ALEKS Math Placement Practice Test.
Free practice exam — 120 verified questions, instant feedback.

Answer: B
Apply the discount: 1,250 × 0.88 = 1,100. Then apply tax to that amount: 1,100 × 1.08 = 1,188 dollars. Simply combining the percents as 12% off and 8% on, a net 4% off, gives 1,250 × 0.96 = 1,200, but percents of different bases cannot be merged that way. Stopping at $1,100 forgets the tax, and $1,242 uses 8% as the discount rate by mistake.

Answer: B
Matching shadows create similar triangles, so height over shadow is constant: h / 37.5 = 6 / 9. Then h = 6 × 37.5 ÷ 9 = 25 feet. Inverting the ratio gives 9 × 37.5 ÷ 6 = 56.25 feet, which would make the tree taller than its own proportions allow. Subtracting shadow lengths gives 28.5, and 22.5 comes from using 10 in place of the 9-foot shadow.

Answer: B
Start at -8 and move up 15: -8 + 15 = 7 degrees. On a number line, rising 15 from 8 below zero crosses zero and lands 7 above it. Getting -7 comes from subtracting 15 - 8 and keeping the negative sign, 23 adds the sizes 8 + 15 as if both were positive, and -23 treats the rise as a further drop.

Answer: D
Square first: (-3)^2 = 9 because a negative times a negative is positive. Then -4(-2) = +8, since subtracting a negative product means adding. So 9 + 8 = 17. Getting 1 comes from computing 9 - 8 by mishandling the double negative, and -1 or -17 come from also treating (-3)^2 as -9.

Answer: D
Reversing a charge adds money back: -14 + 6 = -8, so he is still 8 dollars overdrawn. Subtracting the 6 instead gives -20, which would mean the bank charged the fee again rather than removing it. The positive values 8 and 20 would require the reversal to wipe out more than the whole overdraft, which a $6 credit cannot do.

Answer: A
Add 7 to both sides: 4x = 28. Divide by 4: x = 7. Check: 4(7) - 7 = 28 - 7 = 21. The value 3.5 comes from subtracting 7 first, giving 14/4, instead of adding it. The answer -7 is a sign slip when moving the 7 across, and 5 comes from an arithmetic error in the final division.

Answer: D
Distribute first: 2x - 6 = 5x + 9. Subtract 2x: -6 = 3x + 9. Subtract 9: -15 = 3x, so x = -5. Check: 2(-5 - 3) = -16 and 5(-5) + 9 = -16. The value -4 comes from forgetting to multiply the 3 by 2, leaving 2x - 3 = 5x + 9. The values 5 and -1 come from sign slips while collecting terms.

Answer: A
Subtract 3x from both sides: 4x - 4 = 16. Add 4: 4x = 20, so x = 5. Check: 7(5) - 4 = 31 and 3(5) + 16 = 31. The value 3 comes from computing 16 - 4 = 12 and dividing by 4, subtracting the 4 instead of adding it. The value 2 divides 20 by 10 after adding the x-terms instead of subtracting, and -5 is a sign slip.

Answer: A
Add 4 to both sides: 2x/3 = 14. Multiply both sides by 3/2: x = 14 × 3/2 = 21. Check: 2(21)/3 - 4 = 14 - 4 = 10. The value 9 comes from subtracting 4 to get 2x/3 = 6. The value 9 1/3 multiplies 14 by 2/3 instead of the reciprocal 3/2, and 18 halves 14 × 3 incorrectly by rounding.

Answer: A
Model the bill: 25 + 0.10t = 38.50. Subtract 25: 0.10t = 13.50. Divide by 0.10: t = 135 texts. Check: 25 + 0.10(135) = 25 + 13.50 = 38.50. Dividing the whole bill by 0.10 gives 385 and ignores the monthly base charge. Adding the base charge instead of subtracting gives 635, and 14 rounds 13.50 without ever dividing by the per-text rate.

Answer: C
Subtract 5 from both sides: -2x > 6. Dividing by -2 reverses the inequality: x < -3. Check with x = -4: 5 - 2(-4) = 13, and 13 > 11 is true. The answer x > -3 forgets to flip the sign when dividing by a negative; testing x = 0 gives 5 > 11, which is false. The versions with 3 drop the negative during division.

Answer: B
Subtract 4 from both sides: x/3 ≤ 2. Multiply both sides by the positive number 3: x ≤ 6. Check the boundary: 6/3 + 4 = 6, and 6 ≤ 6 holds. The answer x ≤ 2 stops before multiplying by 3. Flipping to x ≥ 6 is wrong because multiplying by a positive number never reverses an inequality, and x ≤ 30 multiplies by 3 twice by scaling the 4 as well.

Answer: A
Write the budget inequality: 40 + 0.25m ≤ 65. Subtract 40: 0.25m ≤ 25. Divide by 0.25: m ≤ 100 miles. Check: 40 + 0.25(100) = 65 exactly. Dividing the full $65 by 0.25 gives 260 and pretends the daily fee is free. The 160 uses a wrong base fee of $25, and 10 comes from dividing 25 by 2.5 in a decimal slip.

Answer: C
Work inside the bars first: 7 - 12 = -5, and |-5| = 5. Also |-3| = 3. Then 5 - 3 = 2. Adding the two absolute values gives 8, which ignores the subtraction between them. The result -8 flips both signs incorrectly, and -2 comes from treating |7 - 12| as -5 rather than its distance from zero, which is always nonnegative.

Answer: C
An absolute value of 11 means the inside equals 11 or -11. Case 1: 2x - 5 = 11 gives 2x = 16, x = 8. Case 2: 2x - 5 = -11 gives 2x = -6, x = -3. Both check: |16 - 5| = 11 and |-6 - 5| = 11. Keeping only one case misses half the solution set, and the pair 3 and -8 fails the check since |2(3) - 5| = 1, not 11.

