Your FREE Postsecondary Education Readiness Test (PERT) Practice Test 2026 – 450+ Q&A

Correct answer: x = 6

Add 7 to both sides to get 3x = 18, then divide both sides by 3: x = 6. The value x = 11/3 ignores the -7 and solves 3x = 11. The value x = 4 subtracts 7 instead of adding it (3x = 4) and then never divides by 3. The value x = 32/3 divides only the 11 by 3 and then adds 7, but dividing by 3 has to reach every term, the -7 included.

Correct answer: x = 7

Distribute first: 5x - 10 = 3x + 4. Subtract 3x and add 10 to get 2x = 14, so x = 7. The value x = 3 multiplies only the x by 5 and leaves the -2 alone (5x - 2 = 3x + 4). The value x = -3 writes 5 times -2 as +10, which leads to 2x = -6. The value x = 14 reaches 2x = 14 and never divides by 2.

Correct answer: x > 4

Subtract 5 from both sides to get 2x > 8, then divide by 2: x > 4. Dividing by a positive number keeps the direction of the sign. The answer x > 9 adds 5 instead of subtracting it (2x > 18). The answer x > 8 stops after subtracting 5 and never divides by 2. The answer x > 6 subtracts 2 from 8 instead of dividing by 2.

Correct answer: x ≥ -4

Divide both sides by -3. Dividing by a negative number reverses the sign, so -3x ≤ 12 becomes x ≥ -4. The answer x ≤ -4 divides correctly but keeps the original direction. The answer x ≤ 15 reads -3x as x - 3 and adds 3 to 12. The answer x ≥ 15 makes that misreading and also reverses the sign, although adding 3 to both sides never reverses an inequality.

Correct answer: x = 16

Subtract 3 from both sides to get x/4 = 4, then multiply both sides by 4: x = 16. The value x = 40 adds 3 instead of subtracting it, so x/4 = 10. The value x = 4 stops at x/4 = 4 and never clears the fraction. The value x = 25 multiplies only the 7 by 4 (x + 3 = 28) and then subtracts 3; multiplying by 4 has to reach every term.

Correct answer: x = 2 or x = 3

Find two numbers that multiply to 6 and add to -5: -2 and -3. The equation factors as (x - 2)(x - 3) = 0, so x = 2 or x = 3. The pair x = -2 or x = -3 takes the numbers from the factors without switching their signs; it solves (x + 2)(x + 3) = 0. The pair x = 6 or x = 4 uses the quadratic formula but skips the division by 2: 5 ± 1 gives 6 or 4, and halving those gives 3 or 2. The pair x = -6 or x = -4 skips that division and also uses -5 in place of -b.

Correct answer: x = 3 or x = -3

Add 9 to both sides to get x^2 = 9, then take the square root with both signs: x = 3 or x = -3, since 3^2 = 9 and (-3)^2 = 9. The pair x = 9 or x = -9 stops at x^2 = 9 and never takes the root. The pair x = 4.5 or x = -4.5 divides 9 by 2, treating the exponent as a factor, instead of taking the square root. The pair x = 18 or x = -18 multiplies 9 by 2 instead of taking the square root.

Correct answer: x = -3 or x = -4

Find two numbers that multiply to 12 and add to 7: 3 and 4. The equation factors as (x + 3)(x + 4) = 0, so x = -3 or x = -4. The pair x = 3 or x = 4 reads the numbers off the factors without switching their signs. The pair x = -8 or x = -6 uses the quadratic formula, -7 ± 1, but drops the division by 2. The pair x = 8 or x = 6 drops that division and also loses the minus sign on -b.

Correct answer: r = sqrt(A/pi)

Divide both sides by pi to get r^2 = A/pi, then take the positive square root of the whole quotient: r = sqrt(A/pi). The answer r = sqrt(A)/pi takes the root of A but not of pi. The answer r = sqrt(A/2pi) divides by 2pi, mixing up the area formula with the circumference formula C = 2(pi)r. The answer r = sqrt(A)/2pi makes both slips: it roots only A and divides by 2pi.

Correct answer: y = (12 - 2x)/3

Subtract 2x from both sides to get 3y = 12 - 2x, then divide the whole right side by 3: y = (12 - 2x)/3. The answer y = (12 + 2x)/3 moves 2x across without changing its sign. The answer y = (12 - 2x)/2 divides by 2, the coefficient of x, instead of by 3, the coefficient of y. The answer y = (12 + 2x)/2 makes both slips: the sign of 2x is wrong and the divisor is wrong.

Correct answer: x = 10 or x = -2

An absolute value equal to 6 means the expression inside is 6 or -6. From x - 4 = 6, x = 10; from x - 4 = -6, x = -2. So x = 10 or x = -2. The pair x = 2 or x = -10 solves |x + 4| = 6, taking the wrong sign on the 4. The pair x = 6 or x = -6 ignores the 4 and solves |x| = 6. The pair x = 6 or x = -2 solves the second case but forgets to add 4 in the first, so it keeps x = 6.

Correct answer: x = -3

Subtract 4 from both sides to get -2x = 6, then divide by -2: x = -3. The value x = -7 adds 4 to the 10 instead of subtracting it, giving -2x = 14. The value x = -12 multiplies 6 by -2 instead of dividing by -2. The value x = -28 makes both slips: it reaches -2x = 14 and then multiplies 14 by -2.

Correct answer: x < 9

Add 1 to both sides to get x/3 < 3, then multiply both sides by 3: x < 9. The answer x < 3 stops at x/3 < 3 and never multiplies by 3. The answer x < 5 multiplies only the left side by 3 (x - 3 < 2). The answer x < 7 multiplies x/3 and the 2 by 3 but not the 1 (x - 1 < 6).

Correct answer: x = 5

Subtract 2x from both sides and add 4 to both sides: 4x = 20, so x = 5. The value x = 4 drops the -4 altogether and solves 4x = 16. The value x = 2.5 adds 2x to the left instead of subtracting it, giving 8x = 20. The value x = 1.5 makes that same slip and also subtracts 4 from 16 instead of adding it, giving 8x = 12.

Correct answer: x = 2 ± sqrt(3)

With a = 1, b = -4 and c = 1, the formula gives x = (4 ± sqrt(16 - 4))/2 = (4 ± sqrt(12))/2. Since sqrt(12) = 2 sqrt(3), dividing both terms by 2 gives x = 2 ± sqrt(3). The answer x = -2 ± sqrt(3) uses b instead of -b, so the 4 becomes -4. The answer x = 2 ± sqrt(6) halves the 12 under the root instead of halving sqrt(12); sqrt(12)/2 is sqrt(3), not sqrt(6). The answer x = -2 ± sqrt(6) makes both slips.

Correct answer: x = 12

Multiply both sides by 3/2, the reciprocal of 2/3. Since 8 times 3/2 is 12, x = 12. The value x = 16/3 multiplies 8 by 2/3 instead of dividing by it. The value x = 22/3 subtracts 2/3 from 8, as if the 2/3 were added to x rather than multiplying it. The value x = 16 multiplies 8 by the 2 and never deals with the 3; the 2 has to divide, not multiply.

Correct answer: h = 3V/B

Multiply both sides by 3 to get 3V = Bh, then divide both sides by B: h = 3V/B. The answer h = V/(3B) divides by 3 instead of multiplying, as if the 1/3 were a 3. The answer h = 3B/V turns the fraction upside down; dividing by B puts B in the denominator. The answer h = B/(3V) makes both slips: it divides by 3 and also turns the fraction upside down.

Correct answer: x ≤ -5

Distribute: -2x - 2 ≥ 8. Add 2 to get -2x ≥ 10, then divide by -2 and reverse the sign: x ≤ -5. The answer x ≥ -5 forgets to reverse the sign when dividing by a negative number. The answer x ≤ -3 writes -2 times 1 as +2, giving -2x ≥ 6. The answer x ≥ -3 makes that same sign slip and also keeps the original direction.

Correct answer: x = 5

Subtract 1.5 from both sides to get 0.5x = 2.5, then divide by 0.5: x = 5. The value x = 2.5 stops at 0.5x = 2.5 and never divides by 0.5. The value x = 6.5 doubles 0.5x and the 4 to clear the decimal but not the 1.5 (x + 1.5 = 8). The value x = 1 doubles the left side but not the 4 (x + 3 = 4).

Correct answer: x = 3 or x = -3

Divide both sides by 2 to get x^2 = 9, then take the square root with both signs: x = 3 or x = -3. The pair x = 9 or x = -9 divides by 2 but never takes the root. The pair x = 6 or x = -6 multiplies by 2 instead of dividing, giving x^2 = 36, and then takes the root. The pair x = 36 or x = -36 makes both slips: it multiplies by 2 and never takes the root.

Correct answer: 12x^5

Multiply the coefficients (3 times 4 = 12) and add the exponents of x (2 + 3 = 5): 12x^5. The answer 12x^6 multiplies the exponents instead of adding them. The answer 7x^6 adds the coefficients and multiplies the exponents, reversing both rules. The answer 12x subtracts the exponents, which is the rule for dividing powers, not multiplying them.

Correct answer: 3x^2 - x - 3

Combine like terms: 2x^2 + x^2 = 3x^2, 3x - 4x = -x and -5 + 2 = -3, giving 3x^2 - x - 3. The answer 3x^2 + x - 3 gets the sign of the x-term wrong; 3x - 4x is -x, not +x. The answer 3x^4 - x - 3 adds the exponents of the x^2 terms, but adding like terms never changes an exponent. The answer 3x^4 + x - 3 makes both slips: it adds the exponents and gets the sign of the x-term wrong.

Correct answer: 3x^2 - 5x + 5

Change the sign of every term in the second polynomial, then combine: 5x^2 - 2x + 1 - 2x^2 - 3x + 4 = 3x^2 - 5x + 5. The answer 3x^2 - 5x - 3 changes the signs of the first two terms but not the -4. The answer 7x^2 - 5x + 5 misses the sign change on the 2x^2 term and adds it instead. The answer 7x^2 - 5x - 3 makes both slips: it adds the 2x^2 and leaves the -4 unchanged.

Correct answer: x^2 - 2x - 15

Multiply each term by each term: x^2 - 5x + 3x - 15 = x^2 - 2x - 15. The answer x^2 + 2x - 15 gets the sign of the middle term wrong; -5x + 3x is -2x. The answer x^2 - 2x - 2 adds 3 and -5 for the constant instead of multiplying them. The answer x^2 + 2x - 2 makes both slips: the middle sign is wrong and the constant is a sum, not a product.

Correct answer: 6x^2 + 5x - 4

Multiply each term by each term: 6x^2 + 8x - 3x - 4 = 6x^2 + 5x - 4. The answer 6x^2 - 5x - 4 gets the sign of the middle term wrong; 8x - 3x is +5x. The answer 6x^2 + 5x + 3 adds -1 and 4 for the constant instead of multiplying them. The answer 6x^2 - 5x + 3 makes both slips: the middle sign is wrong and the constant is a sum, not a product.

Correct answer: (x - 4)(x + 4)

x^2 - 16 is a difference of squares, a^2 - b^2 = (a - b)(a + b) with b = 4, since 4^2 = 16; so it factors as (x - 4)(x + 4). The answer (x - 8)(x + 8) halves 16 instead of taking its square root, and it gives x^2 - 64. The answer (x - 16)(x + 16) uses 16 itself instead of its square root, giving x^2 - 256. The answer (x - 8)(x - 8) halves 16 and also repeats the minus sign, giving x^2 - 16x + 64.

Correct answer: (x + 3)(x + 5)

Find two numbers that multiply to 15 and add to 8: 3 and 5, so the trinomial factors as (x + 3)(x + 5). The answer (x - 3)(x - 5) has the right numbers with the wrong signs; it gives -8x. The answer (x + 2)(x + 4) swaps the roles, using numbers that multiply to 8; it gives x^2 + 6x + 8. The answer (x - 2)(x - 4) makes both slips: it swaps the roles and uses the wrong signs, giving x^2 - 6x + 8.

Correct answer: 3x^2(2x + 3)

The greatest common factor of 6x^3 and 9x^2 is 3x^2. Dividing each term by it gives 6x^3/(3x^2) = 2x and 9x^2/(3x^2) = 3, so the answer is 3x^2(2x + 3). The answer 3x^2(2x + 9) takes out 3x^2 but leaves the 9 undivided; it multiplies out to 6x^3 + 27x^2. The answer 3x^2(3x + 2) puts the two quotients in the wrong places; it multiplies out to 9x^3 + 6x^2. The answer 3x^2(9x + 2) makes both slips: the quotients are swapped and the 9 is left undivided, giving 27x^3 + 6x^2.

Correct answer: 4x^2

Divide the coefficients (8/2 = 4) and subtract the exponents (4 - 2 = 2): 4x^2. The answer 4x^6 adds the exponents, which is the rule for multiplying powers. The answer 16x^6 multiplies the two terms instead of dividing them. The answer 16x^2 handles the exponents correctly but multiplies 8 by 2 instead of dividing.

Correct answer: 2x^2 - 3x

Divide each term by 3x: 6x^3/(3x) = 2x^2 and -9x^2/(3x) = -3x, giving 2x^2 - 3x. The answer 2x^2 - 3 divides the second term by 3x^2 instead of 3x. The answer 2x^4 - 3x divides 6 by 3 but multiplies x^3 by x instead of dividing it. The answer 2x^4 - 3 makes both slips: the first term gains a power of x and the second term loses its x.

Correct answer: 21

Square first, then multiply: 2(4^2) - 3(4) + 1 = 32 - 12 + 1 = 21. The value 53 squares 2x as a unit, (2 times 4)^2 = 64, and then subtracts 12 and adds 1. The value 33 drops the -3x term and adds only 32 + 1. The value 45 adds 3x instead of subtracting it: 32 + 12 + 1.

Correct answer: x^12

A power raised to a power multiplies the exponents: 3 times 4 = 12, so the result is x^12. The answer x^7 adds the exponents, which is the rule for multiplying two powers of x. The answer x^81 raises 3 to the 4th power instead of multiplying 3 by 4. The answer x^64 raises 4 to the 3rd power (4^3 = 64) instead of multiplying 3 by 4.

Correct answer: x^4

When dividing powers of the same base, subtract the exponents: 7 - 3 = 4, so the result is x^4. The answer x^21 multiplies the exponents, which is the rule for a power of a power. The answer x^(7/3) divides the exponents instead of subtracting them. The answer x^10 adds the exponents, which is the rule for multiplying powers, not dividing them.

Correct answer: 8x^6 y^3

Raise every factor inside the parentheses to the 3rd power: 2^3 = 8, (x^2)^3 = x^6 because the exponents multiply (2 times 3), and y^3, giving 8x^6 y^3. The answer 6x^6 y^3 multiplies the coefficient 2 by 3 instead of cubing it. The answer 8x^8 y^3 cubes the exponent 2 (2^3 = 8) instead of multiplying it by 3. The answer 6x^8 y^3 swaps the two rules: it multiplies the coefficient by 3 and cubes the exponent.

