.
- 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 with all three dots
- A triangle with the two bottom dots only
- A triangle with only the top dot
- A triangle with no dots
Correct answer: A triangle with only the top dot
The answer is a triangle with only the top dot. The square loses its bottom corner dots and keeps the top ones, so removing the two bottom dots from the triangle leaves only the top dot. The rule removes the lower elements while keeping the upper element.
- In a figure analogy, a medium circle is to a circle four times its area, as a medium square is to ___.
- A square half its area
- A medium circle
- A square four times its area
- A medium square
Correct answer: A square four times its area
The answer is a square four times its area. The first pair enlarges the circle to four times its area, so the same scaling enlarges the square to four times its area. The rule multiplies the size by the same factor while keeping the shape.
- 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-shape rotated a quarter turn clockwise
- The J-shape mirrored
- A larger J-shape
- The J-shape unchanged
Correct answer: The J-shape rotated a quarter turn clockwise
The answer is the J-shape rotated a quarter turn clockwise. The first pair turns the L a quarter turn clockwise, so the same quarter turn clockwise must be applied to the J. Matching the exact rotation preserves the relationship.