Answer: B
Subtract 2l from both sides: P - 2l = 2w. Divide by 2: w = (P - 2l)/2, which can also be written as P/2 - l. Test with l = 6 and w = 4, so P = 20: (20 - 12)/2 = 4, correct. Adding 2l instead gives (20 + 12)/2 = 16, multiplying by 2 gives 28, and skipping the division gives 8; none of them return the true width 4.

Answer: C
Let the integers be n, n + 1, and n + 2. Then 3n + 3 = 87, so 3n = 84 and n = 28. The integers are 28, 29, 30, and the largest is 30. Check: 28 + 29 + 30 = 87. The middle integer 29 equals 87 ÷ 3, a common stopping point, and 28 is the smallest. The value 31 would make the sum 90, which is too big.

Answer: A
Let a be adult tickets, so 24 - a are child tickets: 9a + 5(24 - a) = 160. Expand: 9a + 120 - 5a = 160, so 4a = 40 and a = 10. Check: 10 adults bring $90 and 14 children bring $70, totaling $160. The 14 is the number of child tickets, not adult ones. Selling 12 of each would earn $168, and 8 adults would earn only $152.

Answer: A
For negative numbers, the value farthest from zero is the smallest. Compare magnitudes: 3.55 > 3.5 > 3.05 > 3.005, so -3.55 is the coldest reading. Choosing -3.005 confuses more decimal digits with a larger magnitude, when in fact -3.005 is the warmest of the four because it is closest to zero.

Answer: D
Use distance = rate × time, so time = distance ÷ rate: 288 ÷ 64 = 4.5 hours. Check: 64 × 4.5 = 288. The value 4.8 comes from dividing 288 by 60, using minutes-per-hour instead of his actual speed. The whole numbers 4 and 5 are estimates that fail the check, since 64 × 4 = 256 falls short and 64 × 5 = 320 overshoots.

Answer: C
An average of 85 over four exams needs a total of 4 × 85 = 340 points. Her current total is 82 + 76 + 91 = 249, so she needs 340 - 249 = 91. Check: (249 + 91)/4 = 85 exactly. Scoring the target 85 itself gives an average of only 83.5 because her earlier scores drag it down, and 88 still falls short at 84.25. A 95 works but is more than the minimum.

Answer: C
In -4^2 the exponent applies before the negative sign: -(4^2) = -16. In (-4)^2 the parentheses make the base -4, so (-4)^2 = +16. The sum is -16 + 16 = 0. Reading both terms as positive gives 32, reading both as negative gives -32, and 16 keeps only one term. The parentheses are what change the base of the exponent.

Answer: D
Slope is the change in y over the change in x: (13 - 5)/(6 - 2) = 8/4 = 2. The fraction 1/2 flips the ratio and measures run over rise instead. The value -2 subtracts the coordinates in opposite orders, mixing up the direction of change, and 8 is only the change in y before dividing by the change in x.

Answer: D
In a linear model the coefficient of the variable is the rate of change, so each extra mile adds $0.12 to the fare: going from 100 to 101 miles raises the cost from $57.00 to $57.12. The flat fee is the constant 45, not 0.12. A 12-mile trip costs 45 + 1.44 = $46.44, and the first mile costs the same $0.12 on top of the base as every other mile.

Answer: A
Use y = 3x + b and substitute the point: -1 = 3(2) + b, so -1 = 6 + b and b = -7. The line is y = 3x - 7; checking, 3(2) - 7 = -1. The intercept 5 comes from computing -1 + 6 with the wrong sign when moving the 6. Using the point's y-value as the intercept gives y = 3x - 1, which passes through (0, -1) instead of (2, -1).

Answer: D
The slope is (10 - 4)/(3 - 1) = 6/2 = 3. Substitute (1, 4): 4 = 3(1) + b, so b = 1 and the line is y = 3x + 1. Both points check: 3(3) + 1 = 10. The intercept -1 is a sign slip. The equation with slope 1/3 uses run over rise. The line y = 2x + 2 does pass through (1, 4) but gives 2(3) + 2 = 8, not 10, so it misses the second point.

Answer: D
On the x-axis every point has y = 0. Substitute: 2x + 3(0) = 12, so 2x = 12 and x = 6, giving the point (6, 0). The point (0, 4) is the y-intercept, found by setting x = 0 instead. The point (4, 0) swaps the roles of the two coefficients, and (0, 6) puts the correct x-value on the wrong axis.

Answer: B
Parallel lines share the same slope, so the new line keeps slope 2. Since it passes through (0, 5), its y-intercept is 5, giving y = 2x + 5. The slope -1/2 is the perpendicular slope, not the parallel one. The line y = 2x - 3 is the original line itself, which passes through (0, -3), and y = -2x + 5 reverses the direction of the line entirely.

Answer: A
Perpendicular slopes are negative reciprocals: flip 3/4 to 4/3 and change the sign, giving -4/3. Check: (3/4) × (-4/3) = -1, which confirms perpendicularity. The slope 4/3 flips without negating, -3/4 negates without flipping, and 3/4 would make the channel parallel to the path rather than perpendicular.

Answer: C
Substitute y = 2x - 1 into the second equation: 3x + 2x - 1 = 14, so 5x = 15 and x = 3, then y = 2(3) - 1 = 5. Both equations check: 3(3) + 5 = 14. The point (5, 3) swaps the coordinates and fails y = 2x - 1. The points (2, 3) and (4, 7) each lie on the first line but give 9 and 19 in the second equation, so they satisfy only half the system.