Correct answer: x^2 + 8x + 16

(x + 4)^2 means (x + 4)(x + 4) = x^2 + 4x + 4x + 16 = x^2 + 8x + 16. The answer x^2 - 8x + 16 gets the sign of the middle term wrong; it is (x - 4)^2. The answer x^2 + 4x + 16 forgets to double the middle term: 4x + 4x is 8x, not 4x. The answer x^2 - 4x + 16 makes both slips: the middle term is not doubled and its sign is wrong.

Correct answer: (2x + 1)(x + 3)

Check by multiplying: (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. The answer (2x + 3)(x + 1) places the 3 and the 1 the other way round and gives 2x^2 + 5x + 3. The answer (2x + 6)(x + 1) takes the 6 from the split 6x + x as a constant and gives 2x^2 + 8x + 6. The answer (x + 1)(x + 6) factors x^2 + 7x + 6, the trinomial with a times c as its constant, and never divides the 2 back out.

Correct answer: 3x^3 - 6x^2

Multiply 3x^2 by each term inside: 3x^2 times x = 3x^3 and 3x^2 times -2 = -6x^2, giving 3x^3 - 6x^2. The answer 3x^3 + 6x^2 gets the sign wrong; a positive times a negative is negative. The answer 3x^3 - 6x drops a power of x; 3x^2 times -2 is -6x^2, the x^2 is kept, not reduced. The answer 3x^3 + 6x makes both slips: the sign is wrong and the second term loses a power of x.

Correct answer: 12

Substitute and square first: a^2 = (-2)^2 = 4, and 4 times 3 = 12. The value -12 squares the 2 and keeps the minus sign outside, as if a^2 were -(2^2). The value 36 squares b as well as a: (-2)^2 times 3^2 = 36. The value -36 makes both slips: -(2^2) times 3^2 = -36.

Correct answer: 2

Slope is the change in y over the change in x: (8 - 2)/(4 - 1) = 6/3 = 2. 4 subtracts inside each point instead of between the points, (8 - 4)/(2 - 1); 1/2 inverts the ratio, dividing the change in x by the change in y, 3/6; 1/4 makes both slips, subtracting inside each point and inverting the ratio, (2 - 1)/(8 - 4).

Correct answer: -3

In slope-intercept form y = mx + b, the slope is m, the coefficient of x, so the slope is -3. 5 is the y-intercept b, not the slope; 5/3 is the x-intercept (set y = 0: 3x = 5), a point on the axis rather than a slope; -5/3 is that x-intercept with its sign also slipped.

Correct answer: 7

The y-intercept is where x = 0: 2(0) + y = 7, so y = 7. 5 substitutes x = 1 instead of x = 0 (2 + y = 7); 7/2 is the x-intercept (set y = 0: 2x = 7), not the y-intercept; 5/2 makes both slips, subtracting the 2 as if x = 1 and then halving as if solving for the x-intercept.

Correct answer: y = 2x - 3

A slope of 2 and a y-intercept of -3 (the line passes through (0, -3)) give y = 2x - 3. y = 2x + 3 flips the sign of the intercept, so it passes through (0, 3) instead; y = -2x - 3 flips the sign of the slope, so the line falls instead of rising; y = -2x + 3 makes both sign slips at once.

Correct answer: 5

By the distance formula, d = sqrt((3 - 0)^2 + (4 - 0)^2) = sqrt(9 + 16) = sqrt(25) = 5. 7 adds the legs, 3 + 4, without squaring them; 25 is d squared, 9 + 16, before the square root is taken; 14 doubles each leg instead of squaring it, 6 + 8, and takes no root.

Correct answer: (4, 7)

The midpoint averages the matching coordinates: ((2 + 6)/2, (4 + 10)/2) = (4, 7). (8, 14) adds the coordinates but never divides by 2; (2, 3) halves the differences 6 - 2 and 10 - 4 instead of averaging the coordinates; (4, 14) averages the x-values correctly but forgets to halve the sum of the y-values.

Correct answer: x = 6, y = 4

Adding the equations gives 2x = 12, so x = 6; subtracting them gives 2y = 8, so y = 4: x = 6, y = 4. x = 4, y = 6 swaps the two values and fails x - y = 2; x = 12, y = 8 stops at 2x = 12 and 2y = 8 without dividing by 2 and fails x + y = 10; x = 8, y = 12 skips both divisions and also swaps the values.

Correct answer: x = 2, y = 3

Substituting y = x + 1 into 2x + y = 7 gives 3x + 1 = 7, so 3x = 6, x = 2, and y = 2 + 1 = 3: x = 2, y = 3. x = 3, y = 4 subtracts 3 from 6 instead of dividing 6 by 3, then adds 1 correctly; x = 2, y = 5 finds x correctly but doubles it when using y = x + 1, computing 2(2) + 1; x = 3, y = 7 makes both slips.

Correct answer: (3, 5)

Substitute each x-value into y = 2x - 1: at x = 3, y = 2(3) - 1 = 5, so (3, 5) lies on the line. (5, 3) reverses the coordinates, and x = 5 gives y = 9; (4, 9) fits y = 2x + 1, but x = 4 gives y = 7 on this line; (-1, -2) fits y = 2x and ignores the - 1, but x = -1 gives y = -3.

Correct answer: 4

Parallel lines have equal slopes, so the slope is 4, the coefficient of x. -1/4 is the negative reciprocal, which belongs to a perpendicular line; 2 is the y-intercept of the given line; -1/2 is its x-intercept (set y = 0: 4x + 2 = 0), a point on the axis rather than a slope.

Correct answer: -1/2

A perpendicular slope is the negative reciprocal of the given slope 2: flip it and change the sign, giving -1/2. 1/2 flips the number but keeps the positive sign, and a positive slope cannot be perpendicular to another positive slope; -1 takes the negative reciprocal of the intercept 1 instead of the slope 2; 1 makes both slips, using the intercept and keeping the positive sign.

Correct answer: Quadrant II

Quadrant II holds every point with a negative x-coordinate and a positive y-coordinate, so (-3, 5) lies there, up and to the left of the origin. Quadrant III needs both coordinates negative; Quadrant I needs both positive; Quadrant IV pairs a positive x with a negative y.

Correct answer: 4

The x-intercept is where y = 0: 3x - 6(0) = 12, so 3x = 12 and x = 4. -2 sets x = 0 instead and finds the y-intercept (-6y = 12); 6 substitutes y = 1 instead of y = 0 (3x - 6 = 12, so 3x = 18); 9 subtracts 3 from 12 instead of dividing 12 by 3.

Correct answer: y = -x + 5

Point-slope form gives y - 3 = -1(x - 2), so y - 3 = -x + 2 and y = -x + 5. y = -x + 1 multiplies the -1 into x but not into -2 (y - 3 = -x - 2); y = -x - 1 subtracts the 3 instead of adding it; y = -x + 3 takes the point's y-coordinate as the intercept, but the point has x = 2, not x = 0.

Correct answer: $30

25% of $40 is 0.25 times 40 = 10, so the sale price is $40 - $10 = $30. $10 is the amount of the discount, not the price after it; $37.50 misplaces the decimal point, taking 25% of 40 as 2.5 and subtracting $2.50; $39.75 subtracts 25 cents instead of 25 percent.

Correct answer: $16

The unit price is 6 divided by 3 = 2 dollars per pound, so 8 pounds cost 8 times 2 = $16. $11 adds the 5 extra pounds to the $6 as if they were dollars; $24 multiplies the 8 pounds by 3, the number of pounds in the given price, instead of by the $2 unit price; $14 adds the 8 pounds to the $6 price instead of scaling it.

Correct answer: 33

15% of 220 is 0.15 times 220 = 33 (10% is 22 and 5% is 11). 37 finds 10% (22) and then adds 15 instead of 5%; 27 finds 10% (22) and then adds 5 instead of 5%; 30 rounds 220 down to 200 before taking 15%.

Correct answer: 60 mph

Average speed is distance divided by time: 180 miles / 3 hours = 60 mph. 45 mph counts the hours as 4 (hour 0 through hour 3) and divides 180 by 4; 90 mph counts only 2 hours (from hour 1 to hour 3) and divides 180 by 2; 540 mph multiplies the distance by the time (180 times 3) instead of dividing it.

Correct answer: 12

The ratio 3:2 has 3 + 2 = 5 parts, so each part is 30/5 = 6 students and the girls number 2 times 6 = 12. 18 is the number of boys (3 parts of 6); 20 takes 2/3 of 30, treating the part-to-part ratio 2:3 as a fraction of the whole class; 15 splits the class in half and ignores the ratio.

Correct answer: 15

If n + 8 = 23, subtract 8 from both sides: n = 23 - 8 = 15. 31 adds 8 instead of subtracting it; 25 comes from a borrowing slip, taking 3 from 8 in the ones place instead of regrouping; 16 counts back 8 from 23 but counts 23 itself as the first step.

Correct answer: 5 cups

The batch grows by a factor of 30/12 = 2.5, so the flour is 2 times 2.5 = 5 cups. 4.5 cups adds the factor 2.5 to the 2 cups instead of multiplying; 9 cups divides the 18 extra cookies by the 2 cups instead of scaling the recipe; 2.5 cups stops at the scale factor and never multiplies it by 2.

Correct answer: 25%

Percent increase is the change divided by the original price: (100 - 80)/80 = 0.25 = 25%. 20% divides the $20 change by the new price, $100, instead of the original $80; 80% is the old price as a percent of the new one, 80/100, not the increase; 22% divides the $20 change by the average of the two prices, $90 (20/90 is about 22%), instead of by the original price.

Correct answer: 11

Add 5 to both sides: 2n = 22; divide by 2: n = 11. 8.5 halves 17 and ignores the 5 altogether; 3.5 halves 17 first and then subtracts 5 (8.5 - 5); 6 subtracts 5 from 17 instead of adding it (2n = 12).

Correct answer: $630

Weekly pay is the hourly rate times the hours: 18 times 35 = 630, so $630. $590 drops the carried 4 when multiplying 18 by 5, writing 50 instead of 90 for that partial product; $545 multiplies 18 by 30 and then adds 5 instead of 5 times 18; $720 assumes a 40-hour week instead of 35 hours.

Correct answer: 150

If 0.40 times n = 60, then n = 60/0.40 = 150; check: 40% of 150 is 60. 84 increases 60 by 40% instead of finding the whole; 24 takes 40% of 60 instead of finding the number that 60 is 40% of; 120 treats 40% as one half and simply doubles 60.

Correct answer: 60 square centimeters

The area of a rectangle is length times width: 12 times 5 = 60 square centimeters. 34 square centimeters is the perimeter, 2(12 + 5), not the area; 30 square centimeters halves the product, which is the area formula for a triangle; 13 square centimeters is the length of the diagonal, sqrt(12^2 + 5^2), not an area.

Correct answer: 17

Write the integers as n, n + 1 and n + 2: 3n + 3 = 48, so 3n = 45 and n = 15; the integers are 15, 16 and 17, and the largest is 17. 16 is the middle integer (48/3), not the largest; 19 adds the 3 instead of subtracting it (3n = 51, n = 17, largest n + 2 = 19); 14 sets up consecutive even integers, n + (n + 2) + (n + 4) = 48, and reports the smallest of them.

Correct answer: 8

Subtract 3x from both sides: 2x - 7 = 9; add 7: 2x = 16; divide by 2: x = 8. 1 moves the -7 across without changing its sign (2x = 9 - 7); 16 stops at 2x = 16 without dividing by 2; 1/4 adds 3x to 5x instead of subtracting it and also keeps the sign of the 7 (8x = 2).

Correct answer: 13

Distribute: 3x - 12 = 2x + 1; subtract 2x and add 12: x = 13. 5 multiplies only the x by 3 (3x - 4 = 2x + 1); -11 distributes the -4 as +12 (3x + 12 = 2x + 1); 2.6 moves 2x across without changing its sign (5x = 13).

Correct answer: x > 4

Subtract 5: 4x > 16; divide by the positive 4: x > 4, with the inequality sign unchanged. x < 4 reverses the sign, a step needed only when dividing by a negative number; x > 16 never divides by 4; x < 16 makes both slips, skipping the division and reversing the sign.

Correct answer: x ≤ -3

Subtract 8: -2x ≥ 6; dividing by -2 reverses the inequality sign and makes the 6 negative, so x ≤ -3. x ≥ -3 divides by -2 but forgets to reverse the sign; x ≤ 3 reverses the sign but divides 6 by 2 instead of -2, losing the negative; x ≥ 3 makes both slips, solving 2x ≥ 6 as if the coefficient were positive.

Correct answer: 18

Subtract 4: x/3 = 6; multiply by 3: x = 18. 26 multiplies by 3 first but leaves the 4 unmultiplied (x + 4 = 30); 34 multiplies 10 by 3 and then adds the 4 instead of subtracting it; 30 drops the + 4 altogether and solves x/3 = 10.

Correct answer: x = 2 or x = 3

Factor: (x - 2)(x - 3) = 0, so x = 2 or x = 3; check: 2^2 - 5(2) + 6 = 0 and 3^2 - 5(3) + 6 = 0. x = -2 or x = -3 are the roots of (x + 2)(x + 3), which expands to x^2 + 5x + 6; x = 1 or x = 5 swaps the roles of 5 and 6, finding numbers that multiply to 5 and add to 6, and (x - 1)(x - 5) = x^2 - 6x + 5; x = -1 or x = -5 makes both slips, and (x + 1)(x + 5) = x^2 + 6x + 5.

Correct answer: x = ±5

Divide by 2: x^2 = 25; a positive number has two square roots, so x = ±5. x = ±7 takes the square root of 50 first (about 7) and never divides by 2, but 2(49) = 98, not 50; x = ±48 subtracts 2 from 50 instead of dividing by 2 and then skips the square root; x = ±24 subtracts 2 from 50 instead of dividing and then halves 48 instead of taking its square root.

Correct answer: 5

Subtract 2y and add 12: 15 = 3y, so y = 5; check: 2(5) + 3 = 13 and 5(5) - 12 = 13. 4 leaves out the + 3 on the left and solves 3y = 12; 1 leaves out the - 12 on the right and solves 3y = 3; 3 moves the + 3 across without changing its sign (3y = 12 - 3).

Correct answer: -2 < x ≤ 3

Subtract 1 from all three parts: -4 < 2x ≤ 6; divide each part by 2: -2 < x ≤ 3. -2 ≤ x < 3 swaps which end is included, while the original keeps < on the left and ≤ on the right; -4 < x ≤ 6 stops before dividing by 2; -4 ≤ x < 6 makes both slips.

Correct answer: x = 10 or x = -4

An absolute value of 7 means x - 3 = 7 or x - 3 = -7, so x = 10 or x = -4; check: |10 - 3| = 7 and |-4 - 3| = 7. x = 10 or x = 4 slips a sign in the negative case (-7 + 3 is -4, not 4); x = -10 or x = -4 slips a sign in the positive case, moving the 3 and the 7 the wrong way (x = -7 - 3); x = -10 or x = 4 makes both slips, which solves |x + 3| = 7 instead.

Correct answer: 11

Add 1.5: 0.5x = 5.5; divide by 0.5, which is the same as multiplying by 2: x = 11. 5.5 stops before dividing by 0.5; 7 doubles the left side but not the right (x - 3 = 4); 9.5 doubles the 4 but not the 1.5 (x - 1.5 = 8).

