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Digital SAT Math · Problem-Solving & Data Analysis

Percentages

Digital SAT Math · Problem-Solving and Data Analysis

Nothing here is hard arithmetic. What costs points is answering a different question than the one asked: a percent of something when the stem asked for a percent change, a change measured against the wrong number, or a price rebuilt by adding back what was taken off. Get the base right and this skill turns into one multiplication.

On the test

DomainProblem-Solving and Data Analysis (score report)
What it looks likeA short real-world sentence — a price, a survey count, a population, a rate — with one number missing
Often asked“What percent of…?”, “By what percent did it increase?”, “What was the original price?”, “Which of the following is closest to…?”
FormatMultiple choice and student-produced response
CalculatorAllowed throughout; the arithmetic is usually calculator-trivial, which is exactly why the trap lives in the setup

Recognition cues: of, is what percent, increased by, marked down, after a discount, originally, percentage points.

Pattern recognition

Four shapes cover nearly every percent item on the test, and the first move is always to decide which one you are looking at.

  1. Percent of — a rate applied to a known whole. “35% of 240 members.”
  2. What percent — two amounts given, the rate is missing. “48 out of 64 points.”
  3. Percent change — a before and an after, and the question is how big the jump was relative to the before.
  4. Reverse — the after is given and the before is missing. “After a 20% discount it costs $68.”

Method

  1. Classify the shape — percent of, what percent, percent change, or reverse. One glance at whether the stem gives you a before-and-after pair settles it.
  2. Name the whole. Write it down. On a change item, the whole is the starting value; on “A is what percent of B”, it is B; on a reverse item, the whole is the thing you are solving for.
  3. Convert the rate to a decimal — move the point two places left. 35%=0.3535\% = 0.35, 6%=0.066\% = 0.06, 140%=1.4140\% = 1.4.
  4. Use the multiplier for any change. An increase of rr means ×(1+r)\times(1+r); a decrease means ×(1−r)\times(1-r). Down 30% is ×0.7\times 0.7, up 8% is ×1.08\times 1.08.
  5. Reverse means divide. If the printed number is the after, divide it by the multiplier. Never subtract the percent back off, and never add it back on.
  6. Answer the asked quantity — a dollar amount, a count, a percent, or a percent change. Then sanity-check the direction: a discount answer must be smaller than the original, a markup answer larger.
Words in the stemWhat it means
is, areequals
ofmultiply — and the number after it is the whole
what percentthe rate is unknown: part÷whole\text{part} \div \text{whole}
increased by 12%×1.12\times 1.12
marked down 30%, 30% off×0.7\times 0.7
after the discount it costs…that is the after — divide
percentage pointssubtract the two rates; do not divide

Worked example 1 — what percent (translate is and of)

Stem. A theater has 480 seats, and 372 of them were filled for a matinee. What percent of the seats were filled?

Step 1 — shape. Two amounts, missing rate: what percent.

Step 2 — name the whole. The seats in the theater, 480. The part is 372.

Step 3 — divide.

372480=0.775=77.5%\frac{372}{480} = 0.775 = 77.5\%

Check. Run it forwards: 0.775×480=3720.775 \times 480 = 372. Under 100%, as a filled-seat count must be. Answer: 77.5%.

Trap watch. 480/372≈1.29480/372 \approx 1.29 inverts the fraction and reports 129%, which no seating chart allows. And 108 is the number of empty seats — a real number, not the asked one.

Worked example 2 — reverse percent (divide, never add back)

Stem. A tent is marked down 35% and now sells for $91. What was the price before the markdown?

Step 1 — shape. The stem hands you the after and asks for the before: reverse.

Step 2 — multiplier. 35% off means the customer pays 65% of the original, so the multiplier is 0.650.65.

Step 3 — divide by it.

0.65×original=91⇒original=910.65=1400.65 \times \text{original} = 91 \quad\Rightarrow\quad \text{original} = \frac{91}{0.65} = 140

Check. 140×0.65=91140 \times 0.65 = 91, and 140>91140 > 91 as an original price must be. Answer: $140.

