Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.
Math · Problem-Solving and Data Analysis~2 per testHard tier
Uses the new value as the base for percent change.
Adds two percentages instead of multiplying.
Treats a reverse-percent problem as a forward one.
Question 1Hard
The number y is 200% greater than the number x. The number x is 92% less than 350. What is the value of y?
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Why A is right
It's given that x is 92% less than 350, so x=(1−0.92)(350) = 0.08(350) = 28. It's also given that y is 200% greater than x, so y=(1+2.00)(28) = 3.00(28) = 84.
Why the others are wrong
BThis results from adding the percentages (200% + 92% = 292%) and applying this to 350 directly, then rounding incorrectly.
CThis results from treating '200% greater' as an 80% decrease (confusing 200% with 20%), getting y = (1 - 0.80)(28) = 0.20(28) = 5.6, then making a calculation error.
DThis results from incorrectly applying 200% to 28 as 2.00(28) = 56, missing the +1 in the formula for percent increase.
Question 2Hard
The number x is 25% less than the number y. The number y is 40% greater than 120. What is the value of x?
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Why C is right
It's given that y is 40% greater than 120. Therefore, y=(1+40/100)(120) = 1.40(120) = 168. It's also given that x is 25% less than y. Therefore, x=(1−25/100)(168) = 0.75(168) = 126.
Why the others are wrong
AThis results from incorrectly finding 25% less than 120 instead of finding 25% less than y.
BThis results from incorrectly adding the percentages (40 - 25 = 15) and finding 15% greater than 120, rather than applying the percent changes multiplicatively.
DThis results from incorrectly making x 25% greater than y instead of 25% less than y.
Question 3Hard
72% of the employees at a company work full-time. Of those, 40% have been with the company for more than 5 years. Of the full-time employees with more than 5 years tenure, 35% hold management positions. Which of the following is closest to the percentage of employees at the company who are not full-time managers with more than 5 years tenure?
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Why D is right
Let the total number of employees be x. The full-time employees are 0.72x. Of those, 40% have more than 5 years tenure: 0.40(0.72x)=0.288x. Of those, 35% hold management positions: 0.35(0.288x)=0.1008x=10.08%. Therefore, 100% - 10.08% = 89.92% are not in all three categories.
Why the others are wrong
AThis is the percentage who ARE full-time managers with more than 5 years tenure (72% × 40% × 35% = 10.08%), not the percentage who are NOT.
BThis results from computing an intermediate product such as 72% × 40% = 28.8% and incorrectly treating it as a final answer or complement.
CThis results from treating the percentages additively or taking complements at incorrect stages, such as (100% - 72%) + something = 52.9%.
Question 4Hard
22% of the students in a school play a sport. Of those, 30% also play a musical instrument. Of the students who play a sport and a musical instrument, 60% are also in the debate club. Which of the following is closest to the percentage of the students in the school that are not in the debate club among those who play a sport and a musical instrument?
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Why B is right
Let x be the total number of students. Students who play a sport: 0.22x. Students who play a sport and an instrument: 0.30(0.22x)=0.066x. Students who play a sport, an instrument, and are in debate: 0.60(0.066x)=0.0396x. Students NOT in this group: x−0.0396x=0.9604x, which is 96.04%.
Why the others are wrong
AThis is the percentage of students who ARE in all three categories (sport, instrument, debate), not those who are NOT.
CThis results from computing 100% - (22% + 30% + 60%) / some factor ≈ 88%, treating the percentages additively.
DThis results from a sign error where the student adds 0.0396x instead of subtracting it: x + 0.0396x = 1.0396x, yielding 103.96%.
Question 5Hard
A store increased the price of a television by 25% and then decreased the new price by 40%. The final price is what percent less than the original price?
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Why B is right
Let the original price be P. After a 25% increase, the price becomes 1.25P. After a 40% decrease from this new price, the final price is (1 - 0.40)(1.25P)=0.60(1.25P)=0.75P. The final price is 0.75P, which is 25% less than the original price P.
Why the others are wrong
AThis results from incorrectly adding the percentages: 40% - 25% = 15%.
CThis results from applying the 40% decrease to the original price (getting 0.60P) but then subtracting the 25% increase incorrectly, or from reversing the operations.