Answer: B
Let the numbers be a and b with a + b = 41 and a - b = 9. Adding the equations gives 2a = 50, so a = 25 and b = 16. Check: 25 + 16 = 41 and 25 - 16 = 9. The 16 is the smaller number. The value 32 comes from adding 41 and 9 then dividing incorrectly by absorbing the difference twice, and 20.5 is just half the sum, which ignores the difference condition.

Answer: D
Substitute 4 for every x: f(4) = 4^2 - 3(4) + 2 = 16 - 12 + 2 = 6. The value 30 comes from flipping the middle sign to 16 + 12 + 2, the same slip as evaluating at -4. The value -2 treats x^2 as 2x, computing 8 - 12 + 2, and 2 groups the last two terms as 16 - (12 + 2), subtracting the constant instead of adding it.

Answer: A
The y-values climb 6 for every 2 steps in x, so the slope is 6/2 = 3, and the table starts at y = 5 when x = 0, giving y = 3x + 5. All three rows check: 3(2) + 5 = 11 and 3(4) + 5 = 17. The slope 6 uses the jump in y without dividing by the jump in x; it predicts y = 17 at x = 2. The other forms misplace the sign of 5 or swap the slope and intercept.

Answer: A
A fraction is undefined when its denominator equals zero: x - 4 = 0 at x = 4, so 4 must be excluded and every other real number works. Excluding -4 solves x + 4 = 0, the wrong sign. Excluding 0 confuses the input x with the denominator value, and allowing all real numbers overlooks the division entirely, since g(4) would require dividing by zero.

Answer: A
A square root needs a nonnegative radicand: 2x + 6 ≥ 0, so 2x ≥ -6 and x ≥ -3. At the boundary, f(-3) = √0 = 0, which is defined, so -3 is included. Requiring x ≥ 3 drops the sign of the boundary. Using a strict inequality x > -3 wrongly excludes the endpoint, and x ≤ -3 points the interval in the direction where the radicand is negative.

Answer: A
Break-even means cost equals revenue: 500 + 4x = 9x. Subtract 4x: 500 = 5x, so x = 100. Check: costs are 500 + 400 = 900 and revenue is 9(100) = 900. The 56 comes from dividing 500 by the price 9 alone, and 125 divides by the cost 4 alone; both ignore that each sale nets only 9 - 4 = 5 dollars toward the fixed cost. The 500 is the fixed cost, not a quantity.

Answer: C
Work from the inside out: g(3) = 3^2 = 9, then f(9) = 2(9) + 1 = 19. Composing in the reverse order gives g(f(3)) = g(7) = 49, a different composition. The value 36 comes from doubling 3 first and then squaring, which is (2 × 3)^2, and 21 adds 3 instead of 1 after doubling 9. Order matters when one function feeds into another.

Answer: D
When multiplying powers with the same base, add the exponents: x^5 · x^3 = x^(5+3) = x^8. Test x = 2: 32 × 8 = 256 = 2^8. Multiplying the exponents gives x^15, which is the rule for a power raised to a power. Subtracting gives x^2, the rule for division, and 2x^8 invents a coefficient that the product never had.

Answer: C
Raise every factor inside to the fourth power: 2^4 = 16 and (a^3)^4 = a^12, so the result is 16a^12. Test a = 2: (2 × 8)^4 = 65,536 and 16 × 4,096 = 65,536. The 8 uses 2^3 instead of 2^4, the exponent 7 adds 3 + 4 instead of multiplying, and 2a^12 forgets to raise the coefficient at all.

Answer: A
Subtract exponents when dividing like bases: y^(4-9) = y^-5, and a negative exponent means a reciprocal, so the answer is 1/y^5. Test y = 2: 16/512 = 1/32 = 1/2^5. Writing y^5 subtracts in the wrong order, 1/y^13 adds the exponents, and -y^5 misreads the negative exponent as a negative sign on the expression itself.

Answer: C
Any nonzero base to the zero power is 1, and 4^(-1) is the reciprocal 1/4. So the sum is 1 + 1/4 = 5/4. Getting 1/4 treats 4^0 as 0, but the zero exponent gives 1, not 0. Getting -3 reads 4^(-1) as -4, confusing a negative exponent with a negative number, and 3/4 subtracts the reciprocal instead of adding it.

Answer: B
Multiply the coefficients: 3 × 2 = 6. Add the exponents of ten: 4 + (-6) = -2. The product is 6 × 10^-2, which equals 0.06. Getting 10^2 drops the negative sign when adding exponents, 10^-24 multiplies the exponents instead of adding them, and a coefficient of 5 comes from adding 3 + 2 rather than multiplying.

Answer: C
Combine matching powers: 3x^2 + x^2 = 4x^2, then -2x + 4x = 2x, then 5 - 7 = -2, giving 4x^2 + 2x - 2. Check x = 2: (12 - 4 + 5) + (4 + 8 - 7) = 18, and 16 + 4 - 2 = 18. Writing -2x keeps the first middle term's sign without adding 4x, dropping to 3x^2 forgets the second x^2, and +2 flips the constant subtraction.

Answer: A
Distribute the minus sign to every term of the second polynomial: 5x^2 - 2x^2 = 3x^2, x - (-4x) = 5x, and -3 - 6 = -9, so the difference is 3x^2 + 5x - 9. Check x = 1: (5 + 1 - 3) - (2 - 4 + 6) = 3 - 4 = -1, and 3 + 5 - 9 = -1. Subtracting only the first term's sign gives -3x + 3, and adding the squared terms gives 7x^2.