Trap watch. Adding 35% back to the sale price gives 91×1.35=122.8591 \times 1.35 = 122.85 — wrong, because 35% of 91 is smaller than 35% of 140. The percent was taken off the bigger number, so it cannot be put back onto the smaller one. Dividing by 0.350.35 instead of 0.650.65 gives 260, which is the other half of this trap: the rate is not the multiplier.

Worked example 3 — successive changes (multiply the factors)

Stem. A fund worth $12,000 gains 25% in its first year, then loses 8% of its new value in the second. What is it worth after two years, and what is the overall percent change?

Step 1 — one multiplier per change. Up 25% is ×1.25\times 1.25; down 8% is ×0.92\times 0.92.

Step 2 — chain them.

12,000×1.25=15,00015,000×0.92=13,80012{,}000 \times 1.25 = 15{,}000 \qquad 15{,}000 \times 0.92 = 13{,}800

Step 3 — overall factor.

1.25×0.92=1.15⇒a 15% increase overall1.25 \times 0.92 = 1.15 \quad\Rightarrow\quad \text{a 15\% increase overall}

Check. 13,800−12,00012,000=1,80012,000=0.15\frac{13{,}800 - 12{,}000}{12{,}000} = \frac{1{,}800}{12{,}000} = 0.15, which agrees with the factor. Answer: $13,800, a 15% increase.

Trap watch. 25%−8%=17%25\% - 8\% = 17\% is the sum-the-percents error, and it gives $14,040 — close enough to look right in a choice list. The percents apply to different wholes (the second to $15,000, not to $12,000), so they can only be combined by multiplying their factors.

Practice

Answer before you open the explanation. Two items are student-produced response (type the number, no choices), matching the real test. Every wrong choice below is a specific error with a name — when you miss one, log the name, not the item.

12 questions — 10 multiple choice, 2 student-produced response. Every wrong choice has its own explanation.

Question 1 Warm-up

A marine biology club has 240 members. In a club survey, 35% of the members reported that they had volunteered for a beach cleanup. How many members reported that they had volunteered?

Show the answer Choice C

Why it is right

This is a plain percent-of item: part = rate x whole. The whole is the number that follows the word 'of', the 240 club members, and the rate is 35% = 0.35. So the part is 0.35 x 240 = 84 members. A quick sanity check: 10% of 240 is 24, so 30% is 72, and another 5% (12) brings the total to 84. The answer is a count of members, so a whole number under 240 is exactly what we should expect.

Why each other choice fails

Choice A
Decimal-point slip: 35% was entered as 0.035 instead of 0.35, giving 8.4. That is 3.5% of the club, and a survey count cannot come out as a fraction of a person anyway.
Choice B
Reads the rate as if it were the count, reporting 35 members because the stem said 35%. The percent still has to be applied to the 240 members before it means anything.
Choice D
Answers the complement: 240 - 84 = 156 is the number of members who did not report volunteering, which is 65% of the club, not the 35% asked for.

Question 2 Standard

A podcast episode was downloaded 45,000 times during its first week and 63,000 times during its second week. By what percent did the number of downloads increase from the first week to the second week?

Show the answer Choice B

Why it is right

Percent change is (new - old) divided by old, and the old value here is the first week, 45,000. The increase is 63,000 - 45,000 = 18,000 downloads, so the percent increase is 18,000 / 45,000 = 0.4 = 40%. Check it forwards: 45,000 x 1.40 = 63,000, which matches the second week exactly. The base of a change is always the amount you started from, never the amount you ended with.

Why each other choice fails

Choice A
Divides the 18,000 increase by the new value instead of the old one: 18,000 / 63,000 = 0.2857, about 28.6%. This is the wrong-base error, and it always understates an increase.
Choice C
Computes 45,000 / 63,000 = 0.714, which says the first week was about 71.4% of the second. That compares the two totals instead of measuring the change between them.
Choice D
Computes 63,000 / 45,000 = 1.40 and reports 140%. The second week is 140% of the first, but the increase is only the extra 40% on top of the original 100%.