DThis results from incorrectly calculating 40% of the increased price as 40% of the original, getting 40% of P as the decrease, then subtracting from the 25% increase base.
Question 6Hard
The value of a certain investment increased by 20% from the end of 2021 to the end of 2022. The value then decreased by 30% from the end of 2022 to the end of 2023. If the value of the investment at the end of 2023 was $16800, what was the value of the investment, in dollars, at the end of 2021?
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Why B is right
Let x be the value at the end of 2021. After a 20% increase, the value at the end of 2022 was 1.20x. After a 30% decrease from that value, the value at the end of 2023 was (0.70)(1.20x)=0.84x. Setting this equal to 16800 gives 0.84x=16800, so x=20000.
Why the others are wrong
AThis results from treating the percent changes as additive rather than multiplicative: subtracting 10% (the net of +20% and -30%) from 16800.
CThis results from reversing the order of operations, applying -30% first then +20%, which gives (1.20)(0.70)(16800) instead of working backwards from the final value.
DThis results from applying the percent changes in reverse with incorrect direction: increasing 16800 by 30% then by 20%, using the wrong base for the compound calculation.
Question 7Hard
48% of the students in a class play a sport. Of those, 25% also play a musical instrument. Of the students who play a sport and a musical instrument, 50% are in the honor society. Which of the following is closest to the percentage of students in the class who are not in all three groups?
Show the answer and explanation
Why D is right
Let the total number of students be x. The students who play a sport are 0.48x. Of those, 25% also play an instrument: 0.25(0.48x)=0.12x. Of those, 50% are in the honor society: 0.50(0.12x)=0.06x=6%. Therefore, 100% - 6% = 94% are not in all three groups.
Why the others are wrong
AThis is the percentage who ARE in all three groups (48% × 25% × 50% = 6%), not the percentage who are NOT.
BThis results from adding the percentages (48% + 25% + 50% = 123%) and incorrectly manipulating them, or treating them additively instead of multiplicatively.
CThis results from computing the complement of some intermediate step incorrectly, such as 100% - 23% = 77%, confusing which complement to take.
Question 8Hard
32% of the books in a library are fiction. Of those, 25% are also bestsellers. Of the fiction bestsellers, 50% are available for checkout. Which of the following is closest to the percentage of books in the library that are not fiction bestsellers available for checkout?
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Why D is right
Let x be the total number of books. Fiction books: 0.32x. Fiction bestsellers: 0.25(0.32x)=0.08x. Fiction bestsellers available: 0.50(0.08x)=0.04x, which is 4%. Books not in this category: 100% - 4% = 96%.
Why the others are wrong
AThis calculates the percentage that are fiction bestsellers available (4%) instead of those that are not.
BThis adds the percentages: 32% - 25% = 7%.
CThis calculates 100% - 0.25(0.32) = 100% - 0.08 = 92%, then rounds to 93%.
Question 9Hard
The number a is 40% greater than the positive number b. The number c is 25% less than a. The number c is how many times b?
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Why A is right
It's given that a is 40% greater than b, so a=1.40b. It's also given that c is 25% less than a, so c=0.75a=0.75(1.40b)=1.05b. Therefore c is 1.05 times b.
Why the others are wrong
BThis results from computing 1 + 0.40 - 0.25 = 1.15, then incorrectly manipulating to get 0.65.
CThis results from computing c = (1 - 0.25)(1 - 0.40)b = 0.75(0.60)b = 0.45b, treating the first percentage change as a decrease, or using 0.40 as the factor for 'less than'.
DThis results from computing c = 0.75a but forgetting to multiply by 1.40: c = 0.75(b) + 0.40b = 1.15b, then rounding or making an error to 1.40, or simply using a = b and getting c = 1.40b directly.
Question 10Hard
The number r is 150% greater than the positive number s. The number t is 60% less than r. The number t is how many times s?
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Why B is right
It's given that r is 150% greater than s. Therefore, r=(1+1.50)s=2.50s. It's also given that t is 60% less than r. Therefore, t=(1−0.60)r=0.40r=0.40(2.50s)=1.00s. Thus t is 1 times s.
Why the others are wrong
AThis results from adding the percentages (150% - 60% = 90%) and incorrectly treating the result as a reduction yielding 0.10s, then computing 0.10s/s = 0.10, or by other additive errors.