Answer: A
Multiply every pair of terms: x·x = x^2, x·(-4) = -4x, 6·x = 6x, and 6·(-4) = -24. The middle terms combine to 2x, giving x^2 + 2x - 24. Check x = 5: 11 × 1 = 11 and 25 + 10 - 24 = 11. The version with -2x subtracts the middle terms in the wrong order, +24 misses that 6 times -4 is negative, and x^2 - 24 skips the middle terms entirely.

Answer: B
Use (a - b)^2 = a^2 - 2ab + b^2 with a = 2x and b = 5: 4x^2 - 2(2x)(5) + 25 = 4x^2 - 20x + 25. Check x = 3: (6 - 5)^2 = 1 and 36 - 60 + 25 = 1. Writing 4x^2 + 25 squares each term but forgets the middle product, 4x^2 - 25 is the pattern for (2x - 5)(2x + 5), and -10x uses only one copy of the cross term instead of two.

Answer: B
Look for two numbers that multiply to -24 and add to -5: they are -8 and +3, so the factorization is (x - 8)(x + 3). Expanding confirms x^2 - 5x - 24. Swapping the signs gives a middle term of +5x. The pair -6 and +4 adds to -2, and the pair -12 and +2 adds to -10, so those products produce the wrong middle terms even though each multiplies to -24.

Answer: D
Pull out the greatest common factor first: 2(x^2 + 5x + 6). Then factor the trinomial with numbers that multiply to 6 and add to 5, namely 2 and 3: 2(x + 2)(x + 3). Expanding restores 2x^2 + 10x + 12. The pair 1 and 6 adds to 7 and gives a middle term of 14x. The other candidates expand to x^2 + 8x + 12 and 2x^2 + 11x + 12, both mismatches.

Answer: A
This is a difference of squares, since 9x^2 = (3x)^2 and 49 = 7^2: it factors as (3x - 7)(3x + 7). Expanding gives 9x^2 + 21x - 21x - 49 = 9x^2 - 49. The square (3x - 7)^2 expands to 9x^2 - 42x + 49 with the wrong sign and an unwanted middle term. The other pairings scatter the 9 and 49 incorrectly and produce leftover x-terms.

Answer: D
Group in pairs: x^2(x + 3) - 4(x + 3) = (x + 3)(x^2 - 4). The factor x^2 - 4 is itself a difference of squares, so the complete factorization is (x + 3)(x + 2)(x - 2). Check x = 1: 1 + 3 - 4 - 12 = -12 and 4 × 3 × (-1) = -12. The version with x^2 + 4 has the wrong sign and does not factor further, and the others flip a sign or square a factor incorrectly.

Answer: C
Area is the product of the sides: (x + 5)(2x - 3) = 2x^2 - 3x + 10x - 15 = 2x^2 + 7x - 15. Check x = 4: 9 × 5 = 45 and 32 + 28 - 15 = 45. The middle term -7x subtracts the partial products in the wrong order, +13x adds 3x and 10x instead of -3x and 10x, and dropping the middle term entirely leaves 2x^2 - 15.

Answer: B
Substitute carefully: (-2)^3 = -8 and -2(-2)^2 = -2(4) = -8, so P(-2) = -8 - 8 + 4 = -12. The value 4 flips the sign of the middle term, computing -8 + 8 + 4, which is the same slip as evaluating the second term with a positive square. The value 20 makes both terms positive, 8 + 8 + 4, and -4 comes from an addition error at the last step.

Answer: B
Factor the numerator as a difference of squares: x^2 - 9 = (x + 3)(x - 3). The common factor x + 3 cancels, leaving x - 3. Check x = 5: (25 - 9)/8 = 2 and 5 - 3 = 2. The answer x + 3 cancels the wrong factor, x^2 - 3 subtracts 3 only from the constant under the square, and x - 9 divides just the constants instead of factoring.

Answer: D
Factor top and bottom: (x + 2)(x + 3) over (x - 2)(x + 2). Cancel the common factor x + 2 to get (x + 3)/(x - 2). Check x = 3: (9 + 15 + 6)/(9 - 4) = 6 and 6/1 = 6. Keeping x + 2 in the denominator cancels the wrong factor. Crossing out the x^2 terms to leave (5x + 6)/(-4) is not valid, because only whole factors cancel, never individual terms.

Answer: C
Multiply across: 3x^2 / 6x. Divide top and bottom by 3x: the result is x/2. Check x = 4: (3/4)(16/6) = 2 and 4/2 = 2. The answer 2/x inverts the result, x^2/2 cancels the 3 into the 6 but forgets to reduce x^2 by the x below, and 1/2 cancels the x^2 completely even though only one factor of x appears in the denominator.

Answer: B
Use the common denominator x(x + 1): the first fraction becomes 2(x + 1) and the second becomes 3x over that denominator. Add the tops: 2x + 2 + 3x = 5x + 2, giving (5x + 2)/(x(x + 1)). Check x = 1: 2 + 3/2 = 3.5 and 7/2 = 3.5. Adding tops and bottoms separately gives 5/(2x + 1), multiplying the tops gives 6, and 5x + 3 drops the 2 from distributing.

Answer: D
Cross-multiply: 3(x + 2) = 4(x - 1), so 3x + 6 = 4x - 4. Subtract 3x and add 4: x = 10. Check: 12/4 = 3 and 9/3 = 3. The answer -11 comes from cross-multiplying the wrong pairs, 4(x + 2) = 3(x - 1), which pairs each numerator with its own denominator instead of the opposite one. The small values 2 and -2 fail the check outright.