Question 3 Standard

During a clearance sale, a jacket is marked down 20% from its regular price. The sale price of the jacket is $68. What was the regular price of the jacket?

Show the answer Choice A

Why it is right

The $68 is the price after the markdown, so this is a reverse-percent item. Taking 20% off means the shopper pays 80% of the regular price, so the multiplier is 0.80 and 0.80 x (regular price) = 68. Dividing gives 68 / 0.80 = 85. Check it forwards: 20% of 85 is 17, and 85 - 17 = 68. The regular price must be larger than the sale price, which rules out anything below $68 immediately.

Why each other choice fails

Choice B
The add-it-back fallacy: 68 x 1.20 = 81.60. The 20% was removed from the larger regular price, so 20% of the smaller sale price is not the same amount of money and cannot restore it.
Choice C
Applies the 20% discount a second time (68 x 0.80 = 54.40) instead of undoing it. The discount has already happened; the missing price is above $68, not below it.
Choice D
Divides by the rate rather than the multiplier: 68 / 0.20 = 340. The 0.20 is the fraction of the price that was taken off; the 0.80 is the fraction that remained and was actually paid.

Question 4 Standard

A chemistry quiz is worth a total of 64 points. Priya earned 48 of those points. What percent of the possible points did Priya earn?

Show the answer Choice C

Why it is right

Two amounts are given and the rate is missing, so the setup is rate = part / whole. The whole is the total possible, 64 points, and the part is what Priya earned, 48 points: 48 / 64 = 0.75 = 75%. Check it forwards: 0.75 x 64 = 48. Because the part is a piece of the whole here, the answer has to land between 0% and 100%, which eliminates any choice above 100 without arithmetic.

Why each other choice fails

Choice A
Reports the 16 points Priya missed (64 - 48) as though the raw point gap were already a percent. A count of points and a percent of points are different quantities.
Choice B
Answers the complement: the 16 missed points are 16 / 64 = 25% of the quiz, so 25% is the percent lost, not the percent earned.
Choice D
Inverts the fraction: 64 / 48 = 1.333, about 133%. That says the quiz total is 133% of Priya's score, which is true but is not what was asked, and no one can earn more than 100% of the possible points.

Question 5 Standard

In 2023, 24% of a city's household waste was recycled. In 2025, 30% of the city's household waste was recycled. Which of the following correctly describes the change in the city's recycling rate from 2023 to 2025?

Show the answer Choice C

Why it is right

The quantity that changed is itself a percent, so both descriptions are available and they mean different things. The gap between the two rates is 30 - 24 = 6 percentage points. The relative growth of the rate is a percent change with the old rate as the base: (30 - 24) / 24 = 6 / 24 = 0.25, a 25% increase. Choice C states both correctly and links them, which is why it survives.

Why each other choice fails

Choice A
Calls the 6-point gap '6%'. A 6% increase in the rate would raise 24% to 24 x 1.06 = 25.44%, not to 30%, so the two statements are not interchangeable.
Choice B
Uses the new rate as the base: 6 / 30 = 0.20. Percent change is always measured against the value you started from, which is the 2023 rate of 24%.
Choice D
Mixes the two labels the other way round, attaching 'percentage points' to the relative 25% figure. The rates differ by 6 points; a 25-point rise would have taken the city from 24% to 49%.

Question 6 Harder

A wildlife survey counted 800 frogs at a pond. One year later the count was 20% greater than that. The following year the count was 15% less than the count one year later. How many frogs were counted in the final year?