CThis is the value of r/s, not t/s. The student correctly found the relationship between r and s but stopped before applying the second transformation.
DThis results from incorrectly treating the 60% less as 60% more when finding t, yielding t = 1.60(2.50s) = 4s.
Question 11Hard
30% of the items in a warehouse are labeled. Of those, 60% are also fragile. Of the labeled fragile items, 40% are also refrigerated. Which of the following is closest to the percentage of the items in the warehouse that are not fragile labeled refrigerated items?
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Why D is right
Let x represent the total items. Then 0.30x are labeled. Of those, 0.60(0.30x)=0.18x are also fragile. Of those, 0.40(0.18x)=0.072x are also refrigerated. Therefore 7.2% are fragile labeled refrigerated items, so 100% - 7.2% = 92.8% are not.
Why the others are wrong
AThis is the percentage that ARE fragile labeled refrigerated items, not the percentage that are NOT.
BThis results from multiplying the first two percentages correctly to get 18%, then stopping, treating that as the final percentage of items that are fragile labeled refrigerated items.
CThis results from adding the percentages (30% + 60% + 40% = 130%) then treating that sum incorrectly, yielding an estimate around 82% remaining.
Question 12Hard
A quantity decreases by 60% and then increases by 75%. The final quantity is what percent of the original quantity?
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Why B is right
Let the original quantity be x. After a 60% decrease: (1 - 0.60)x=0.40x. After a 75% increase from that: (1 + 0.75)(0.40x)=1.75(0.40x)=0.70x. This is 70% of the original.
Why the others are wrong
AThis adds the percentages: 75% - 60% = 15%.
CThis applies the 75% increase to the original instead of to the decreased amount: 0.40x + 0.75x = 1.15x, then incorrectly gets 85%.
A retailer increased the price of an item by 50% and then decreased the new price by 30%. The final price is what percent greater than the original price?
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Why A is right
Let the original price be P. After a 50% increase, the price becomes 1.50P. After a 30% decrease from this new price, the final price is (1 - 0.30)(1.50P)=0.70(1.50P)=1.05P. The final price is 1.05P, which is 5% greater than the original price P.
Why the others are wrong
BThis results from calculating 30% of the increased price incorrectly as a percentage of the original price.
CThis results from incorrectly subtracting the percentages: 50% - 30% = 20%.
DThis results from applying the 30% decrease to the original price (getting 0.70P) and then adding the 50% increase incorrectly.
Question 14Hard
The number h is 90% greater than the number k. The number k is 75% less than 160. What is the value of h?
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Why C is right
It's given that k is 75% less than 160. Therefore, k=(1−0.75)(160) = 0.25(160) = 40. It's also given that h is 90% greater than k. Therefore, h=(1+0.90)(40) = 1.90(40) = 76.
Why the others are wrong
AThis results from adding the percentages (90% + 75% = 165%) and applying 165% as a combined reduction factor instead of compounding the two changes.
BThis is the value of k, not h. The student correctly found k but stopped before applying the 90% increase.
DThis results from incorrectly treating the 75% less as 75% more when finding k, yielding k = 280, then applying the 90% increase to that incorrect value, giving h = 1.90(280) ≈ 532, or from other sign errors leading to this value.
Question 15Hard
56% of the items in a warehouse are boxes. Of those, 30% are labeled fragile. Of the boxes labeled fragile, 25% are also refrigerated. Which of the following is closest to the percentage of items in the warehouse that are not refrigerated fragile boxes?
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Why D is right
Let the total number of items be x. The items that are boxes are 0.56x. Of those, 30% are fragile: 0.30(0.56x)=0.168x. Of those, 25% are refrigerated: 0.25(0.168x)=0.042x=4.2%. Therefore, 100% - 4.2% = 95.8% are not refrigerated fragile boxes.
Why the others are wrong
AThis is the percentage who ARE refrigerated fragile boxes (56% × 30% × 25% = 4.2%), not the percentage who are NOT.
BThis results from computing an intermediate product such as 56% × 30% = 16.8%, then taking 100% - 16.8% - some other error, or treating the percentages additively.
CThis results from computing the complement at an incorrect stage, such as 100% - (30% × 25%) = 100% - 7.5% or other base confusion.