Answer: A
Multiply everything by x - 3: x = 2(x - 3) + 3 = 2x - 3, so x = 3. But x = 3 makes the original denominators zero, so it must be rejected, leaving no solution. Accepting x = 3 ignores the domain restriction. Testing x = 6 gives 6/3 = 2 on the left but 2 + 1 = 3 on the right, and x = -3 gives 1/2 on the left but 3/2 on the right, so neither works.

Answer: A
Factor out the largest perfect square: 72 = 36 × 2, so √72 = √36 × √2 = 6√2, about 8.49. Writing 2√6 swaps the roles of the root and the leftover factor and equals about 4.9. The form 36√2 forgets to take the square root of 36, and 8√3 equals √192, which comes from mixing up the factor pair.

Answer: B
First simplify √27 = √(9 × 3) = 3√3. Then add like radicals: 3√3 + 2√3 = 5√3, about 8.66. Adding the radicands to get √39 treats unlike quantities as addable inside the root, which is never valid. The answer 6√3 multiplies 3 by 2 instead of adding, and 3√6 moves the 2 under the radical incorrectly.

Answer: B
Multiply coefficients and radicands separately: 2 × 4 = 8 and √3 × √6 = √18. Since √18 = 3√2, the product is 8 × 3√2 = 24√2, about 33.9. The answer 8√3 keeps one radicand and ignores the other, 12√2 uses the right radical but the wrong coefficient, and 24√3 carries the 3 under the radical after it has already been extracted.

Answer: A
Multiply top and bottom by √3: (6√3)/(√3 × √3) = 6√3/3 = 2√3, about 3.46. The value 6√3 multiplies the numerator but forgets to divide by the new denominator 3. The answer 3√2 shuffles the numbers into the wrong positions, and √3/2 inverts the final division, giving a value less than 1 when the original fraction is clearly bigger than 3/2.

Answer: B
Square both sides to remove the radical: x + 5 = 49. Subtract 5: x = 44. Check: √(44 + 5) = √49 = 7. The answer 49 squares correctly but forgets to subtract the 5. The answer 2 comes from computing 7 - 5 without squaring anything, and 54 adds the 5 after squaring instead of subtracting it from both sides.

Answer: D
Square both sides: 2x - 1 = (x - 2)^2 = x^2 - 4x + 4, which rearranges to x^2 - 6x + 5 = 0 and factors as (x - 1)(x - 5) = 0. Test each candidate: for x = 5, √9 = 3 and 5 - 2 = 3, valid. For x = 1, √1 = 1 but 1 - 2 = -1, and a square root cannot equal a negative number, so 1 is extraneous. Only x = 5 survives, so the pair, the lone 1, and no-solution are all wrong.

Answer: C
A cube root of a negative number is negative, because an odd power keeps the sign: (-4)^3 = -64, so ∛(-64) = -4. The answer 4 drops the sign, but 4^3 = +64. The answer -8 halves the number instead of taking a root, since (-8)^3 = -512. Unlike square roots, cube roots of negatives are real, so declaring it not real confuses even and odd roots.

Answer: D
A fractional exponent means root then power: 16^(3/4) = (fourth root of 16)^3 = 2^3 = 8. The answer 12 multiplies 16 by 3/4, treating the exponent like a factor. The answer 64 takes the square root instead of the fourth root, computing (√16)^3 = 4^3. The answer 2 stops after the fourth root and never applies the cube.

Answer: C
Add the rates: the first pipe empties 1/6 of the tank per hour and the second 1/3 per hour, together 1/6 + 2/6 = 1/2 of the tank per hour. So the job takes 2 hours. Averaging the times to get 4.5 ignores that rates, not times, combine. Adding the times to get 9 would mean cooperation is slower than either pipe alone, and 1/2 is the combined hourly rate, not the time.

Answer: C
Isolate the square: 3x^2 = 48, so x^2 = 16. Taking the square root of both sides gives both roots, x = 4 and x = -4, since each squares to 16. Keeping only the positive root loses a solution: (-4)^2 = 16 as well. The pair with 16 forgets to take the square root at the end, and the pair with 8 divides 48 by 6 instead of 3.

Answer: C
Find numbers multiplying to +12 and adding to -7: they are -3 and -4, so the equation factors as (x - 3)(x - 4) = 0 and the solutions are x = 3 and x = 4. Both check: 9 - 21 + 12 = 0 and 16 - 28 + 12 = 0. The all-negative pair flips the factor signs, since (x + 3)(x + 4) would need +7x. Testing x = 2 gives 4 - 14 + 12 = 2, not 0, ruling out the 2-and-6 pair.

Answer: A
With a = 2, b = 3, c = -5, the discriminant is 3^2 - 4(2)(-5) = 9 + 40 = 49, and √49 = 7. Then x = (-3 ± 7)/4, giving x = 4/4 = 1 or x = -10/4 = -5/2. Both check in the equation: 2 + 3 - 5 = 0. The other pairs flip signs on one or both roots or forget to divide by 2a = 4; for instance 2(2)^2 + 3(2) - 5 = 9, not 0.

Answer: A
The discriminant decides: b^2 - 4ac = 16 - 4(1)(7) = 16 - 28 = -12. A negative discriminant means the square root in the quadratic formula is not real, so the equation has no real solutions. Two real solutions would need a positive discriminant and one repeated solution would need exactly zero. The discriminant settles the count by itself, so graphing is unnecessary.

Answer: A
The x-coordinate of the vertex is -b/(2a) = 6/2 = 3. Substitute back: y = 9 - 18 + 11 = 2, so the vertex is (3, 2). Nearby inputs confirm a minimum there: at x = 2 and x = 4, y = 3. The point (-3, 38) uses the wrong sign in -b/(2a). The point (6, 11) doubles the x-coordinate, and (3, 11) grabs the constant term instead of evaluating y at x = 3.