Show the answer Choice B

Why it is right

Each change gets its own multiplier applied to the count it acts on. Up 20% is x 1.20, so the middle count is 800 x 1.20 = 960 frogs. Down 15% is x 0.85, and it acts on that 960, not on the original 800: 960 x 0.85 = 816. Equivalently, the two-year factor is 1.20 x 0.85 = 1.02, so the final count is 800 x 1.02 = 816, a net increase of only 2%.

Why each other choice fails

Choice A
Swaps which change is which, taking 20% off first and adding 15% afterwards: 800 x 0.80 x 1.15 = 736. Order does not matter to multiplication, but the sizes attached to each direction do.
Choice C
Adds the percents: 20% - 15% = 5%, giving 800 x 1.05 = 840. The 15% decrease applies to the larger 960, so it removes more than 15% of 800 and the two rates cannot simply be combined.
Choice D
Stops after the first year. 960 is the middle count, and the question asks for the count after the second change has been applied.

Question 7 Harder

An online order totals $265.00. The total consists of the price of the items plus a shipping surcharge equal to 6% of the price of the items. What was the price of the items, before the surcharge?

Show the answer Choice D

Why it is right

The surcharge is 6% of the item price, so the total is the item price plus 6% of itself, that is 1.06 x (item price) = 265. This is a reverse-percent item, so divide: 265 / 1.06 = 250. Check it forwards: 6% of 250 is 15, and 250 + 15 = 265 exactly. Notice the actual surcharge is $15, not 6% of the total, because the 6% was charged on the smaller item price.

Why each other choice fails

Choice A
Computes 6% of the total, 0.06 x 265 = 15.90, and reports it. That answers a different question (roughly the fee) and is not even the right fee, since the fee was 6% of $250, which is $15.00.
Choice B
Takes 6% off the total: 265 x 0.94 = 249.10. The 6% was applied to the item price, so subtracting 6% of the larger total removes too much.
Choice C
Adds the surcharge again: 265 x 1.06 = 280.90. The surcharge is already inside the $265, so applying it a second time moves further from the item price instead of back to it.

Question 8 Harder Student-produced response

A library recorded 1,250 weekday visits in March and 950 weekday visits in April. By what percent did the number of weekday visits decrease from March to April? (Enter your answer without the percent sign.)

Show the answer 24

Why it is right

Percent change uses the value you started from as the base, and March is the starting month. The drop is 1,250 - 950 = 300 visits, so the percent decrease is 300 / 1,250 = 0.24, or 24%. Check: 1,250 x 0.76 = 950, which is April's count.

Answers students type instead

76
Computes 950 / 1,250 = 76%, the share of March's visits that remained in April. That is the percent left, not the percent of the decrease.
300
Reports the raw drop in visits. The question asks for a percent, so the 300 still has to be divided by the March total.
31.6
Divides the 300-visit drop by April's 950 instead of March's 1,250. The new value is never the base of a percent change.

Question 9 Harder

At a school with 1,400 students, 45% of the students are enrolled in a world language course. Of the students enrolled in a world language course, 20% are enrolled in Japanese. How many students at the school are enrolled in Japanese?

Show the answer Choice A

Why it is right

The second percent is taken of a smaller group, not of the school. First find the world-language group: 0.45 x 1,400 = 630 students. Then take 20% of that group: 0.20 x 630 = 126 students in Japanese. Equivalently, Japanese students are 0.20 x 0.45 = 0.09, or 9% of the school, and 0.09 x 1,400 = 126. The phrase 'of the students enrolled in a world language course' is what resets the whole.

Why each other choice fails

Choice B
Applies the 20% to the entire school: 0.20 x 1,400 = 280. That would be the answer only if one student in five school-wide took Japanese, but the 20% was quoted within the world-language group.
Choice C
Stops after the first step. 630 is the number of world-language students, of whom only a fifth take Japanese.
Choice D
Adds the two percents: 45% + 20% = 65%, and 0.65 x 1,400 = 910. Nested percents multiply; adding them describes two separate groups instead of a group inside a group.