Question 16Hard
The price of an item increased by 25% in January. In February, the new price decreased by 20%. The price after both changes is what percent of the original price?
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Why B is right
Let the original price be P. After a 25% increase, the price becomes 1.25P. After a 20% decrease from this new price, the price becomes (1 - 0.20)(1.25P)=0.80(1.25P)=1.00P, which is 100% of the original price.
Why the others are wrong
AThis results from computing 20% of the original price instead of 20% of the increased price: 125% - 20% = 105%, then an arithmetic error.
CThis results from adding the percentage changes instead of compounding them: 25% - 20% = 5%, so 100% + 5% = 105%.
DThis results from applying only the first increase and ignoring the decrease, or from computing the decrease in the wrong direction.
Question 17Hard
The number w is 200% greater than the positive number v. The number z is 55% less than w. The number z is how many times v?
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Why B is right
It's given that w is 200% greater than v. Therefore, w=(1+2.00)v=3.00v. It's also given that z is 55% less than w. Therefore, z=(1−0.55)w=0.45w=0.45(3.00v)=1.35v. Thus z is 1.35 times v.
Why the others are wrong
AThis results from adding the percentages (200% - 55% = 145%) and treating the result as 1.45, then computing an incorrect ratio or by other additive errors.
CThis is the value of w/v, not z/v. The student correctly found the relationship between w and v but stopped before applying the second transformation.
DThis results from incorrectly treating the 55% less as 55% more when finding z, yielding z = 1.55(3.00v) = 4.65v.
Question 18Hard
In a company, 45% of the employees work in sales. Of those who work in sales, 80% have at least 3 years of experience. Of the sales employees with at least 3 years of experience, 25% speak a second language. What percent of all employees in the company are sales employees with at least 3 years of experience who speak a second language?
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Why A is right
Let the total number of employees be E. Then 0.45E work in sales. Of those, 0.80(0.45E)=0.36E have at least 3 years of experience. Of those, 0.25(0.36E)=0.09E speak a second language. Therefore, 0.09E/E=0.09=9% of all employees fit all three criteria.
Why the others are wrong
BThis results from computing 25% of the 45% who work in sales (0.25 × 45% = 11.25%), using the wrong base by skipping the 80% filter.
CThis results from adding 45% - 80% - 25% in some fashion, or computing only the first two layers: 0.45 × 0.80 = 0.36 = 36%.
DThis results from adding all three percentages instead of multiplying: 45% + 80% + 25% = 150%.
Question 19Hard
The number p is 100% greater than the positive number q. The number r is 40% less than p. The number r is how many times q?
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Why A is right
It's given that p is 100% greater than q, so p=2.00q. It's also given that r is 40% less than p, so r=0.60p=0.60(2.00q)=1.20q. Therefore r is 1.20 times q.
Why the others are wrong
BThis results from computing r = 0.60p but treating p = q (forgetting the 100% increase), so r = 0.60q.
CThis results from computing r = (1 - 0.40)(1 + 0.00)q = 0.60q, then adding 0.20 from misapplying the 100% increase, or computing (1 - 0.40) + 0.20 = 0.80.
DThis results from computing r = 0.60p but then using p = q + 1.00q + some error factor, such as r = 0.60(2q) + 1.20q = 2.40q, or treating 'less than' as 'greater than' in the second step.
Question 20Hard
The number f is 75% greater than the positive number g. The number j is 40% less than f. The number j is how many times g?
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Why B is right
It's given that f is 75% greater than g. Therefore, f=(1+0.75)g=1.75g. It's also given that j is 40% less than f. Therefore, j=(1−0.40)f=0.60f=0.60(1.75g)=1.05g. Thus j is 1.05 times g.
Why the others are wrong
AThis results from adding the percentages (75% - 40% = 35%) and treating the result incorrectly as 0.35 or by other additive errors yielding 0.70.
CThis is the value of f/g, not j/g. The student correctly found the relationship between f and g but stopped before applying the second transformation.
DThis results from incorrectly treating the 40% less as 40% more when finding j, yielding j = 1.40(1.75g) = 2.45g.
Question 21Hard
28% of the students in a school play an instrument. Of those, 35% also sing in the choir. Of the students who play an instrument and sing in the choir, 60% also participate in drama club. Which of the following is closest to the percentage of the students in the school that are not students who play an instrument, sing in the choir, and participate in drama club?