Correct answer: 3

Add 2x to both sides to get 7 = 5x - 8, then add 8 to get 15 = 5x, so x = 3. Moving both terms across without changing their signs gives 7 - 8 = 3x - 2x, so -1. Dividing 15 by 3 instead of by the coefficient 5 gives 5. Subtracting 2x on the right instead of adding it leaves 7 = x - 8, so 15.

Correct answer: x = b/a

Add b to both sides to get ax = b, then divide both sides by a (allowed because a is not 0): x = b/a. Moving b across without changing its sign gives ax = -b, so x = -b/a. Dividing the wrong way, a by b, gives x = a/b. Making both slips, the sign and the division, gives x = -a/b.

Correct answer: y = 2

Substitute x = 3 into 2x + 3y = 12: 6 + 3y = 12, so 3y = 6 and y = 2. Subtracting in the wrong order (6 - 12) gives 3y = -6, so y = -2. Dropping the 2x term and solving 3y = 12 gives y = 4. Making both slips gives 3y = -12, so y = -4.

Correct answer: x > 3

Subtract 5 from both sides to get -x < -3. Dividing both sides by -1 reverses the inequality sign, so x > 3. Forgetting to reverse the sign gives x < 3. Adding 5 instead of subtracting it gives -x < 7, which reverses to x > -7. Changing the sign on the x side only, without reversing, gives x < -3.

Correct answer: 12x^3y^4

Multiply the coefficients (3 times 4 = 12) and add the exponents of like bases: x^2 times x = x^3 and y times y^3 = y^4, giving 12x^3y^4. Multiplying the exponents instead of adding them gives 12x^2y^3. Adding the coefficients and also multiplying the y exponents gives 7x^3y^3. Adding the coefficients and also multiplying the x exponents gives 7x^2y^4.

Correct answer: x^2 + 3x - 10

Multiply every pair of terms: x times x = x^2, x times -2 = -2x, 5 times x = 5x, and 5 times -2 = -10. Combining -2x + 5x = 3x gives x^2 + 3x - 10. Swapping the two signs, as if multiplying (x - 5)(x + 2), gives x^2 - 3x - 10. Reading (x - 2) as (x + 2) multiplies (x + 5)(x + 2) and gives x^2 + 7x + 10. Reading (x + 5) as (x - 5) multiplies (x - 5)(x - 2) and gives x^2 - 7x + 10.

Correct answer: 7x - 4

Remove the parentheses and combine like terms: 2x + 5x = 7x and 3 - 7 = -4, giving 7x - 4. Taking 3 - 7 as +4 gives 7x + 4. Treating 2x + 5x as a product of x terms gives 7x^2 - 4, but adding like terms never changes the exponent. Making both slips gives 7x^2 + 4.

Correct answer: 3x^2 - 4x + 8

Change the sign of every term in the second polynomial and add: 4x^2 - x + 6 - x^2 - 3x + 2 = 3x^2 - 4x + 8. Changing the sign of the x^2 term only gives 3x^2 + 2x + 4. Changing the sign of the 3x term only gives 5x^2 - 4x + 4. Changing the sign of the -2 only gives 5x^2 + 2x + 8.

Correct answer: (x - 3)(x + 3)

x^2 - 9 is a difference of squares, a^2 - b^2 = (a - b)(a + b) with a = x and b = 3, so it factors as (x - 3)(x + 3). Using 9 itself instead of its square root gives (x - 9)(x + 9), which multiplies to x^2 - 81. Squaring 9 instead of taking its root gives (x - 81)(x + 81), which multiplies to x^2 - 6561. Writing each factor in the reverse order gives (3 - x)(3 + x), which multiplies to 9 - x^2, the opposite of the expression.

Correct answer: 2x(x + 4)

The greatest common factor of 2x^2 and 8x is 2x; dividing each term by 2x leaves x and 4, so 2x^2 + 8x = 2x(x + 4). The other choices match the 8x term but not the x^2 term, the result of checking only the last term: x(x + 8) multiplies to x^2 + 8x, 4x(x + 2) multiplies to 4x^2 + 8x, and 8x(x + 1) multiplies to 8x^2 + 8x.

Correct answer: 4x^2 - 12x + 9

Write the square as a product: (2x - 3)(2x - 3) = 4x^2 - 6x - 6x + 9 = 4x^2 - 12x + 9. Losing the minus sign squares (2x + 3) instead and gives 4x^2 + 12x + 9. Forgetting to double the middle term (2 times 2x times -3 = -12x) gives 4x^2 - 6x + 9. Making both slips, the lost minus sign and the undoubled middle term, gives 4x^2 + 6x + 9.

Correct answer: (x + 3)(x + 4)

Find two numbers whose product is 12 and whose sum is 7: 3 and 4, so x^2 + 7x + 12 = (x + 3)(x + 4). (x + 2)(x + 6) has the right product, 12, but the pair adds to 8, giving x^2 + 8x + 12. (x + 1)(x + 6) has the right sum, 7, but the pair multiplies to 6, giving x^2 + 7x + 6. (x + 2)(x + 3) multiplies to x^2 + 5x + 6, so both the sum and the product are wrong.

Correct answer: 6x^3 - 12x^2 + 15x

Multiply 3x by each term: 3x times 2x^2 = 6x^3, 3x times -4x = -12x^2, and 3x times 5 = 15x, giving 6x^3 - 12x^2 + 15x. Multiplying every term by 3 and forgetting the x gives 6x^2 - 12x + 15. Losing the negative sign on -4x gives 6x^3 + 12x^2 + 15x. Making both slips gives 6x^2 + 12x + 15.

Correct answer: 3x^2y

Divide the coefficients (12/4 = 3) and subtract the exponents of like bases: x^(4 - 2) = x^2 and y^(2 - 1) = y, giving 3x^2y. Subtracting the coefficients (12 - 4) instead of dividing them gives 8x^2y. Ignoring the unwritten exponent 1 on the y in the denominator leaves y^2 unchanged and gives 3x^2y^2. Making both slips gives 8x^2y^2.

Correct answer: (3x - 1)(x - 2)

Check each choice by multiplying: (3x - 1)(x - 2) = 3x^2 - 6x - x + 2 = 3x^2 - 7x + 2. Swapping the constants gives (3x - 2)(x - 1), which has the right first and last terms but multiplies to 3x^2 - 5x + 2. Flipping the sign in the second factor gives (3x - 1)(x + 2), which multiplies to 3x^2 + 5x - 2. Making both slips gives (3x - 2)(x + 1), which multiplies to 3x^2 + x - 2.

Correct answer: -6x - 10

Distribute both factors: 2x - 6 - 8x - 4 = -6x - 10. Multiplying the +1 by 4 but not carrying the minus sign to it gives 2x - 6 - 8x + 4 = -6x - 2. Losing the minus sign on the 8x term gives 2x - 6 + 8x - 4 = 10x - 10. Making both slips, which is the same as adding 4(2x + 1) instead of subtracting it, gives 2x - 6 + 8x + 4 = 10x - 2.

Correct answer: x^3 + 2x^2 - 7x + 4

Multiply each term of x + 4 by each term of the trinomial: x^3 - 2x^2 + x + 4x^2 - 8x + 4, which combines to x^3 + 2x^2 - 7x + 4. Multiplying 4 by the last term only gives x^3 - 2x^2 + x + 4. Multiplying x by x^2 only, then 4 by every term, gives x^3 + 4x^2 - 8x + 4. Dropping the product x times 1 gives x^3 + 2x^2 - 8x + 4.

Correct answer: 2

Slope is the change in y over the change in x: (11 - 3)/(6 - 2) = 8/4 = 2. Subtracting the y-values in one order and the x-values in the other gives 8/(2 - 6) = -2. Stopping at the change in y without dividing by the change in x gives 8. Making both slips, subtracting 3 - 11 and not dividing, gives -8.

Correct answer: 7

In y = mx + b form, b is the y-intercept, the value of y when x = 0: y = -3(0) + 7 = 7. Reading the size of the slope, 3, as the intercept gives 3, but the number multiplying x is the slope. Setting y = 0 finds the x-intercept instead: 3x = 7, so 7/3. Making that switch and also dropping the sign when solving -3x + 7 = 0 gives -7/3.

Correct answer: (2, -2)

Average each pair of coordinates: ((-2 + 6)/2, (4 + (-8))/2) = (2, -2). Adding the coordinates without dividing by 2 gives (4, -4). Halving the differences instead of the sums, (6 - (-2))/2 and (-8 - 4)/2, gives (4, -6). Halving the differences taken in the other order gives (-4, 6).

Correct answer: 5

Use the distance formula: sqrt((4 - 1)^2 + (6 - 2)^2) = sqrt(9 + 16) = sqrt(25) = 5. Adding the two changes, 3 + 4, without squaring gives 7, the path along the grid lines rather than the straight-line distance. Squaring only the first difference gives sqrt(9 + 4) = sqrt(13). Adding the coordinates instead of subtracting them gives sqrt(5^2 + 8^2) = sqrt(89).

Correct answer: y = 2x + 1

Parallel lines have equal slopes and different intercepts. The given line has slope 2, and y = 2x + 1 also has slope 2, so the two never meet. y = x/2 + 1 uses the reciprocal slope, 1/2, so it crosses the given line. y = 3x - 5 shares the intercept -5, but sharing an intercept does not make lines parallel; its slope is 3, so it crosses the given line at (0, -5). y = x/2 - 5 also shares the intercept and has slope 1/2, so it meets the given line at (0, -5) too.

Correct answer: -4/3

Perpendicular slopes are negative reciprocals: flip 3/4 to 4/3 and change the sign, giving -4/3. Flipping without changing the sign gives 4/3. Changing the sign without flipping gives -3/4. Keeping 3/4 unchanged gives the slope of a parallel line, not a perpendicular one.

Correct answer: (5, 0)

The x-intercept is where y = 0: 2x + 5(0) = 10, so 2x = 10, x = 5, and the point is (5, 0). Setting x = 0 instead finds the y-intercept (5y = 10), which is (0, 2). Subtracting 2 instead of dividing by 2 gives x = 8, so (8, 0). Putting x = 5 in the y-coordinate gives (0, 5), a point on the y-axis.

Correct answer: y = 3x - 4

A line through (0, -4) crosses the y-axis at -4, so b = -4; with m = 3, y = mx + b becomes y = 3x - 4. Flipping the sign of the intercept gives y = 3x + 4, a line that crosses the y-axis at 4. Flipping the sign of the slope gives y = -3x - 4, a line that falls instead of rising. Making both sign slips gives y = -3x + 4.

Correct answer: Third quadrant

Both coordinates of (-5, -2) are negative, and the quadrant where x < 0 and y < 0 is the third quadrant. The first quadrant needs both coordinates positive. The second quadrant has x negative but y positive. The fourth quadrant has x positive but y negative.

Correct answer: -1

Use point-slope form: y - 5 = -2(x - 1), which simplifies to y = -2x + 7. At x = 4, y = -8 + 7 = -1. Stopping at the change in y, -2(4 - 1), gives -6. Solving y - 5 = -6 by moving the 5 with the wrong sign gives -11. Finding the intercept with a sign slip (b = -7 instead of 7) gives y = -2x - 7, and at x = 4 that is -15.

Correct answer: 420 miles

The rate is 240/4 = 60 miles per hour, so in 7 hours the car travels 60 times 7 = 420 miles. Doubling 240, as if 7 hours were 8, gives 480 miles. Estimating 7 hours as about 1 2/3 times the 4-hour trip (240 times 5/3) gives 400 miles, but 7/4 is 1 3/4, not 1 2/3. Adding the first 240 miles on top of the 7-hour distance, as if the car drove 7 more hours, gives 660 miles.

Correct answer: 25

Let the integers be n, n + 1 and n + 2. Then 3n + 3 = 72, so 3n = 69, n = 23, and the largest is 25. Dividing 72 by 3 gives 24, which is the middle integer, not the largest. Adding 3 instead of subtracting it gives 3n = 75, n = 25, and a largest of 27. Setting up consecutive even integers (n, n + 2, n + 4) gives 3n + 6 = 72, and stopping at n gives 22.

Correct answer: $12.00

The unit price is $4.50/3 = $1.50 per pound, so 8 pounds cost 8 times $1.50 = $12.00. Treating $4.50 as the price per pound gives 8 times $4.50 = $36.00. Multiplying $4.50 by 3 prices 9 pounds and gives $13.50. Adding the 3 pounds to the 8 and pricing 11 pounds gives $16.50.

Correct answer: 84 cm^2

The area of a rectangle is length times width: 12 times 7 = 84 cm^2. Adding all four sides, 2(12 + 7) = 38, finds the perimeter, a length rather than an area, so 38 cm^2 is wrong. Squaring the width (7 times 7) gives 49 cm^2. Adding the length and the width (12 + 7) gives 19 cm^2.

Correct answer: 90

The average is the sum divided by the number of tests: (85 + 90 + 95)/3 = 270/3 = 90. Averaging only the first two scores gives (85 + 90)/2 = 87.5. Averaging only the last two scores gives (90 + 95)/2 = 92.5. Giving the highest score, 95, ignores the other two tests.

Correct answer: 5 cups

Each cup of flour makes 12/2 = 6 cookies, so 30 cookies need 30/6 = 5 cups. Finding flour for only the 18 extra cookies gives 18/6 = 3 cups. Dividing 30 by 12 and forgetting to multiply by the 2 cups gives 2.5 cups. Adding that factor of 2.5 to the 2 cups instead of multiplying gives 4.5 cups.

Correct answer: 9

Write n + 18 = 3n. Subtract n from both sides to get 18 = 2n, so n = 9. Dropping the n on the left and solving 18 = 3n gives 6. Reporting three times the number instead of the number itself gives 27. Dropping the n on the left and then multiplying 18 by 3 instead of dividing gives 54.

Correct answer: $30

Simple interest is principal times rate times time: $200 times 0.05 times 3 = $30. One year of interest alone is $10. Adding the interest to the principal gives the ending balance, $230, not the interest earned. Adding just one year of interest to the principal gives $210, which is neither the interest nor the 3-year balance.

Correct answer: 225 miles

From 9:00 AM to 1:30 PM is 4 hours 30 minutes, or 4.5 hours, so the train travels 50 times 4.5 = 225 miles. Counting 9, 10, 11, 12 and 1 as five hours gives 5.5 hours and 275 miles. Subtracting 1.5 from 9 instead of finding the time from 9:00 to 1:30 gives 7.5 hours and 375 miles. Rounding 4 hours 30 minutes up to 5 hours gives 250 miles.

Correct answer: x = 6

The correct answer is x = 6. Subtract 4x from both sides to get 3x + 2 = 20, subtract 2 to get 3x = 18, and divide by 3. "x = 2" comes from adding the x terms instead of subtracting them and adding the constants as well, which gives 11x = 22. "x = 22/3" comes from adding 2 to 20 instead of subtracting it, which gives 3x = 22. "x = 20/3" comes from dropping the + 2 altogether, which gives 3x = 20.