Answer: D
The peak occurs at t = -b/(2a) = -64/(-32) = 2 seconds. The height there is h = -16(4) + 64(2) = -64 + 128 = 64 feet. One second earlier or later the ball is at 48 feet, confirming 64 is the top; that 48 is the height at t = 1, not the maximum. The 128 counts only the 64t part at t = 2, and 32 halves the true maximum.

Answer: C
Vertex form shows the vertex directly: (x - 1)^2 is never negative and is multiplied by -2, so y is largest when x = 1, giving y = 8. At x = 0 or x = 2, y drops to 6, which confirms 8 is the peak; the 6 is a nearby output, not the maximum. The 1 is the x-coordinate where the maximum happens, and -2 is the stretch factor that opens the parabola downward.

Answer: C
In 12 hours there are 12 ÷ 3 = 4 doubling periods, so the count is 400 × 2^4 = 400 × 16 = 6,400 cells. Using 3 doublings gives 3,200 and miscounts the periods. Multiplying 400 by 12 gives 4,800, which treats doubling as linear growth. And 1,600 multiplies by 4, the number of periods, instead of by 2 raised to that number.

Answer: B
Each year the truck keeps 80% of its value: 24,000 × 0.8 = 19,200 after one year, and 19,200 × 0.8 = 15,360 after two. The $14,400 subtracts 20% twice from the original, 24,000 × 0.6, but the second year's loss applies to the smaller value, not the purchase price. The $19,200 stops after one year, and $9,600 keeps only 40% in a single step.

Answer: A
The logarithm asks: 2 raised to what power gives 32? Since 2^5 = 32, the answer is 5. Getting 16 divides 32 by 2, treating the log like division. Getting 4 stops one doubling short, since 2^4 = 16, and 6 overshoots, since 2^6 = 64. Listing powers of 2 up to the target is a quick check: 2, 4, 8, 16, 32.

Answer: C
Rewrite in exponential form: the base 3 raised to the exponent 4 equals x, so x = 3^4 = 81. Check: log base 3 of 81 is 4 because 3 × 3 × 3 × 3 = 81. The answer 12 multiplies 3 × 4, confusing the log with a product. The answer 64 computes 4^3, swapping the base and the exponent, and 7 adds them.

Answer: D
The product rule combines the logs: log 8 + log 125 = log(8 × 125) = log 1,000. Since 10^3 = 1,000, the value is 3. Writing log 133 adds the numbers inside the logs, but the rule multiplies them, never adds. The answer 1,000 stops at the product and forgets to take its log, and 8 just repeats a number from the problem.

Answer: D
Write both sides with base 2: 32 = 2^5, so 2^(x+1) = 2^5. Equal bases force equal exponents: x + 1 = 5, giving x = 4. Check: 2^5 = 32. The answer 5 solves for the whole exponent but forgets to subtract the 1. The answer 16 halves 32 as if dividing out one factor of 2 solved the equation, and 3 subtracts 2 instead of 1.

Answer: C
Both sides are powers of 3: 9 = 3^2 and 27 = 3^3, so (3^2)^x = 3^3 means 3^(2x) = 3^3. Then 2x = 3 and x = 3/2. Check: 9^(3/2) = (√9)^3 = 27. The answer 3 divides 27 by 9, treating the equation as multiplication. The reciprocal 2/3 solves 3x = 2 after swapping the exponents, and x = 2 gives 9^2 = 81, which overshoots.

Answer: B
Combine with the product rule: log base 2 of x(x - 2) = 3, so x(x - 2) = 2^3 = 8. Expand: x^2 - 2x - 8 = 0, which factors as (x - 4)(x + 2) = 0, giving candidates 4 and -2. Logarithms require positive inputs, and x = -2 would ask for the log of a negative number, so only x = 4 is valid: 4 × 2 = 8 checks. Keeping both candidates, or only -2, ignores the domain, and x = 8 solves x(x - 2) = 8 incorrectly since 8 × 6 = 48.

Answer: A
Sine is opposite over hypotenuse. The side opposite the angle measures 3 and the hypotenuse is 5, so the sine is 3/5. The ratio 4/5 is the cosine, adjacent over hypotenuse. The ratio 3/4 is the tangent, opposite over adjacent, and 5/3 inverts the sine, which would exceed 1 and is impossible for a sine value.

Answer: A
Cosine is adjacent over hypotenuse: cos θ = 8/17. The Pythagorean theorem gives the other leg: √(17^2 - 8^2) = √225 = 15. The ratio 15/17 is sin θ, using the opposite leg. The ratio 8/15 is adjacent over opposite, the reciprocal of the tangent, and 17/8 flips the cosine, producing a value greater than 1, which no cosine can be.

Answer: A
The tangent of the ramp angle is rise over run: tan θ = 6/(6√3) = 1/√3, which is the exact tangent of 30 degrees. A 60-degree angle has tangent √3, the reciprocal, and comes from putting the run over the rise. A 45-degree angle would need an equal rise and run, and 15 degrees halves the correct angle without justification.

Answer: B
In a 30-60-90 triangle the side opposite the 30-degree angle is half the hypotenuse, so sin 30 = 1/2. The value √3/2 is cos 30, the ratio for the longer leg. The value √2/2 belongs to the 45-degree angle, where both legs are equal. And 1/3 confuses the 30 in the angle with a fraction; checking, sin 30 = 0.5 exactly.

Answer: B
The angle 210 degrees lies in the third quadrant, 30 degrees past 180, so its reference angle is 30 degrees. Cosine of 30 is √3/2, and in the third quadrant cosine is negative: cos 210 = -√3/2. The positive √3/2 ignores the quadrant sign. The values with 1/2 use the sine of the reference angle instead of the cosine.