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Why D is right
Let x represent the total number of students. The number who play an instrument is 0.28x. Of those, 35% also sing: 0.35(0.28x)=0.098x. Of those, 60% also do drama: 0.60(0.098x)=0.0588x. Therefore, 5.88% meet all three criteria, so 94.12% do not.
Why the others are wrong
AThis is the percentage that DO meet all three criteria, not the percentage that do NOT.
BThis results from subtracting the sum of the percentages (28% + 35% + 60% = 123%) from 100%, getting -23%, then incorrectly adjusting to 77%.
CThis results from using an incorrect base, applying the 60% to the original instrument-playing students (0.28x) instead of to those who also sing (0.098x).
Question 22Hard
A quantity increases by 25%, then decreases by 20%. Which of the following is closest to the overall percent change in the quantity?
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Why A is right
Let the original quantity be x. After a 25% increase, the quantity becomes 1.25x. Then decreasing by 20% gives (1 - 0.20)(1.25x)=0.80(1.25x)=x. The overall change is 0%, since the final value equals the original.
Why the others are wrong
BThis results from adding the percent changes: 25% - 20% = 5%, which incorrectly treats successive changes as additive.
CThis results from computing the decrease on the wrong base, such as taking 20% of the original x instead of 20% of 1.25x.
DThis results from adding the absolute values of the percentages: 25% + 20% = 45%, confusing the direction of change.
Question 23Hard
The number d is 140% greater than the number e. The number e is 80% less than 350. What is the value of d?
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Why B is right
It's given that e is 80% less than 350. Therefore, e=(1−0.80)(350) = 0.20(350) = 70. It's also given that d is 140% greater than e. Therefore, d=(1+1.40)(70) = 2.40(70) = 168.
Why the others are wrong
AThis is the value of e, not d. The student correctly found e but stopped before applying the 140% increase.
CThis results from incorrectly treating the 80% less as 80% more when finding e, yielding e = 630, then applying the 140% increase to that incorrect value, or from other sign errors.
DThis results from adding the percentages (140% + 80% = 220%) and treating 220% as a combined reduction factor instead of compounding the two changes.
Question 24Hard
32% of the students in a school play an instrument. Of those, 25% also participate in a sport. Of the students who play an instrument and participate in a sport, 50% are also in the debate club. Which of the following is closest to the percentage of students in the school who are not in all three activities?
Show the answer and explanation
Why D is right
Let x represent the total number of students. Then 0.32x play an instrument. Of those, 0.25(0.32x)=0.08x play an instrument and participate in a sport. Of those, 0.50(0.08x)=0.04x are in all three activities. Therefore, the percentage not in all three is (1 - 0.04)(100%) = 96%.
Why the others are wrong
AThis gives the percentage who are in all three activities instead of those who are not.
BThis incorrectly subtracts the individual percentages (100 - 32 + 25 - 50) instead of compounding them.
CThis represents the students who play an instrument and participate in a sport but doesn't account for the final 50% filter.
Question 25Hard
35% of the books in a library are fiction. Of those, 20% are also hardcover. Of the fiction hardcover books, 50% are also bestsellers. Which of the following is closest to the percentage of the books in the library that are not hardcover fiction bestseller books?
Show the answer and explanation
Why B is right
Let x be the total number of books. Fiction books: 0.35x. Fiction hardcover books: 0.20(0.35x)=0.07x. Fiction hardcover bestseller books: 0.50(0.07x)=0.035x. Books that are NOT fiction hardcover bestsellers: x−0.035x=0.965x, which is 96.5%.
Why the others are wrong
AThis is the percentage of books that ARE fiction hardcover bestsellers, not those that are NOT.
CThis results from computing 100% - (35% + 20% + 50%) / some factor, treating the percentages additively rather than multiplicatively.
DThis results from a sign error where the student adds 0.035x instead of subtracting it: x + 0.035x = 1.035x, yielding 103.5%.
These 25 are a sample, not a study plan
These 25 come from a bank of 17,599 questions across all 31 SAT skills and three difficulty tiers. Inside SAT Climb you get the rest of Percentages, a diagnostic that finds which skills are actually costing you points, and a grid that shows what to drill next.
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