Correct answer: x < 5

The correct answer is x < 5. Add 8 to both sides to get 3x < 15, then divide by 3; dividing by a positive number keeps the inequality sign as it is. "x > 5" reverses the sign, which is needed only when you multiply or divide by a negative number. "x < -1/3" comes from subtracting 8 instead of adding it, which gives 3x < -1. "x > -1/3" makes that same subtraction error and also reverses the sign.

Correct answer: x < -5

The correct answer is x < -5. Divide both sides by -4 and reverse the inequality sign, because dividing by a negative number flips the inequality: 20 divided by -4 is -5. "x > -5" divides correctly but forgets to reverse the sign. "x > -80" multiplies 20 by -4 instead of dividing and keeps the sign. "x < -80" reverses the sign but still multiplies by -4 instead of dividing.

Correct answer: x = 7

The correct answer is x = 7. Distribute to get 6x - 2 = 4x + 12, subtract 4x to get 2x - 2 = 12, add 2 to get 2x = 14, and divide by 2. "x = 5" comes from moving the -2 across without changing its sign, which gives 2x = 10. "x = 6" comes from losing the -2 while distributing, which gives 6x = 4x + 12. "x = 4" comes from halving both sides but turning 2(x + 3) into 2x + 3 instead of 2x + 6.

Correct answer: x = 2 or x = 5

The correct answer is x = 2 or x = 5. Two numbers that multiply to 10 and add to -7 are -2 and -5, so x^2 - 7x + 10 = (x - 2)(x - 5) = 0, and setting each factor equal to zero gives 2 and 5. The quadratic formula agrees: x = (7 ± 3)/2. "x = -2 or x = -5" gives -2 and -5, the numbers used in the factors, not the roots; it is also what you get by starting the formula from -7 instead of 7. "x = 4 or x = 10" uses the formula correctly but leaves out the division by 2, stopping at 7 ± 3. "x = -4 or x = -10" makes both slips: it starts from -7 instead of 7 and also skips the division by 2.

Correct answer: x = ±5

The correct answer is x = ±5, that is, x = 5 or x = -5. Add 25 to both sides to get x^2 = 25; both 5 and -5 square to 25, which the difference of squares (x - 5)(x + 5) = 0 also shows. "x = 5" forgets the negative square root. "x = ±25" stops at x^2 = 25 and never takes the square root. "x = 25" makes both slips: it skips the square root and also gives only the positive answer.

Correct answer: x = -5 or x = 2

The correct answer is x = -5 or x = 2. Look for two numbers that multiply to -10 and add to 3: 5 and -2. So the equation factors as (x + 5)(x - 2) = 0; x + 5 = 0 gives -5, and x - 2 = 0 gives 2. "x = -2 or x = 5" factors correctly but flips the signs when setting each factor equal to zero. "x = 4 or x = -10" comes from the quadratic formula, (-3 ± 7)/2, without the division by 2. "x = 1 or x = -10" uses the factor pair 10 and -1, whose product is -10 but whose sum is 9, not 3.

Correct answer: b = P/2 - L

The correct answer is b = P/2 - L. Subtract 2L from both sides to get P - 2L = 2b, then divide every term by 2: P/2 - 2L/2 = P/2 - L, which is the same as (P - 2L)/2. "b = P/2 + L" adds 2L instead of subtracting it when moving it across the equals sign, then divides correctly. "b = P/2 - 2L" divides only P by 2 and leaves 2L undivided. "b = P/2 + 2L" makes both slips: it adds 2L instead of subtracting it and divides only P by 2.

Correct answer: x = 5 or x = -4

The correct answer is x = 5 or x = -4. An absolute value equal to 9 means the expression inside is 9 or -9: 2x - 1 = 9 gives 2x = 10 and x = 5, and 2x - 1 = -9 gives 2x = -8 and x = -4. "x = 4 or x = -5" subtracts 1 in each case instead of adding it, as if the equation were |2x + 1| = 9. "x = 5 or x = -5" solves the positive case and then simply attaches a minus sign to that answer instead of solving 2x - 1 = -9. "x = 4 or x = -4" makes both slips: it subtracts the 1 in the positive case instead of adding it and then just negates that answer.

Correct answer: x ≥ 6

The correct answer is x ≥ 6. Multiply both sides by 4 to get x + 6 ≥ 12, then subtract 6; multiplying by a positive 4 does not change the direction of the sign. "x ≤ 6" reverses the sign, which is needed only when multiplying or dividing by a negative number. "x ≥ 2" reaches x + 6 ≥ 12 but then divides 12 by 6 instead of subtracting 6. "x ≤ 2" makes that same division error and also reverses the sign.

Correct answer: x = -4

The correct answer is x = -4. Subtract 5x from both sides to get 3x + 5 = -7, subtract 5 to get 3x = -12, and divide by 3. "x = 4" reaches 3x = -12 but drops the negative sign. "x = -36" reaches 3x = -12 but multiplies by 3 instead of dividing by 3. "x = 36" makes both slips: it multiplies by 3 instead of dividing and also drops the negative sign.

Correct answer: x = -3 ± √5

The correct answer is x = -3 ± √5. With a = 1, b = 6 and c = 4, the discriminant is 36 - 16 = 20, so x = (-6 ± √20)/2; since √20 = 2√5, this simplifies to -3 ± √5. "x = -6 ± √5" divides the root by 2 but forgets to divide the -6 as well. "x = -3 ± √13" adds 16 to 36 instead of subtracting it, so the root becomes √52 divided by 2, which is √13. "x = -6 ± √13" makes both slips: it uses 36 + 16 under the root and leaves the -6 undivided.

Correct answer: x = 20

The correct answer is x = 20. Multiply both sides by the reciprocal 5/3: 12 × 5/3 = 60/3 = 20. "x = 7.2" multiplies 12 by 3/5 itself instead of by its reciprocal 5/3. "x = 60" multiplies by 5 but never divides by 3. "x = 2.4" divides 12 by 5 and ignores the 3 altogether.

Correct answer: r = d ÷ t

The correct answer is r = d ÷ t. In d = rt, r is multiplied by t, so divide both sides by t to undo the multiplication and isolate r. "r = d × t" multiplies d by t instead of dividing by it. "r = d + t" undoes the multiplication with addition, as if the formula were d = r - t. "r = d - t" undoes it with subtraction, as if the formula were d = r + t.

Correct answer: x > -1

The correct answer is x > -1. Distribute to get -3x + 6 < 9, subtract 6 to get -3x < 3, then divide by -3 and reverse the sign, because dividing by a negative number flips the inequality. "x < -1" divides by -3 but forgets to reverse the sign. "x > -5" distributes -3 times -2 as -6 instead of +6, which gives -3x < 15. "x < -5" makes that same distribution error and also forgets to reverse the sign.

Correct answer: x = 14

Subtract 2 from both sides to get 0.25x = 3.5, then divide by 0.25 (the same as multiplying by 4) to get x = 14. x = 20 divides 5.5 by 0.25 first and only then subtracts the 2, but the 2 must come off before dividing; x = 22 divides 5.5 by 0.25 without taking away the 2 at all; x = 30 adds 2 to 5.5 instead of subtracting it.

Correct answer: x = ±4

Divide both sides by 3 to get x^2 = 16, then take the square root, which gives two solutions: x = ±4. x = ±8 halves 16 instead of taking its square root; x = ±12 multiplies 48 by 3 instead of dividing, so it solves x^2 = 144; x = ±144 multiplies by 3 and then never takes the square root at all.

Correct answer: 10x^7

Multiply the coefficients (5 times 2 = 10) and add the exponents (3 + 4 = 7) to get 10x^7. 10x^12 multiplies the exponents instead of adding them; 7x^12 adds the coefficients and multiplies the exponents, reversing both rules; 10x^-1 subtracts the exponents (3 - 4), which is the rule for dividing powers, not multiplying them.

Correct answer: 6x^2 + 4x + 3

Combine like terms: 4x^2 + 2x^2 = 6x^2, -x + 5x = 4x, and 6 - 3 = 3, giving 6x^2 + 4x + 3. 6x^4 + 4x + 3 adds the exponents, but adding like terms never changes the exponent; 6x^2 + 6x + 3 reads -x as +x, so -x + 5x becomes 6x instead of 4x; 6x^4 + 6x + 3 makes both slips, adding the exponents and reading -x as +x.

Correct answer: 4x^2 + 6x - 6

Distribute the minus sign to every term of the second polynomial: 7x^2 + 2x - 1 - 3x^2 + 4x - 5. Then 7x^2 - 3x^2 = 4x^2, 2x + 4x = 6x, and -1 - 5 = -6, giving 4x^2 + 6x - 6. 4x^2 - 2x + 4 applies the minus sign only to the first term; 4x^2 + 6x + 4 changes the sign of the x term but not of the 5; 4x^2 + 2x - 6 leaves out the 4x term of the second polynomial.

Correct answer: x^2 - 4x - 12

FOIL gives x^2 + 2x - 6x - 12, and 2x - 6x = -4x, so the product is x^2 - 4x - 12. x^2 + 4x - 12 flips both signs, multiplying (x + 6)(x - 2); x^2 - 8x + 12 misreads the +2 as -2 and multiplies (x - 6)(x - 2); x^2 + 8x + 12 misreads the -6 as +6 and multiplies (x + 6)(x + 2).

Correct answer: 6x^2 - 11x - 10

FOIL gives 6x^2 - 15x + 4x - 10, and -15x + 4x = -11x, so the product is 6x^2 - 11x - 10. 6x^2 - 19x - 10 combines -15x and 4x as if both were negative; 6x^2 - 15x - 10 leaves out the inner product 4x; 6x^2 - 19x - 3 makes the same sign error and also adds 2 and -5 for the constant instead of multiplying them.

Correct answer: (x - 7)(x + 7)

x^2 - 49 is a difference of squares, x^2 - 7^2, so it factors as (x - 7)(x + 7): the middle terms -7x and +7x cancel. (x - 49)(x + 49) uses 49 itself instead of its square root and multiplies out to x^2 - 2401; (7 - x)(7 + x) puts the terms in the wrong order and multiplies out to 49 - x^2, the negative of the expression; (x - 1)(x + 49) uses the factor pair 1 and 49, which leaves a middle term and gives x^2 + 48x - 49.

Correct answer: (x - 5)(x + 3)

You need two numbers that multiply to -15 and add to -2: -5 and 3, so the factors are (x - 5)(x + 3). (x + 5)(x - 3) has the signs reversed and gives a middle term of +2x; (x - 15)(x + 1) uses the pair -15 and 1, whose sum is -14; (x + 15)(x - 1) uses the same wrong pair with the signs reversed as well, so its sum is +14.

Correct answer: 4x^2(2x^2 - 3)

The greatest common factor of 8x^4 and 12x^2 is 4x^2 (the largest number dividing 8 and 12, times the lowest power of x), which leaves 4x^2(2x^2 - 3). 2x^2(4x^2 - 6) takes out only 2x^2 and leaves a common factor of 2 inside, so it is not the greatest common factor; 4x^2(2x^2 + 3) loses the minus sign and multiplies back to 8x^4 + 12x^2; 2x^2(4x^2 + 6) makes both slips, taking out too small a factor and losing the minus sign.

Correct answer: 3x^3

Divide the coefficients (15 / 5 = 3) and subtract the exponents (5 - 2 = 3) to get 3x^3. 3x^7 adds the exponents, which is the rule for multiplying powers; 10x^3 subtracts the coefficients (15 - 5) instead of dividing them; 10x^7 makes both of those mistakes, subtracting the coefficients and adding the exponents.

Correct answer: 3x^2 - 2x

Divide each term by 4x^2: 12x^4 / 4x^2 = 3x^2 and -8x^3 / 4x^2 = -2x, giving 3x^2 - 2x. 3x^2 + 2x drops the minus sign when dividing the second term; 3x^2 - 8x divides the power of x in the second term but never divides 8 by 4; 3x^2 + 8x makes both slips, leaving 8 undivided and dropping the minus sign.

Correct answer: 21

Substitute x = -2: 3(4) - 2(-2) + 5 = 12 + 4 + 5 = 21. 13 treats -2(-2) as -4 instead of +4; -3 squares -2 as -4 instead of 4, giving -12 + 4 + 5; 45 squares 3x as a whole, (3 times -2)^2 = 36, instead of squaring only the x.

Correct answer: 9x^8y^4

Raise every factor to the power 2: 3^2 = 9, (x^4)^2 = x^8 and (y^2)^2 = y^4, giving 9x^8y^4. 6x^8y^4 multiplies the 3 by 2 instead of squaring it; 9x^16y^4 squares the exponent 4 instead of multiplying it by 2; 6x^16y^4 makes both slips, doubling the 3 and squaring the exponent 4.

Correct answer: x^2 - 10x + 25

(x - 5)^2 means (x - 5)(x - 5) = x^2 - 5x - 5x + 25, which is x^2 - 10x + 25. x^2 + 10x + 25 gives the middle term the wrong sign, since 2(x)(-5) = -10x; x^2 - 5x + 25 counts only one of the two -5x products instead of doubling it; x^2 + 5x + 25 makes both slips, counting the middle product once and giving it the wrong sign.

Correct answer: -3

Slope is the change in y over the change in x: (-8 - 4) / (3 - (-1)) = -12 / 4 = -3. -1/3 divides the change in x by the change in y (4 / -12) instead of the change in y by the change in x; -1 slips on the sign in the rise, working -8 - 4 as -4 instead of -12, so it divides -4 by 4; 1 reads the second point as (3, 8), giving (8 - 4) / 4.

Correct answer: 10

Substitute x = 3: 2(9) - 3(3) + 1 = 18 - 9 + 1 = 10. 16 forgets to multiply the 3 by x, giving 18 - 3 + 1; 28 squares 2x as a whole, (2 times 3)^2 = 36, then works out 36 - 9 + 1; 46 squares 2x the same way and also adds the 9 instead of subtracting it.

Correct answer: x = 2, y = 3

Subtracting the second equation from the first gives 3y - (-y) = 4y and 13 - 1 = 12, so y = 3; then 2x - 3 = 1 gives 2x = 4, so the solution is x = 2, y = 3. x = -1, y = 3 moves the 3 to the wrong side, writing 2x = 1 - 3; x = 5, y = 3 substitutes y = 3 into the first equation as 2x + 3 = 13, forgetting to multiply 3 by 3, so 2x = 10; x = 11, y = 3 substitutes into the first equation but adds 9 to 13 instead of subtracting it.

Correct answer: y = -4

At the y-intercept x = 0, so -2y = 8 and y = -4. y = 4 divides 8 by 2 instead of -2 and loses the sign; y = 2 is where the line crosses the x-axis (set y = 0 to get 4x = 8), not the y-axis; y = -2 reads the coefficient of y as the intercept.

Correct answer: x + 3

Factor the numerator as (x - 3)(x + 3), then cancel the common factor, leaving x + 3. x - 3 keeps the factor that cancels instead of the one that remains; x - 9 divides only x^2 by x and leaves the -9 untouched, but single terms cannot be cancelled out of a sum; x + 9 makes the same piecewise division and also flips the sign of the 9.