Answer: A
From sin^2 θ + cos^2 θ = 1: cos^2 θ = 1 - 9/25 = 16/25, so cos θ = 4/5, positive because θ is acute. This matches the 3-4-5 right triangle. The ratio 3/4 is the tangent of the same angle, not the cosine. The value 5/4 is greater than 1, impossible for a cosine, and 2/5 comes from subtracting 3/5 from 1 without squaring.

Answer: D
Multiply by π/180: 150 × π/180 = 15π/18 = 5π/6 radians. As a check, 5π/6 is just short of π, matching an angle just short of 180 degrees. The value 5π/3 equals 300 degrees, doubling the answer. The value 3π/4 is 135 degrees, a different common angle, and 5π/12 equals 75 degrees, half the correct measure.

Answer: B
The pole, ground, and wire form a right triangle with legs 24 and 10. By the Pythagorean theorem, the wire is √(24^2 + 10^2) = √(576 + 100) = √676 = 26 feet. Adding the legs gives 34, which ignores the right-angle relationship. About 21.8 comes from subtracting the squares as if the wire were a leg, and 25 misremembers the 24-10-26 triple as 3-4-5 scaled wrong.

Answer: C
The Pythagorean identity gives 1 - cos^2 θ = sin^2 θ. Dividing by sin θ leaves sin^2 θ / sin θ = sin θ. A numeric check at θ = 60 degrees: (1 - 0.25)/(√3/2) = 0.75/0.866 ≈ 0.866 = sin 60. The result cos θ picks the wrong identity partner, tan θ would need a cosine in the denominator, and 1 would require the numerator and denominator to be identical, which they are not.

Answer: B
The height satisfies tan 60 = height/shadow, so height = 10 × tan 60 = 10√3 ≈ 17.3 feet. The value 10√3/3 ≈ 5.8 divides by √3 instead of multiplying, which computes the shadow for a 30-degree sun instead. The 20 doubles the shadow without using the tangent ratio at all, and 5 halves it, mixing up which side is opposite the 60-degree angle.

Answer: A
Look at the tens digit to round to the nearest hundred. The number 47,362 has a 6 in the tens place, and 6 is 5 or more, so the hundreds digit rounds up: 47,400. Rounding down gives 47,300, rounding to the nearest thousand gives 47,000, and rounding to the nearest ten gives 47,360.

Answer: C
Divide the total length by the piece length: 8.4 ÷ 0.35. Multiply both numbers by 100 to clear the decimals: 840 ÷ 35 = 24 pieces. The other results come from misplacing the decimal point when clearing it; 2.4 and 240 are each off by a factor of ten, and 0.24 is off by a factor of one hundred.

Answer: D
The next shared departure is the least common multiple of 12 and 18. Multiples of 12: 12, 24, 36. Multiples of 18: 18, 36. The LCM is 36 minutes. The number 6 is the greatest common factor, not the LCM; 216 is the product 12 × 18, which overshoots; and 30 is just the sum of the two intervals.

Answer: A
A fraction of a fraction means multiply: 2/3 × 3/5 = 6/15 = 2/5 of all students. Adding the fractions gives 19/15 = 1 4/15, which is more than the whole school and cannot be right. Dividing 2/3 by 3/5 gives 1 1/9, and dividing in the other order gives 9/10; both mistake the word of for division.

Answer: C
Compare place by place after lining up the decimal points: 0.8 = 0.800, which beats 0.480, 0.405, and 0.058 in the tenths place. A longer decimal is not automatically larger; 0.405 has three digits but is smaller than 0.48, and 0.058 has a zero in the tenths place, making it the smallest of the four.

Answer: D
Multiply the monthly charge first, then add the one-time fee: 22 × 7 = 154, and 154 + 35 = 189 dollars. Adding the fee to the monthly rate before multiplying gives (35 + 22) × 7 = 399, which charges the enrollment fee every month. Forgetting the fee entirely gives $154, and multiplying the fee by 7 while adding one month gives $267.

Answer: B
Add the scores and divide by how many there are: 78 + 85 + 92 + 69 = 324, and 324 ÷ 4 = 81. The value 81.5 is the median (the average of the middle two scores 78 and 85), not the mean. The 324 is just the sum before dividing, and 80 comes from an addition slip.

Answer: A
The decimal 0.375 means 375 thousandths: 375/1000. Divide the top and bottom by 125: 375 ÷ 125 = 3 and 1000 ÷ 125 = 8, so 0.375 = 3/8 inch. Misplacing the decimal by one position leads to 3/80, the fraction 5/8 equals 0.625, and 3/7 is a repeating decimal near 0.4286, so none of those match.

Answer: A
Three shelves use 3 × 2 3/4 = 8 1/4 feet, since 3 × 2 = 6 and 3 × 3/4 = 2 1/4. Then 12 - 8 1/4 = 3 3/4 feet remain. Subtracting only one shelf gives 9 1/4 feet, and the other values come from multiplying or borrowing incorrectly when handling the mixed number.

Answer: A
Divide 137 by 12: since 12 × 11 = 132 and 12 × 12 = 144 is too big, the farm fills 11 cartons and has 137 - 132 = 5 eggs left. The 11 is the number of full cartons, not the leftover count. The values 7 and 1 come from stopping at 12 × 10 = 120 incorrectly or from subtraction slips.

Answer: C
The discount is 35% of 240: 0.35 × 240 = 84 dollars off. The value $156 is the sale price after the discount (240 - 84), not the discount itself. Subtracting 35 as if it were dollars gives $205, and $8.40 comes from misplacing the decimal when converting 35% to 0.35.