Correct answer: y = -2x + 5

Slope-intercept form is y = mx + b. The slope gives m = -2, and the point (0, 5) lies on the y-axis, so b = 5 and the line is y = -2x + 5. y = 2x + 5 drops the minus sign on the slope; y = -2x - 5 gives the intercept the wrong sign, as if the line crossed the y-axis at (0, -5); y = 2x - 5 makes both slips, dropping the minus sign on the slope and flipping the sign of the intercept.

Correct answer: x = 4

Distribute to get 4x + 4 - 2 = 2x + 10, so 4x + 2 = 2x + 10, then 2x = 8 and x = 4. x = 3 distributes neither side (4x + 1 - 2 = 2x + 5); x = 1.5 distributes the 4 on the left but not the 2 on the right, reaching 2x = 3; x = 5.5 distributes the 2 on the right but not the 4 on the left, reaching 2x = 11.

Correct answer: 5/x^2

The answer is 5/x^2. A negative exponent moves only its own base to the denominator: x^(-2) becomes 1/x^2, while the coefficient 5 stays in the numerator, giving 5/x^2. The choice 1/(5x^2) moves the coefficient 5 into the denominator too, but the exponent -2 applies only to x. The choice -5/(x^2) moves x correctly but makes the coefficient negative; a negative exponent never changes the sign of the expression. The choice -5x^2 makes two slips: it treats the negative exponent as a negative sign on the whole expression, and it leaves x in the numerator.

Correct answer: 20%

The answer is 20%. The decrease is 600 - 480 = 120, and percent decrease is the change divided by the original price: 120/600 = 0.20, or 20%. The choice 25% divides the change by the new price instead: 120/480 = 0.25. The choice 0.20% stops at the decimal 0.20 and attaches a percent sign without multiplying by 100. The choice 0.25% makes both mistakes: it divides by the new price and skips multiplying by 100.

Correct answer: x = 5, y = 10

The answer is x = 5, y = 10. Substitute y = 2x into 3x + y = 25 to get 3x + 2x = 5x = 25, so x = 5; then y = 2(5) = 10. Check: 3(5) + 10 = 25. The choice x = 4, y = 8 fits y = 2x but fails the second equation: 3(4) + 8 = 20, not 25. The choice x = 5, y = 15 multiplies by 3, borrowing the coefficient from the other equation. The choice x = 5, y = 20 drops the 3 when substituting back, writing 5 + y = 25.

Correct answer: 36pi cm^2

The answer is 36pi cm^2. The area of a circle is A = pi r^2; with r = 6, A = pi(6^2) = 36pi cm^2. The choice 144pi cm^2 uses the diameter, 12, in place of the radius: pi(12^2) = 144pi. The choice 12pi cm^2 uses the circumference formula, 2 pi r, or squares 6 as 6 x 2. The choice 24pi cm^2 uses the circumference formula with the diameter as the radius, 2 pi(12).

Correct answer: Refrigerated rail cars made American food habits more alike nationwide

The passage builds to its last sentence: regional diets that once depended on local farms "began to look much the same from coast to coast." That nationwide change is the main idea: Refrigerated rail cars made American food habits more alike nationwide. Refrigerated rail cars made Americans rely more heavily on local farms reverses the passage, which says diets depended on local farms before the rail cars, not after. Refrigerated rail cars let Chicago beef reach distant cities still fresh is a true supporting detail, but it is one example and too narrow to be the main idea. Refrigerated rail cars made grocery bills more costly for most Americans is never stated; the passage says nothing about prices.

Correct answer: Honeybees matter chiefly because they pollinate growing farm crops

The paragraph opens by saying honeybees "do far more than make honey," explains that the pollen they carry lets plants produce fruit and seeds, and ends with farmers renting hives "simply to ensure their crops will grow." So the main idea is: Honeybees matter chiefly because they pollinate growing farm crops. Honeybees matter chiefly because they make honey from blooming crops contradicts the opening sentence, which sets honey aside as the lesser role. Honeybees matter chiefly because they make the seeds crops grow from misreads the passage: the bees carry pollen, and the plants themselves produce the seeds. Honeybees matter chiefly because farmers rent their hives annually mistakes a supporting detail for the main idea; farmers rent hives to get pollination, which is the real reason bees matter.

Correct answer: Yawning may help cool the brain, not just signal boredom

Yawning may help cool the brain, not just signal boredom is the central point: the passage sets the common belief that yawning is “simply a sign of boredom” against studies suggesting that yawning “may help cool the brain,” and it concludes that the evidence “points to a physical purpose rather than a purely emotional one.” Yawning may only signal boredom, not help cool the brain restates the popular belief the passage argues against. In cool rooms, people may stop yawning entirely stretches one finding: people yawn “more often” in warm rooms than in cool ones, which means fewer yawns in cool rooms, not none. Yawning may help ease mild fear, not just signal boredom swaps in an emotional purpose the passage never mentions; its evidence points to a physical one instead.

Correct answer: Volunteering tends to help the volunteers and the recipients equally

Correct answer: Volunteering tends to help the volunteers and the recipients equally. Explanation: This is the closest summary of the whole passage. The passage opens by saying the benefits "reach beyond the people being helped" and closes by saying that giving time "often rewards the giver as well." The other choices fail: one turns the minor detail about new friendships and jobs into the main point, one overstates "report lower stress" as reliably eliminating stress, and one reverses the passage's point that the volunteer benefits too.

Correct answer: The woman was thoroughly prepared before her interview

The woman was thoroughly prepared before her interview is the implied main idea: every detail points to readiness, from arriving “an hour early” and reviewing her notes three times to researching the company and rehearsing “answers to likely questions.” The woman was privately terrified before her interview reads anxiety into the extra review, but she meets the interviewer “with a steady smile.” The woman was excessively relaxed before her interview ignores how much effort she put in the night before and on the train. The woman was quietly indifferent before her interview cannot be squared with someone who researched the company and rehearsed her answers.

Correct answer: Coral Reefs: Small Habitats of Outsized Importance

Coral Reefs: Small Habitats of Outsized Importance fits the whole passage, which says reefs “cover less than one percent of the ocean floor” yet “support roughly a quarter of all marine species,” and concludes that protecting them “protects a vast web of ocean life.” Coral Reefs: Overlooked Casualties of Warmer Water is too narrow; damage from warming water appears in only one sentence, as a reason the dependent animals suffer. Coral Reefs: Brilliant Gardens of Tropical Lagoons describes beauty and setting, which the passage never discusses. Coral Reefs: Crowded Nurseries of Juvenile Turtles names an animal and a role the passage never mentions; it speaks of food and shelter for fish, crabs, and smaller creatures.

Correct answer: Adult language learners can overcome the added hurdles

Adult language learners can overcome the added hurdles expresses the main idea: learning a language as an adult is “harder” but “far from impossible,” and “difficulty, in other words, is not the same as defeat.” Adult language learners can rarely master the material contradicts the statement that “many adults reach a high level of fluency.” Adult language learners can skip the constant practice ignores the condition the passage sets, since success comes “with regular practice and exposure.” Adult language learners can acquire the grammar easily contradicts the opening claim that learning a language as an adult is harder than learning one as a child.

Correct answer: Public libraries now double as broad neighborhood hubs

Public libraries now double as broad neighborhood hubs is the main idea: after listing tools, instruments, workshops, tax help, and after-school programs, the passage concludes that “the modern library has quietly become a center for the whole community.” Public libraries now operate as equipment rental shops narrows the point to one example, the lending of tools and telescopes. Public libraries now serve as student homework centers narrows it to the after-school programs, only one service among many. Public libraries now treat books as secondary concerns goes one step past the text, which says libraries offer “much more than books,” not that books matter less.

Correct answer: Sedentary people should scale up workouts progressively

Sedentary people should scale up workouts progressively states the main idea: a “sudden burst of exercise” can harm someone inactive, so doctors advise “starting slowly,” and “building up gradually allows the body to adapt safely.” Sedentary people should emphasize short vigorous bursts recommends exactly what the passage warns against. Sedentary people should shun hard exercise indefinitely overgeneralizes the warning; the passage expects activity to build up over time. Sedentary people should choose gentle walks exclusively turns one suggested starting point, “short walks or gentle stretches,” into a permanent limit.

Correct answer: Beacon automation made keepers redundant

Beacon automation made keepers redundant gives the stated reason: in 1965 “an automatic light was installed that required no human attention,” and with it in place “the keepers were no longer needed.” Lighthouse shutdown left keepers jobless misreads the change; the light kept working automatically, and the passage never says the lighthouse closed. Mainland employment tempted keepers away mistakes where the keepers went, “back to the mainland,” for why they went. Budget shortfalls forced keepers onshore names a cause the passage never mentions.

Correct answer: They have figured out how to open jars to get food

They have figured out how to open jars to get food restates the passage's evidence that octopuses “have learned to open jars to reach food inside,” which shows problem solving. They have been kept in laboratory tanks to be studied is true, since the passage places them “in laboratory tanks,” but where an animal is kept says nothing about how smart it is. They have made their homes in the ocean is also true, but it describes where octopuses live, not anything they work out. They have been watched closely by researchers is true as well, but it describes who studies them, not a problem the octopuses solved.

Correct answer: It is lightweight yet sturdy and bendable

It is lightweight yet sturdy and bendable matches the passage: “despite its light weight, it is remarkably strong and flexible,” and the next sentence says builders use it for scaffolding and homes. It is heavyweight yet elastic and durable contradicts the passage's “light weight.” It is immense yet slender and sustainable mixes in claims about size the passage never makes, and its ecological point, that bamboo “regrows so quickly,” is offered as an extra benefit, not as what suits it to building. It is decorative yet waterproof and sleek lists qualities the passage never mentions.

Correct answer: The market follows a seasonal calendar

The market follows a seasonal calendar is supported by the first sentence: the market opens “every Saturday from May through October,” so it runs only part of the year. The market adopts a weeknight schedule contradicts that sentence, which names Saturday as its day. The market attracts a tourist majority has no support; the only shoppers the passage describes are local families who make the visit “a weekend tradition.” The market employs a professional band goes beyond the text, which says only that “local musicians often perform near the entrance.”

Correct answer: Winds peeled the roofs off several houses

Winds peeled the roofs off several houses is the best evidence of severity, since the passage says “winds tore the roofs off several houses,” damage only a powerful storm causes. Storm landfall began just past midnight gives the timing, “just after midnight,” which says nothing about strength. Storm winds first struck along the coast gives the location, “the coast,” which also says nothing about strength. Trees had stood along the town's Main Street describes the street before the storm; the trees being “uprooted” would show severity, but their having stood along the street does not.

Correct answer: The brain files away recent information during sleep

The brain files away recent information during sleep restates the passage: while we sleep, the brain “sorts through the day's experiences and stores new memories.” The brain erases older painful memories during sleep turns sorting into deleting, which the passage never says. The brain repairs damaged muscle tissue during sleep reassigns a detail; the passage says “the body also repairs tissue,” not the brain. The brain fights off serious infections during sleep both reassigns and overstates the claim that deep sleep “strengthens the immune system”; strengthening the body's defenses is not curing an infection.

Correct answer: Daytime jobs kept employed residents away

Daytime jobs kept employed residents away gives the reason the passage states: “surveys showed that many working adults wanted to visit but could not arrive before closing,” and “in response” the museum added Thursday evening hours. Weak ticket sales prompted extended hours names a money problem the passage never mentions. Tour bus operators requested night visits credits a group the passage never mentions; the surveys concerned working adults. Museum staff members favored later shifts also has no support; the change answered visitors' needs, not the staff's wishes.

Correct answer: Desert tortoises dig burrows partly to avoid extreme temperatures

Desert tortoises dig burrows partly to avoid extreme temperatures is directly supported: the tortoise “digs deep burrows to escape the daytime heat and the cold of winter nights,” and the shelters “also protect the animal from predators,” so temperature is one reason among others. Desert tortoises dig a single burrow and stay for life contradicts the statement that “a single tortoise may dig several burrows.” Desert tortoises spend virtually their whole lives inside burrows overstates the passage, which says only that the tortoise spends “much of its life underground.” Desert tortoises dig deep burrows mainly to store food names a use for burrows the passage never mentions.

Correct answer: Scanty

Scanty fits the context: the farmers “had hoped for full baskets but gathered only a few small ears of corn,” so the harvest was small in amount. Wilted borrows from the drought, but “meager” describes how much was gathered, not the condition of the plants. Lavish is the opposite of the meaning. Unripe misreads “small ears”; the sentence is about the quantity harvested, not whether the corn had matured.

Correct answer: Honest

Honest fits the context: the manager's answer came “instead of hiding behind vague excuses,” and “she admitted the team had missed several deadlines.” Polite describes manners, which the passage never discusses. Hidden borrows the “candid camera” sense of secret or unposed, and the sentence sets candor against hiding. Abrupt describes a curt delivery, but the contrast in the passage is between openness and excuses, not between short and long answers.

Correct answer: Original

Original fits the context: the solution was one “that no one in the company had ever tried before,” so it was new. Imagined borrows the familiar sense of a novel as a work of fiction, which does not fit an engineer's proposal. Workable describes whether a plan would succeed, which the sentence never addresses. Thorough borrows the idea of a long, detailed book; nothing in the sentence describes how complete the plan was.

Correct answer: Worsen

Worsen fits the context: without maintenance, “rust spread across the beams, and cracks appeared in the concrete supports,” so the bridge was declining in condition. Settle describes sinking slightly into place, not rust and cracks. Topple goes too far; the bridge was weakening, and nothing says it fell. Repair is the opposite of the meaning, the work the bridge was not getting.

Correct answer: Plentiful

Plentiful fits the context: in a single afternoon a visitor might spot “dozens of bird species and countless insects,” which signals a large amount of wildlife. Scattered suggests thin, spread-out numbers, nearly the opposite. Protected describes legal status, which the passage never mentions. Nocturnal clashes with the detail that all this wildlife can be seen “in a single afternoon.”

Correct answer: Hesitant

Hesitant fits the context: the child “stood shivering at the edge for several minutes” instead of jumping, which shows an unwillingness to go in. Cheerful points the other way; a cheerful, eager child would have jumped right in. Wretched borrows from the shivering, but the word describes the child's attitude toward jumping, not how the child felt in the cold. Unsteady reads the edge of the pool as a balance problem, which the sentence never mentions.

Correct answer: Enormous

Enormous fits the context: the plain “stretched to the horizon in every direction with no end in sight,” so it was extremely large. Confined is nearly the opposite. Desolate suggests empty and lifeless, which the passage never says; it describes only the plain's size. Treeless borrows a common picture of a plain, but the word describes extent, not what grows there.

Correct answer: Decrease

Decrease fits the context: as the medicine took effect, the patient's temperature “had nearly returned to normal” by evening, so the fever was lessening. Heighten is the opposite. Continue ignores the change the passage describes. Dissolve goes too far; the fever was shrinking, and “nearly” returned to normal shows it had not vanished.

Correct answer: Wise

Wise fits the context: saving part of every paycheck is a habit “that can protect a family during hard times,” which describes sound judgment. Rich confuses the habit with its possible result; saving is sensible at any income. Dull judges the habit as boring, which the sentence never does. Rash is the opposite, describing acting without thought.

Correct answer: The audience was gigantic

The audience was gigantic is what the figurative phrase suggests: “a sea of faces” compares the crowd to an ocean in size, and the next sentence confirms that “thousands of people filled every seat in the arena.” The crowd was tempestuous pushes the comparison to stormy water, but nothing in the passage describes unrest. The stadium was beachside takes the phrase literally; the sea is an image, not a location. The presenter was nervous reads a feeling into the moment that the passage never states.

Correct answer: Tiresome

Tiresome fits the context: copying the long list made the clerk's attention “wander from sheer boredom,” so the work was dull and wearying. Pleasant is nearly the opposite. Careless describes what can happen to a bored worker, not what the work itself was like. Physical borrows from “by hand,” but the sentence stresses boredom, not effort.

Correct answer: Strengthen

Strengthen fits the context: the engineers “added layers of stone and packed earth to make it stronger.” Waterproof borrows from the flood setting, but the sentence names strength, not sealing, as the goal. Straighten describes changing the levee's shape, which the passage never mentions. Camouflage borrows the military sense of fortifications, which does not fit work on a levee.

Correct answer: Maria anticipated wet weather

Maria anticipated wet weather is the inference every clue supports: she “checked the sky,” grabbed her umbrella, rolled up the car windows, and brought in the patio cushions, all ordinary precautions against rain. Maria planned extended travel reads too much into the cushions; she was only “leaving the house,” with nothing to suggest a long trip. Maria suspected local thieves ignores that checking the sky and taking an umbrella have nothing to do with theft. Maria expected harsh sunlight would not explain rolling up the car windows or bringing in the cushions.

Correct answer: The restaurant was exceptionally busy

The restaurant was exceptionally busy is the conclusion the details support: the tables “were full,” a line “stretched out the door,” waiters hurried, and the chef called out orders “without pause.” The restaurant was badly understaffed reads a staffing problem into the hurrying; the passage describes how much business there was, not how many workers. The restaurant was somewhat expensive has no basis; prices are never mentioned. The restaurant was locally celebrated goes beyond one busy evening; nothing describes the restaurant's reputation.

Correct answer: The bike lanes likely boosted ridership and reduced crashes

The bike lanes likely boosted ridership and reduced crashes is the best-supported conclusion: after the lanes were painted, “the number of cyclists tripled within a year,” and “reports of car-bike collisions in the area dropped sharply.” The bike lanes likely ended collisions and tripled cyclists overstates the safety result; collisions “dropped sharply,” not to zero. The bike lanes likely slowed vehicles and angered motorists names effects on drivers that the passage never mentions. The bike lanes likely raised city costs and drained budgets ignores the gains in ridership and safety, and the passage says nothing about cost.

Correct answer: James was profoundly stirred by the letter

James was profoundly stirred by the letter is the inference the clues support: he rereads it “for the fifth time,” his hands are “trembling slightly,” he cannot keep from picking it up again, and he stares out the window “without really seeing the street outside.” James was somewhat perplexed by the letter treats the rereading as a comprehension problem, but confusion does not explain trembling hands or a blank stare. James was mildly entertained by the letter clashes with every sign of strong emotion. James was largely unaffected by the letter is the opposite of what his behavior shows.

Correct answer: The factory had been vacated years earlier

The factory had been vacated years earlier is the inference the details support: it “had stood empty for decades,” its windows were broken, weeds grew through the parking lot, and the sign had faded. The factory had been redone quite recently contradicts that long neglect. The factory had been charred beyond repair reads a fire into the broken windows; the passage never mentions one. The factory had been badly built initially mistakes decay for poor construction; cracks and faded paint come from decades without upkeep.

Correct answer: The neighbor was leaving food out for a stray cat

The neighbor was leaving food out for a stray cat is best supported: each morning she sets a bowl on her back porch, it is empty by evening, and she only “glimpsed a thin gray cat slipping away into the bushes,” which describes a stray, not a pet. The neighbor was setting food out for her pet cat assumes ownership, but her own cat would not need to be glimpsed slipping away into the bushes. The neighbor was putting food out for a few birds ignores the one animal the passage shows near the bowl, the cat. The neighbor was bringing food out for a gray fox borrows the color of the animal she saw, but that animal was a cat, and the passage never mentions a fox.

Correct answer: The policy boosted morale and reduced turnover

The policy boosted morale and reduced turnover is most likely true: after the change “fewer workers quit their jobs,” and surveys showed “most employees felt more satisfied with their work-life balance.” The policy ended departures and raised profits overstates the evidence; fewer workers quit, not none, and profits are never mentioned. The policy harmed teamwork and slowed projects names drawbacks the passage never reports. The policy shortened shifts and improved wages confuses working from home with working fewer hours; the passage changes only where people work.

Correct answer: Were doubtful about their whereabouts

Were doubtful about their whereabouts is the conclusion the passage supports: the trail signs “they had counted on were nowhere to be seen,” and they studied a worn map, “turning it one way and then another.” Were confident about their directions is the opposite of what the map-turning shows. Were curious about their surroundings mistakes studying a map for sightseeing. Were argumentative about their routes reads a disagreement into the scene; the passage never says the hikers disagreed.

Correct answer: The weather was very cold

The weather was very cold is the inference every detail supports: people walked “with their collars turned up and their hands jammed into their pockets,” and “a thin layer of frost coated the parked cars.” The weather was warm, with a strong wind contradicts the frost and the bundled-up walkers, and the passage never mentions wind. The air was very humid misreads the steam, which rose from hot coffee in cold air, not from humid heat. The wind was very strong explains the turned-up collars but not the frost, and the passage never mentions wind.

Correct answer: Advocate regular breakfast habits

Advocate regular breakfast habits is the author's purpose: the passage argues that skipping breakfast “can backfire” and ends with a direct appeal, “before you rush out the door, consider grabbing a quick bite.” Report current breakfast research mistakes an argument for a neutral report; the passage cites no studies and closes by urging action. Narrate hectic breakfast mornings mistakes the busy-morning setting for a story, but no character or events appear. Describe simple breakfast recipes inflates a single example, “fruit and whole grains,” into the purpose of the whole passage.

Correct answer: Tranquil

Tranquil matches the tone, calm and peaceful: the garden is “a quiet marvel,” bees drift “lazily,” and “everything seemed unhurried and at peace.” Mournful mistakes quiet for sadness; nothing in the passage expresses loss or grief. Ecstatic overstates the feeling; “marvel” shows calm appreciation, not overwhelming joy. Detached ignores the warm, appreciative description, which shows the speaker is engaged with the scene.

Correct answer: Teach readers to complete a task

Teach readers to complete a task is the purpose: the passage is a set of ordered instructions for changing a flat tire, moving from "first loosen the lug nuts" through "Next, raise the car" to "Finally, lower the car and tighten the nuts firmly." Urge readers to purchase a spare misreads one step; the passage says to "mount the spare" but never recommends buying anything. Invite readers to follow a story does not fit, because no character or event is narrated; the reader is simply told what to do. Warn readers to expect a blowout adds a warning the passage never gives; it starts with the tire already flat and explains only how to replace it.

Correct answer: Sarcastic

Sarcastic fits: the repeated "once again," the quotation marks around the schedule's promise of a bus "every ten minutes," and riders who "stopped believing the sign" mock the posted schedule. Impartial is wrong because the author is not neutral; the pointed repetition and the quoted promise take a clear, mocking stance. Nostalgic is wrong because nothing looks back fondly on the past; the lateness is an ongoing annoyance. Sorrowful is wrong because the riders merely "shrugged"; the passage is dry and mocking rather than sad.

Correct answer: Convey the immense scale of blue whales

Convey the immense scale of blue whales is the main purpose: every sentence piles up a size fact, from "the largest animal ever known to have lived" to a heart "about the size of a small car." Demand the urgent rescue of blue whales is wrong because the passage never mentions threats to whales or asks readers to act. Trace the early ancestry of blue whales misreads "ever known to have lived"; the passage compares the whale with all animals in history but says nothing about where whales came from. Explain the rapid growth of blue whales misreads "It can grow longer than three school buses"; that phrase describes the adult length the whale reaches, not how fast it grows.

Correct answer: Outraged

Outraged fits: the author calls the missing crossing "shameful," stresses that parents "have begged for a crosswalk for years, yet nothing has been done," and ends with the angry question "How many close calls must we witness before someone acts?" Detached is wrong because the author is plainly involved and upset, not indifferent. Cheerful is wrong because the author is upset about danger to children, not in good spirits. Hesitant is wrong because the author states the complaint forcefully, without any sign of doubt or reluctance.

Correct answer: Promote a vacation at the resort

Promote a vacation at the resort is the purpose: the passage invites the reader to "Imagine waking to the smell of salt air," then advertises "our seaside resort," private balconies, and "fresh seafood at sunset," all to make a stay appealing. Relate a honeymoon at the resort is wrong because no particular trip or couple is described; the reader is asked to imagine a stay, not told about one. Disclose a problem at the resort is wrong because every detail is positive and nothing goes wrong. Criticize a dinner at the resort is wrong because the seafood at sunset is offered as an attraction, not judged as a disappointing meal.

Correct answer: Admiring

Admiring fits: the speaker has gathered people "to honor a teacher," praises her "patience and warmth," and says "Her example will guide us long after she has gone." Scolding is wrong because the passage finds no fault with the teacher. Doubting is wrong because the speaker praises her openly and without reservation. Pleading is wrong because the speaker asks the audience for nothing; the passage honors the teacher rather than appealing for help.

Correct answer: Chronological order

Chronological order is the pattern: the signal words "First," "Next," "Then," and "Finally" walk the seed through each stage in time order, from swelling to unfolding leaves. Order of importance is wrong because the steps are arranged by when they happen, not ranked from most to least important. Cause and effect fits only the opening detail that "the seed absorbs water and swells"; the passage as a whole is organized as a series of steps in time. Spatial description is wrong because, although the root pushes "downward" and the shoot rises "toward the light," the passage arranges these as stages in time, not as a description of where the parts sit.

Correct answer: Cause and effect

Cause and effect is the relationship: the signal phrase "As a result" links the cause, three days of heavy rain, to the effect, the river overflowing and flooding the valley. Rule and example is wrong because the second half is not an instance of a general rule; it is what the rain produced. Problem and cure is wrong because the flood is not solved; the passage stops at the damage. Thesis and proof is wrong because the first half makes no claim to be argued; it reports the weather.

Correct answer: Compare and contrast

Compare and contrast is the pattern: the passage weighs cats against dogs with signal words such as "yet," "while," "Unlike dogs," and "Both, however," showing how the pets differ and what they share. Problem and solution is wrong because no problem is raised and nothing is fixed. Question and answer is wrong because the passage poses no question for the later sentences to answer; it states traits of both pets. Order of importance is wrong because the traits are not ranked from most to least important; they are set side by side.

Correct answer: Problem and solution

Problem and solution is the structure: the passage opens with a problem, students who "struggle to stay organized with their assignments," and then offers "One effective answer," a simple planner. Cause and effect fits only the last sentence, which says the habit "greatly reduces missed deadlines"; that result supports the solution but does not organize the passage. Compare and contrast is wrong because no two things are compared. Order of importance is wrong because the passage does not rank points by importance; it moves from a problem to its fix.

Correct answer: The second sentence offers a specific case of the first.

"The second sentence offers a specific case of the first." is the relationship: the phrase "For example" signals that staying open "until nine on Thursdays" is one instance of the general news that the museum "added later evening hours." "The second sentence contradicts the claim of the first." is wrong because "For example" introduces support for the first sentence, not a contradiction; the two sentences agree. "The second sentence defines a term used in the first." is wrong because the second sentence defines no term; it simply shows one way the hours changed. "The second sentence makes a simple summary of the first." is wrong because it adds new, narrower details (the day and the closing time) rather than restating the first sentence more briefly.

Correct answer: Nevertheless

Nevertheless fits the blank: the team "practiced hard all season," yet the outcome was the opposite of what that effort would lead us to expect, since "they lost in the very first round," and this transition signals that kind of contrast. Consequently would wrongly present the loss as a result of the hard practice. Additionally would treat the loss as one more item in a list rather than an unexpected outcome. Specifically would introduce a detail or example of the first sentence, but losing is not an example of practicing hard.

Correct answer: The Pacific Ocean occupies the largest area

The Pacific Ocean occupies the largest area is a fact: its size can be measured on maps and checked against the other oceans. The Pacific Ocean offers the finest scenery is an opinion, because "finest" is a personal judgment of beauty. The Pacific Ocean boasts the greatest charm is also an opinion; charm cannot be measured, so there is no way to prove which ocean has the most. The Pacific Ocean merits the highest praise is an opinion, because whether a place merits praise, and how much, is a matter of personal judgment.

Correct answer: The sentence about its relative merit

The sentence about its relative merit is the opinion: "In my view, it is the finest building in the entire town" ranks the library against other buildings, and "In my view" marks it as a personal judgment that cannot be proven. The sentence about its inaugural date is a fact, since "The new library opened in March" can be checked. The sentence about its total holdings is a fact, since "It contains over fifty thousand books and twenty computers" can be counted. The sentence about its weekly service is a fact, since "The library is open six days a week" can be verified from its schedule.

Correct answer: Chocolate ice cream is tastier than vanilla

Chocolate ice cream is tastier than vanilla is the opinion: which flavor tastes better depends on personal preference and cannot be proven. Strawberry ice cream is pinker than coconut is a fact, because color can be observed and compared. Gelato is denser than supermarket ice cream is a fact, because density can be measured; gelato is churned with less air. Sherbet is leaner than commercial ice cream is a fact that can be checked on a label: sherbet has only about 1 to 2 percent milkfat, while ice cream must have at least 10 percent. Each of these uses a comparison, but only the claim about taste rests on preference.

Correct answer: Water freezes at zero degrees Celsius at sea level

Water freezes at zero degrees Celsius at sea level is a verifiable fact: the freezing point of water can be measured with a thermometer. Lake water feels nicest for swimming at twenty degrees Celsius is an opinion, because how nice the water feels depends on the swimmer. Tea tastes best when made with water at ninety degrees Celsius is an opinion, because which taste is best is a matter of preference even though the temperature is precise. Morning air feels freshest at five degrees Celsius is an opinion as well; how fresh air feels cannot be measured. A precise number does not turn a judgment into a fact.

Correct answer: Park skaters mostly use a separate plaza and rarely touch benches

Park skaters mostly use a separate plaza and rarely touch benches most weakens the argument, which claims that skateboards "always damage public property" and "ruin everything they touch"; if most skaters stay on their own paved area and rarely touch the benches, that sweeping claim fails. A skateboard hit a park bench last week and left a second scratch adds more damage, which supports the author rather than weakening the case. Park neighbors want a steep fine for riders who touch the benches is beside the point; a proposed penalty says nothing about whether skateboards actually damage property. Park crews claim skateboards touch the benches nearly every night strengthens the argument by suggesting that contact with the benches is frequent.