Answer: D
Percent change compares the change to the original amount: (92 - 80) / 80 = 12/80 = 0.15, a 15% increase. Dividing the $12 change by the new price 92 gives about 13%, but percent change always uses the starting value. The raw $12 is a dollar change, not a percent, and 115% is the ratio 92/80 before subtracting the original 100%.

Answer: D
Paying after a 20% discount means paying 80% of the original: 0.80p = 68, so p = 68 ÷ 0.80 = 85 dollars. Adding 20% of 68 back gives $81.60, but the 20% was taken from the original price, not from $68. Taking 80% of 68 gives $54.40, which discounts the price twice, and $88 is just $20 added incorrectly.

Answer: C
Set up a proportion: 3/24 = x/60, so x = 3 × 60 ÷ 24 = 180 ÷ 24 = 7 1/2 cups. Flipping the proportion gives 3 × 24 ÷ 60 = 1 1/5 cups, which is less flour for more cookies and fails a sense check. The answers 8 and 5 come from rounding 60/24 to a whole number before multiplying.

Answer: B
A unit rate divides the first quantity by the second: 322 ÷ 14 = 23 miles per gallon, since 14 × 23 = 322. The nearby values 22 and 24 come from estimating instead of dividing exactly; checking by multiplication rules them out. Dividing 14 by 322 gives about 0.04, which is gallons per mile, the inverted rate.

Answer: A
The ratio 3:5 has 3 + 5 = 8 parts, so each part is 96 ÷ 8 = 12 dollars. The larger share is 5 × 12 = 60 dollars. The smaller share is 3 × 12 = 36 dollars, splitting evenly gives $48 and ignores the ratio, and $32 comes from dividing by 3 instead of using all 8 parts.

Answer: D
The area of a rectangle is length times width: 18 × 12.5 = 225 square feet. The value 61 is the perimeter 2(18 + 12.5), which measures the border, not the surface. Adding the two dimensions gives 30.5, and 450 doubles the area by multiplying by 2 as if computing perimeter and area at once.

Answer: D
Fencing follows the perimeter: P = 2(length + width) = 2(42 + 28) = 2 × 70 = 140 feet. Multiplying 42 × 28 = 1,176 gives the area in square feet, which measures ground coverage rather than border length. The 70 counts only one length and one width, and 112 is four times the width alone.

Answer: C
The area of a circle is π times the radius squared: A = π(6)^2 = 36π square centimeters. The value 12π is the circumference 2πr, a length rather than an area. Using half the radius gives 9π, which treats the 6 as a diameter, and 6π multiplies by the radius without squaring it.

Answer: D
One lap is the circumference: C = πd = π × 50 ≈ 157.1 meters. Using the radius 25 in πd style gives about 78.5, half a lap. Treating 50 as the radius in C = 2πr gives about 314.2, double the true distance. And about 1,963.5 is the area π(25)^2 in square meters, not a walking distance.

Answer: B
Volume of a rectangular box is length × width × height: 8 × 5 × 3.5 = 140 cubic feet. The value 171 is the surface area 2(40 + 28 + 17.5) in square feet, which measures the outside faces rather than the space inside. The 40 is just the floor area 8 × 5, and 16.5 is the sum of the three dimensions.

Answer: B
Complementary angles add to 90 degrees, so the other angle is 90 - 37 = 53 degrees. The value 143 comes from using 180 degrees, which is the rule for supplementary angles, not complementary ones. An angle of 37 would only be right if the two angles were equal, and 63 comes from subtracting from 100 instead of 90.

Answer: C
Angles of a triangle sum to 180 degrees: x + 2x + 3x = 6x = 180, so x = 30. The largest angle is 3x = 90 degrees. The 30 is x itself, the smallest angle, and 60 is the middle angle 2x. Using 360 degrees, the total for a quadrilateral, leads to x = 60 and a largest angle of 120, which is the wrong figure.

Answer: A
Do division and multiplication before addition and subtraction: 6 ÷ 2 = 3 and 4 × 3 = 12. Then work left to right: 18 - 3 + 12 = 27. Subtracting 6 first gives 18, grouping everything after the minus sign gives 3, and working strictly left to right with the 3 applied at the end gives 30; each of those breaks the order of operations.

Answer: D
Use the common denominator 12: 3/4 = 9/12 and 5/6 = 10/12, so 9/12 + 10/12 = 19/12 = 1 7/12 miles. Adding tops and bottoms straight across gives 4/5, keeping the right numerators but multiplying the denominators gives 19/24, and doubling 3/4 gives 1 1/2; none of those follows the common-denominator rule.

Answer: A
Divide the total by the serving size: 4 1/2 ÷ 3/8 = 9/2 × 8/3 = 72/6 = 12 servings. Multiplying 4 1/2 by 3/8 instead of dividing gives 1 11/16, dividing only the whole number 4 and tacking the 1/2 back on gives 11 1/6, and dividing 4 1/2 by 3 gives 1 1/2.

Answer: B
Convert to improper fractions: 5 1/3 = 16/3 = 64/12 and 2 3/4 = 11/4 = 33/12. Then 64/12 - 33/12 = 31/12 = 2 7/12 yards. Subtracting the smaller fraction from the larger one in each column gives 3 5/12, and the other results come from borrowing incorrectly when 1/3 is smaller than 3/4.

Answer: B
Multiply: 12.5 × 3.48 = 43.50, so the fill-up costs $43.50. Since 3.48 has two decimal places and 12.5 has one, the product needs three decimal places before trailing zeros: 43.500. A digit swap produces $43.05, dropping the 8 cents entirely gives $42.50, and misplacing the decimal point gives $435.00.