Correct answer: The author bases a broad verdict on scant evidence

The author bases a broad verdict on scant evidence: two cold meals at one diner are used to conclude that "every restaurant in this city serves cold food," a hasty generalization. The author builds a case on secondhand diner rumor is wrong because the author ate there personally ("I have eaten at that diner twice"); the problem is the amount of evidence, not its source. The author scorns a diner chef on personal grounds is wrong because no cook is mentioned or attacked; the complaint is about the food. The author leans on a noted city restaurant critic is wrong because no critic or expert is cited at all.

Correct answer: Studies linking extra bike lanes to measurably eased congestion

Studies linking extra bike lanes to measurably eased congestion best strengthens the argument, because the author claims that "Cycling reduces traffic," and measured results from added lanes would back that claim directly. Polls tying painted bike lanes to recurrent motorist complaints would hurt the case, since they show public opposition rather than any benefit. Histories tracing early bike lanes to visionary urban designers are irrelevant; where the idea came from says nothing about whether more lanes would help this city now. Audits tying expensive bike lanes to postponed sidewalk repairs point to a cost of the plan, which weakens rather than supports it.

Correct answer: It assumes music lessons cannot aid students' prospects

It assumes music lessons cannot aid students' prospects: the argument treats music only as time taken "away from those subjects" and never considers that music might also help "students' futures," an idea it leaves unsupported. It assumes students' parents cannot afford music tutors is wrong because the argument says nothing about cost or private lessons. It assumes music clubs cannot retain students' interest is wrong because whether students enjoy music plays no part in the reasoning. It assumes students' current math scores cannot improve gets the argument backward; the author wants more time for math and reading precisely because scores could rise.

Correct answer: Coral reefs host a vast share of saltwater life in little space

Coral reefs host a vast share of saltwater life in little space states the main idea: the passage opens by contrasting "less than one percent of the ocean floor" with "roughly a quarter of all marine species," and every later sentence explains or pictures that crowding. Coral reefs cover less than one percent of the ocean floor is a true detail, but it leaves out the point the detail serves, the huge amount of life reefs support. Coral reefs shelter a majority of marine species in the ocean overstates the figure; the passage says "roughly a quarter," not more than half. Coral reefs form a busy city of builders in shallow ocean water is too narrow; it turns the closing image of the reef as "a crowded city built by some of the sea's smallest architects" into the main point, when that image only illustrates how much life reefs hold.

Correct answer: Libraries slowly opened locked collections to patrons

Libraries slowly opened locked collections to patrons is the central idea: the paragraph traces how libraries went from chaining books "to the shelves" to allowing borrowing, a shift that "turned the library from a guarded vault into a community resource." Libraries initially chained costly volumes to shelves is a true detail from the opening, not the idea the whole paragraph develops. Libraries removed chains to satisfy impatient readers gives the wrong reason; the chains came off "As printing made books cheaper," not because readers complained. Libraries hired monastic scribes to duplicate volumes is never mentioned; the passage credits printing, not hand copying, with making books cheaper.

Correct answer: Fast regrowth makes bamboo a favored sustainable material

Fast regrowth makes bamboo a favored sustainable material is the main idea: the passage builds from bamboo's speed and its ability to be "harvested again and again without replanting" to the conclusion that builders "increasingly view bamboo as a renewable alternative." Bamboo grows approximately a meter daily wherever planted overstates a detail; that growth happens only in "Certain species" and "under ideal conditions." Tropical rainfall largely gives bamboo a growth advantage offers a cause the passage never names; it says only that the fastest growth happens "under ideal conditions." Quick growth earns grasses a good construction reputation is too broad; the passage is about bamboo alone and never discusses grasses in general.

Correct answer: Rural fire coverage rests largely on volunteer crews

Rural fire coverage rests largely on volunteer crews captures the main point: volunteers "make up the majority of fire crews in many rural towns," and without them "large areas of the country would have no fire protection at all." Rural volunteer fire crews prepare on spare weekends is a supporting detail ("They train on weekends"), not the point the passage builds to. Rural towns spend heavily on volunteer fire stations is never mentioned; the passage stresses unpaid service, not spending. Rural districts steadily cut back on volunteer crews is unsupported; the passage describes volunteers as holding together "the safety net of small communities," with no sign that towns are reducing them.

Correct answer: Yawning may cool the brain, not just show tiredness

Yawning may cool the brain, not just show tiredness is the main idea: the passage challenges the belief that yawning "simply means a person is tired" and presents studies suggesting it "may help cool the brain," so the yawn "may be doing far more than signaling boredom." Yawning may just signal fatigue, not cool the brain restates the common assumption the passage sets out to question. Yawning may warm the brain, not just reduce boredom reverses the theory; the studies link yawning to cooling an overheated brain, and subjects "yawned less often when their foreheads were chilled." Yawning may cool the airways, not just ease fatigue names the wrong target; the cooling the passage describes is of the brain, and it says nothing about the airways.

Correct answer: The Waggle Dance as a Map to Nectar

The Waggle Dance as a Map to Nectar is the best title: the whole passage explains how the dance tells the hive "both the direction and the distance to a flower patch," so that thousands of bees "can be guided to a single bloom." The Waggle Dance as a Signal to Run is wrong because the dance points bees toward food, not away from danger. The Waggle Dance as a Lure to Mates is wrong because the dance has nothing to do with mating; it communicates "the location of food." The Waggle Length as a Key to Range is too narrow; the length of the waggle "signals how far to fly," but that is only one part of the dance, which also uses its angle to show direction.

Correct answer: Newer devices gradually made voyages more secure and accurate

Newer devices gradually made voyages more secure and accurate summarizes the whole passage, which moves from the stars to the compass, radio beacons, and satellites and concludes that "Each new tool built upon the last, steadily making travel across open water safer and more precise." Satellites alone made ship navigation more reliable and exact credits one tool with the whole change; satellites are only the latest step in a long progression. Sailors followed the stars, and clouds made some nights perilous is a fair inference about the first stage, since the compass later "added a reliable guide even under clouds," but it covers only the opening of the passage, not its main point. Radio beacons proved more precise than the satellites after them gets the progression backward; each new tool built upon the last, so the later satellites, not the beacons, gave the greater precision ("pinpoint positioning").

Correct answer: To prove their craft merited guild membership

To prove their craft merited guild membership is the reason the passage gives: "The guild examined this work to decide whether his skills justified admission." To achieve their guild journeyman rank sooner mixes up the stages; the craftsman had already served "as an apprentice and then a journeyman" before he made the masterpiece. To afford their expensive guild entrance fees is never mentioned; the work was inspected, not sold to raise money. To show their apprentices fresh guild methods is also unsupported; the masterpiece was made for the guild to judge, not to teach anyone.

Correct answer: Their yawning rate dropped

Their yawning rate dropped is what the passage reports: "Researchers noticed that subjects yawned less often when their foreheads were chilled." Their yawning count soared reverses the finding; chilling led to fewer yawns, not more. Their yawning turned quiet is never mentioned; the passage says nothing about how loud the yawns were. Their yawning got stronger is also unsupported; the passage describes how often the subjects yawned, not how forceful each yawn was.

Correct answer: It marks the flight path relative to the sun

It marks the flight path relative to the sun is the detail the passage gives: "The angle of the dance points the way relative to the sun." It gives the trip length relative to the sun confuses the angle with the waggle's length, which is what "signals how far to fly." It fixes the dance speed relative to the sun is never mentioned; the passage says nothing about how fast the dance is performed. It sets the takeoff time relative to the sun is also unsupported; the angle tells the bees which way to go, not when to leave.

Correct answer: Printing cut book costs and eased replacement

Printing cut book costs and eased replacement is the reason given: "As printing made books cheaper, librarians gradually removed the chains," and books "that had once been irreplaceable could now be reproduced in quantity." Printing improved book paper and limited wear names the right cause but the wrong effect; the passage credits printing with lowering cost, not with making books sturdier. Printing guilds donated book copies and funds is never mentioned; no gifts to libraries appear in the passage. Printing created book clubs and public demand is unsupported; the passage mentions no clubs and gives no sign that readers pushed for the change.

Correct answer: They stow equipment in personal vehicles

They stow equipment in personal vehicles is the detail the passage gives: volunteers "keep gear in their cars" so they can respond quickly. They secure gear in firehouse storerooms contradicts that detail; the gear travels with the volunteers rather than staying at a station. They drill weeknights in community halls changes the schedule; the passage says they "train on weekends." They sleep nightly in staffed firehouses is wrong because the volunteers "leave work or sleep to answer alarms," which means they are at home or at work, not living at a station.

Correct answer: It grows back speedily without replanting

It grows back speedily without replanting is stated as a reason: "Because it regrows so quickly after cutting, it can be harvested again and again without replanting," and the passage then says "For these reasons, builders increasingly view bamboo as a renewable resource." It is far stronger than any softwood overstates the passage, which says only that bamboo has "a strength comparable to some softwoods." It spreads very widely without replanting is not stated; the passage describes how fast bamboo grows back where it is cut, not how far it spreads. It endures dry seasons without replanting is also unsupported; the passage mentions growth "under ideal conditions" and says nothing about drought.

Correct answer: It furthered the gains of earlier tools

It furthered the gains of earlier tools matches the passage: "Each new tool built upon the last, steadily making travel across open water safer and more precise." It replaced the duties of earlier tools misreads "built upon"; the compass, for example, "added a guide" rather than doing away with the stars. It erased the benefits of earlier tools is never suggested; each tool made travel safer, not harder. It duplicated the work of earlier tools is wrong because each new tool added something, such as a guide "even under clouds" or "pinpoint positioning," instead of repeating what came before.

Correct answer: Fresh and original

Fresh and original is the meaning here: the engineer's idea was one "no one on the team had considered before," it used cables "in a way the firm had never tried," and colleagues noticed "the freshness of the approach." Long and fictional is the common meaning of the noun "novel," a book-length story, which does not fit a solution. Strong and slender guesses at the cables themselves, which the passage calls only "lightweight"; it does not explain why the solution is called novel. Classic and proven is the opposite of the meaning, since her design departed from older methods that "relied on heavy steel beams."

Correct answer: Moderate and gentle

Moderate and gentle is the meaning here: the climate is "never scorching in summer nor freezing in winter," visitors find it comfortable "year-round," and "the conditions are so mild." Angry and irritable confuses "temperate" with "temper"; the word describes weather, not a mood. Frigid and blustery contradicts the passage, which says the island is never "freezing in winter." Unsettled and rainy is not supported; nothing in the passage mentions changeable weather or rain, and comfortable conditions in nearly every season suggest the opposite.

Correct answer: Extremely thorough and exact

Extremely thorough and exact is the meaning here: the watchmaker checks "each tiny gear three times," leaves "Not a single screw" loose, and polishes every surface "to perfection." Extremely slow and uncertain misreads the repeated checking; the passage shows care, not hesitation or doubt. Extremely admired and famous confuses the word with its result; the watchmaker's "careful attention was famous," but fame is not what "meticulous" means. Extremely casual and relaxed is the opposite of the meaning.

Correct answer: Restrain or lessen

Restrain or lessen is the meaning here: the fees were meant to "discourage businesses from handing out so many disposable bags," and afterward "the volume of discarded plastic had dropped noticeably." Street or roadside reflects the common noun "curb," the edge of a street, which does not fit a verb about waste. Promote or enlarge is the opposite of what the fees were for. Bill or overcharge confuses the fees, which were the tool, with the word "curb," which names the goal of cutting waste.

Correct answer: Dull and unchanging

The fields were identical and nothing changed the view for hours, so 'monotonous' means dull and unchanging. Scenic and charming praises a view the travelers wanted broken by a hill or a town. Exciting and varied is the opposite of mile after mile of identical fields. Sprawling and empty describes the plains themselves, not the quality of the drive that the word names.

Correct answer: Present in large amounts

Bees could hardly fly a few feet without finding another blossom, and the sheer number of flowers amazed the hikers, so 'abundant' means present in large amounts. Blooming in early spring repeats when the flowers appeared, not what the word means. Scattered in tiny groups clashes with a blossom every few feet. Uncommon in open meadows is the opposite of the sheer number the passage stresses.

Correct answer: Worsen

Rust spreading along the beams and concrete crumbling into the river show the bridge's condition declining, so 'deteriorate' means worsen. Spread is what the rust did along the beams, not what the word says about the bridge as a whole. Topple goes too far: engineers warn only that the structure will soon become unsafe, so it is still standing. Darken fixes on the look of the rust, a color change the passage never mentions and not a decline in condition.

Correct answer: Hesitant or unwilling

She stood at the edge for several minutes glancing nervously at the water and jumped only after her coach encouraged her, so 'reluctant' means hesitant or unwilling. Interested or willing is the opposite of her long delay. Confident or cheerful ignores her nervous glances at the water below. Unqualified or unable confuses holding back with lacking skill; nothing says she could not dive.

Correct answer: Customary and traditional

Most architects in the region built this way, and the new firm 'broke from this tradition' with a home 'so unlike the usual designs,' so 'conventional' means customary and traditional. Experimental and original describes the new firm's glass-and-steel home, the opposite of the conventional one. Reinforced and dependable and inexpensive and practical add claims about strength and cost that the passage never makes about brick and timber.

Correct answer: Make plain or clarify

The diagram turned a theory that 'had seemed murky' into one that 'suddenly became clear,' so 'illuminate' means make plain or clarify. Make murkier or cloud is the opposite of what the diagram did for the students. Make bright or light is the literal sense of shining light on something, which does not fit an idea. Make visual or depict describes the diagram itself, the tool the professor used, not what using it achieved.

Correct answer: Probably expand if results stay strong

Lots sit half empty because fewer people drive, shops report steady sales, and officials are studying an extension to weekdays, so the author expects the policy to probably expand if results stay strong. Probably halt if parking remains empty misreads the empty lots, which are a sign of success here. Probably shrink if shop sales decrease has no support, since sales are steady and the author never ties the policy's future to them. Probably add congestion if usage grows contradicts the streets feeling less crowded with cars; more bus riders means fewer cars, not more.

Correct answer: Were facing a dangerous lack of water

They had drained their last bottle, the next stream was miles away, and the sun beat down, so the hikers were facing a dangerous lack of water. Were reaching a fresh source of water ignores that the stream was still marked miles ahead. Were packing a large reserve of water contradicts the drained last bottle. Were rationing a tiny supply of water assumes some water was left, but the last bottle was already empty.

Correct answer: Is truly popular and may soon expand

A line forms before the doors open, regulars arrive early to avoid an hour's wait, and the chef is talking with a contractor about a second dining room, so the restaurant is truly popular and may soon expand. Is poorly managed and may soon close mistakes the long wait for a sign of trouble. Is too cramped and may soon relocate ignores that the chef plans to add a room, not move. Is fairly new and may soon advertise conflicts with the regulars who already know the routine.

Correct answer: Deeply disturbed by the letter

He reads the letter twice, folds it slowly, stares out the window in silence, and leaves rather than say what is wrong, so he is deeply disturbed by the letter. Briefly confused by the letter ignores that he stares out the window for a long while. Secretly pleased by the letter clashes with his silence and the shake of his head when asked what is wrong. Mostly annoyed by the question gets the order wrong: his silence began before his sister spoke.

Correct answer: The city was suddenly destroyed, likely by an erupting volcano

Layers of volcanic ash, belongings left 'as if everyone had fled in an instant,' and charred timbers from sudden, intense heat point to one answer: the city was suddenly destroyed, likely by an erupting volcano. The city was gradually abandoned, likely by an extreme drought clashes with pottery and tools left exactly where their owners set them. The city was intentionally torched, likely by an invading army names attackers the passage never mentions and leaves the volcanic ash unexplained. The city was quickly toppled, likely by an enormous earthquake fits the haste but not the ash or the intense heat.

Correct answer: Persevered after setbacks

He sketched a fourth design the same night his third machine broke down, and each failure 'seemed only to sharpen his determination,' so the inventor persevered after setbacks. Floundered after setbacks is the reverse: he moved straight to a new design instead of struggling helplessly. Complained after setbacks has no support; the passage records no grumbling, only more work. Apologized after setbacks invents someone he let down, though the passage mentions no one but the inventor.

Correct answer: Is in economic decline

A boarded factory, a diner that closed last spring, a train that no longer stops, and homes that have sat unsold for months show a town that is in economic decline. Is in gradual recovery has no support; every detail shows loss, not improvement. Is in total collapse goes too far, since only a few families have put their homes up for sale and the town is still lived in. Is in a building boom misreads the boarded windows, which mark a closed factory, not new construction.

Correct answer: Caused mischief while they were gone

Muddy prints on the carpet, a chewed shoe, a tipped trash can, and a tail tucked low as the dog greets the family all suggest that the dog caused mischief while they were gone. Behaved sweetly while they were gone is contradicted by the mess the family found. Stalked rabbits while they were gone may explain mud but not the chewed shoe or the scattered trash inside. Faced intruders while they were gone has no support; nothing points to anyone breaking in, and the dog's tucked tail fits guilt, not a fight.

Correct answer: Persuade readers to curb consumption

The passage opens 'Recycling is not enough; we must reduce what we consume' and ends by calling on readers to rethink their habits, so its purpose is to persuade readers to curb consumption. Remind readers to continue recycling misses the point that recycling alone falls short; the author wants more than the habit readers already have. Help readers to understand materials turns one supporting detail, the energy and raw materials behind each product, into the whole purpose. Implore readers to abandon recycling overstates the claim: the author calls recycling not enough, not useless.

Correct answer: Sarcastic

'Oh, splendid' and 'I can hardly contain my excitement' about a pointless meeting say the opposite of what the writer means, which makes the tone sarcastic. Heartfelt takes the praise at face value, though the meeting accomplished nothing. Delighted reads 'excitement' literally and misses the irony. Objective ignores the writer's plain irritation at deciding again what was decided last week.

Correct answer: Teach readers to perform a task

The passage moves through ordered steps marked first, next, and finally, which shows the author wrote it to teach readers to perform a task: changing a flat tire. Urge readers to purchase a jack mistakes a tool used in one step for a sales pitch. Remind readers to check a spare turns one item from the steps into a safety reminder the passage never gives. Invite readers to share a story does not fit instructions written in the second person with no past event to recall.

Correct answer: Calm and appreciative

Light that 'spilled gently,' birds that 'called softly,' and a 'peaceful dawn' that makes the world feel new create a tone that is calm and appreciative. Anxious and unsettled clashes with the gentle, peaceful images. Energetic and playful overstates the energy of a quiet scene. Formal and impersonal ignores the warm, appreciative words the writer chooses for the morning.

Correct answer: To share unusual facts about octopus bodies

The passage lists three hearts, blue blood, and a copper-based molecule, then calls these 'strange features,' so its purpose is to share unusual facts about octopus bodies. To raise concern about octopus habitat loss names a threat the passage never mentions. To give thorough detail about octopus gills is too narrow; the gills appear in only one clause. To debunk common myths about octopus colors misreads 'blue blood' as a correction, though no myth is stated or disproved.

Correct answer: Strongly against

Calling the highway a 'so-called improvement,' asking 'at what cost' to the children, and vowing to 'refuse to stay silent' show the author is strongly against it. Mildly skeptical understates the force of a writer who calls the plan a trade of the community's heart. Quietly resigned clashes with refusing to stay silent. Calmly impartial ignores the loaded words the author uses throughout.

Correct answer: Offer the reader a hopeful lesson

The last sentence steps back from the river to a general truth, 'Sometimes patience and effort really can heal what we have harmed,' so it was included to offer the reader a hopeful lesson. Offer the reader a solemn warning reverses its tone, which looks ahead with hope rather than alarm. Offer the reader a sharp contrast describes the sentence before it, which sets children fishing against grandparents holding their noses. Offer the reader a complete recap does not fit a sentence that restates none of the events.

Correct answer: Cause and effect

Each sentence shows one event producing the next, from dumped chemicals to vanished fish to departed birds to a silent valley, and the passage ends on 'a chain of consequences,' so the pattern is cause and effect. Order of events is the trap: the events do happen in sequence, but the passage stresses why each one led to the next, not simply when. Compare and contrast would weigh two subjects side by side, and none are compared. Problem and solution would need a fix, and the passage offers none.

Correct answer: Compare and contrast

The passage sets gas heating against electric heating and gives each one's strengths and weaknesses, so its structure is compare and contrast. Problem and solution misreads the final sentence; the passage offers no fix, only a reminder that the choice depends on a household's priorities. Sequence of events would trace events through time, and none are dated or placed in order. Order of importance would rank points from most to least significant, which the even-handed passage never does.

Correct answer: Sequence of steps

The signal words first, next, then, and finally walk the reader through baking in order, so the structure is a sequence of steps. Order of importance would rank ideas by significance rather than by when they happen. Compare and contrast needs two things weighed against each other, and the recipe describes only one process. Problem and solution needs a difficulty to fix, and the passage presents none.

Correct answer: Problem and solution

The passage names a problem, traffic that costs commuters hours each day, and then offers two answers, longer subway lines and protected bike lanes, so it is organized as problem and solution. Compare and contrast is tempting because two measures appear, but they are offered together, not weighed against each other. Cause and effect fits only the first sentence, not the passage's overall plan. Order of importance would rank the ideas, and the passage presents the two answers as equals.

Correct answer: Spatial description

Details are arranged by location as a visitor moves through the building, ahead, to the left, to the right, and beyond on the upper floor, so the pattern is spatial description. Chronological order is the trap: the walk-through feels like a sequence, but the passage organizes places, not events in time. Compare and contrast mistakes the ancient and modern galleries for subjects being weighed, though the passage only says where each one is. Problem and solution does not apply, since no problem is raised.

Correct answer: The third sentence

The third sentence, 'In my view, it is also the most beautiful body of water in the world,' gives a personal judgment that cannot be proven, so it is the opinion. The first sentence states that the Pacific is the largest ocean on Earth, a measurable fact. The second sentence says the ocean covers more area than all the continents combined, which maps can confirm. The fourth sentence gives an average depth of more than 4,000 meters, a figure that can be measured.

Correct answer: At sea level, water boils at 100 degrees Celsius.

A fact can be checked by measurement, and "At sea level, water boils at 100 degrees Celsius." can be confirmed with a thermometer, so it is the fact. Each of the others pairs a real temperature with a matter of taste: "Brewed at 85 degrees Celsius, green tea is ideal." rests on 'ideal,' "In a sauna, 110 degrees Celsius feels most relaxing." rests on 'most relaxing,' and "Served at 70 degrees Celsius, soup is most pleasant." rests on 'most pleasant.' The numbers can be measured, but those judgments cannot.

Correct answer: The third sentence

The third sentence, 'Honestly, it is the finest building our town has ever constructed,' rests on a personal judgment of what is 'finest,' so it expresses an opinion. The first sentence gives the month the library opened, which records can confirm. The second sentence counts more than fifty thousand books on three floors, a fact that can be checked. The fourth sentence names the funding sources, city taxes and private donations, which can also be verified.

Correct answer: A nearby school banned phones, and its results held steady.

The author blames phones for falling scores and says a ban would 'surely reverse the decline.' The statement "A nearby school banned phones, and its results held steady." shows a ban doing nothing, which most directly weakens that claim. "A rival school allowed phones, and its grades fell sharply." fits the author's view and would support it instead. "In one biology class, students look up answers on phones." shows phones helping in one class but says nothing about whether a ban would raise scores. "The school board has publicly opposed phones in classrooms." reports an opinion about phones, not evidence of their effect on learning.

Correct answer: It treats a shared habit as the root of achievement

The author notices that successful entrepreneurs wake early and concludes that rising early must explain their business success, ignoring every other factor, so the main flaw is that it treats a shared habit as the root of achievement. It treats an expert's opinion as proof of the claim does not apply, since no expert is cited. It treats a strong emotion as the basis of judgment does not fit an argument that makes no emotional appeal. It treats its own conclusion as evidence for itself misses that the argument does offer evidence, a pattern among entrepreneurs; the problem is what it concludes from that pattern.

Correct answer: A study finding tree-lined streets are several degrees cooler and have cleaner air

The author argues that trees lower summer temperatures and absorb pollutants, so a study finding tree-lined streets are several degrees cooler and have cleaner air confirms both claimed benefits and best strengthens the argument. A report finding street trees are easily harmed and have unusually short lifespans would weaken the case for planting, since the benefits depend on trees that survive. A survey showing shady streets are highly desirable and have costlier family homes speaks to popularity and property values, not the public health the author names as the goal. A paper showing trees along streets are noticeably taller and have stronger trunks describes the trees themselves without touching shade, heat, or pollution.

Correct answer: Over long periods, glaciers have carved out valleys and created lakes

The main idea is "Over long periods, glaciers have carved out valleys and created lakes": every sentence shows moving ice shaping the land over thousands of years, ending with "Much of the landscape in northern regions was shaped this way." "As glaciers melt over time, they leave behind rocky ridges and lakes" is one supporting detail about what retreating ice leaves, too narrow to be the main idea; "Over time, glaciers move faster than rivers as they carve out valleys" contradicts the passage, which calls glaciers "slow-moving rivers of ice"; "Glaciers have turned most of the valleys in northern areas into lakes" distorts the passage, which says glaciers carved valleys and left basins that became lakes, not that most valleys turned into lakes.

Correct answer: To avoid drifting off in the water while they sleep

The passage says the kelp "anchors them in place so they do not drift away on the current while they rest," so the reason is "To avoid drifting off in the water while they sleep." "To stay warm in the cold water while they sleep" gives a reason, warmth, that the passage never mentions; "To grab passing food on the current while they sleep" turns the kelp into a way of feeding, which the passage does not say; "To hide from predators in the water while they sleep" names a danger the passage never raises, since the only risk it describes is waking up far from the group.

Correct answer: To persuade readers to visit the aquarium

The passage praises the glass tunnel and the touch pool with excitement and ends with "Buy your tickets today," so its purpose is "To persuade readers to visit the aquarium." "To list rules for a visit to the aquarium" names something the passage never gives, since it states no rules at all; "To describe how the aquarium was restored" mistakes the passing word "renovated" for the topic; "To recount a visit to the old aquarium" treats the passage as a story about a past trip, but it tells no story and speaks directly to the reader about a future one.

Correct answer: Sarcastic and frustrated

"Once again" and "every single morning" show irritation, and "Of course it would choose today" is said mockingly, so the tone is "Sarcastic and frustrated." "Playful and lighthearted" mistakes the sarcasm for fun, but the writer is plainly annoyed; "Nervous and apprehensive" overreads the deadline, since the writer complains rather than worries; "Unbothered and forgiving" ignores the biting "of course," which shows the writer is anything but calm about the printer.

Correct answer: Somebody is going to celebrate a birthday soon

The matches, the cake box, and the fistful of small candles all point to a cake about to be lit, so the best inference is "Somebody is going to celebrate a birthday soon." "The family is getting ready for a power outage" fits the matches and candles but not the cake or the counting out of small candles; "Dad is planning to bake a chocolate cake today" ignores that he carried in a cake box, so the cake is already made; "Guests are about to come for a candlelit feast" fits the candles but leaves the cake box and the counted candles unexplained.

Correct answer: Doubtful

The detective asks the same question three different ways "to see whether the answers would change," which shows he does not yet believe the story, so "skeptical" most nearly means "Doubtful." "Curious" describes wanting to learn more, not distrust; "Confused" would mean he could not follow the story, but he is testing it on purpose; "Trusting" is the opposite of the detective's attitude.

Correct answer: Cause and effect

The signal word "because" links the dam (the cause) to the flooding and the new pond, which in turn draw new animals (the effects), so the structure is "Cause and effect." "Problem and solution" does not fit, since no problem is presented and then solved; "Compare and contrast" would need two things set side by side; "Question and answer" would need the passage to pose a question and then answer it.

Correct answer: Autumn is clearly the most beautiful season of the year

"Clearly the most beautiful" is a personal judgment that cannot be proven true or false, so the opinion is "Autumn is clearly the most beautiful season of the year." "Many people say autumn is the season they like the most" reports what people say, which a survey can check, so it is a fact about their opinions; "In autumn, the length of the day gets shorter each week" and "Autumn is the season when many trees lose their leaves" describe things that can be observed and measured.

Correct answer: Adding stops at useful places can increase overall ridership

The route added stops at the hospital, the community college, and three apartment complexes, and ridership then rose by forty percent, so the passage supports "Adding stops at useful places can increase overall ridership." "Adding stops at the college can double ridership within a month" overstates both the cause and the size of the rise, which was forty percent across five new stops, not just the college; "Adding stops can increase ridership by forty percent each month" turns one month's result into a steady monthly rate; "Adding stops can make a bus route faster for riders than a taxi" compares buses with taxis, which the passage never does.

Correct answer: Instead

The cook could not get fresh basil and used dried basil in its place, so the transition that signals a substitution is "Instead." "Otherwise" means "if not," which does not fit a choice already made; "Also" adds a further point in the same direction rather than a replacement; "Meanwhile" signals something happening at the same time, not a swap.

Correct answer: Intimidating

The trail "looked daunting from the bottom," yet the group "decided to attempt the hike anyway," so the word describes something that makes people hesitate: "Intimidating." "Breathtaking" would make the steep climb sound beautiful, which does not explain the "anyway"; "Unremarkable" would mean the trail looked ordinary, which clashes with steep switchbacks up a cliff; "Exhausting" describes how the hike might feel afterward, not how it looked from the bottom.

Correct answer: Thorough

Lena rechecks "each figure in the budget twice" and reads the file aloud "to catch any final slip," so she is best described as "Thorough." "Anxious" reads a feeling into the passage, which shows careful checking but no worry; "Hesitant" would mean she held back, yet she finishes and sends the file; "Careless" is the opposite of her behavior.

Correct answer: By comparing the strong and weak points of two kinds of books

The passage sets the strengths and weaknesses of hardcovers beside those of paperbacks, so it is organized "By comparing the strong and weak points of two kinds of books." "By explaining why the hardcover costs more than the paperback" fits a cause-and-effect passage, but this one never says why the prices differ; "By describing a problem with paperback books and its solution" picks out one weakness and offers no fix; "By tracing the history of hardcover books through the decades" describes a time sequence the passage does not contain.