Question 10 Hardest

In a laboratory, the mass of sample A is 25% less than the mass of sample B. The mass of sample B is what percent greater than the mass of sample A?

Show the answer Choice B

Why it is right

Write the given relationship with B as the base: A = 0.75B. The question flips the comparison, so now A is the base and the excess is B - A = B - 0.75B = 0.25B. The percent greater is (B - A) / A = 0.25B / 0.75B = 1/3, about 33.3%. Concretely, if B is 100 grams then A is 75 grams, and going from 75 back up to 100 is a gain of 25 on a base of 75. Reversing a comparison changes the base, so the two percents are never equal.

Why each other choice fails

Choice A
Assumes the comparison is symmetric. The 25% was measured against B; measuring the same 25-gram gap against the smaller A gives a larger percent.
Choice C
Reports that A is 75% of B. That is the correct relationship in the original direction but it is a percent-of statement, not the percent by which B exceeds A.
Choice D
Computes B / A = 100 / 75 = 1.333 and calls it 133% greater. B is 133% of A, so it is only about 33% greater once the original 100% is removed.

Question 11 Hardest Student-produced response

A coat is marked down 30% during a sale. A 5% sales tax is then applied to the sale price, and the customer pays $73.50 in total. What was the price of the coat, in dollars, before the markdown? (Disregard the $ sign when entering your answer.)

Show the answer 100

Why it is right

Two multipliers were applied in order: x 0.70 for the 30% markdown, then x 1.05 for the tax, so 0.70 x 1.05 x (original) = 73.50. Undo them in reverse. Remove the tax first: 73.50 / 1.05 = 70, the sale price. Then remove the markdown: 70 / 0.70 = 100. Check forwards: 100 x 0.70 = 70, and 70 x 1.05 = 73.50.

Answers students type instead

91
Removes the tax correctly to reach the $70 sale price, then adds 30% back (70 x 1.3 = 91) instead of dividing by 0.70. The 30% came off the larger original price.
105
Divides out the markdown but never removes the tax: 73.50 / 0.70 = 105. The $73.50 is a taxed amount, so the tax factor has to come off first.
245
Divides by the discount rate rather than the multiplier: 73.50 / 0.30 = 245. The 0.30 is the part removed; 0.70 is the part actually paid.

Question 12 Hardest

Last year a foundation approved 15% of the 4,000 grant applications it received. This year it approved 12% of the 6,000 applications it received. Which of the following correctly compares the number of applications approved this year with the number approved last year?

Show the answer Choice D

Why it is right

The question is about counts, so convert both rates into counts before comparing. Last year: 0.15 x 4,000 = 600 approvals. This year: 0.12 x 6,000 = 720 approvals. The change is 720 - 600 = 120 more approvals, measured against last year's 600: 120 / 600 = 0.20, a 20% increase. The approval rate fell while the number approved rose, because the pool of applications grew by half.

Why each other choice fails

Choice A
Compares the two rates instead of the two counts: (15 - 12) / 15 = 0.20, so the approval rate fell 20%. That is a true statement about the rate and the wrong answer to a question about how many applications were approved.
Choice B
Subtracts the rates, 15% - 12% = 3%, and labels the gap a percent change. Those 3 percentage points are also applied to different pool sizes, so they say nothing directly about the counts.
Choice C
Divides the increase of 120 by this year's 720 rather than by last year's 600. Percent change always uses the earlier value as the base.

Common mistakes

  1. Measuring change against the new value — the denominator of a percent change is the old amount. Using the new one makes every increase look smaller than it was.
  2. The “add it back” fallacy — undoing a 20% discount by adding 20% to the sale price. The discount came off a bigger number; reversing it takes a division.
  3. Dividing by the rate instead of the multiplier — 68/0.2068/0.20 rather than 68/0.8068/0.80. The 20% is what you lose; the 80% is what the price is.
  4. Adding successive percents — two changes of +20%+20\% and −15%-15\% are not +5%+5\%. Multiply 1.20×0.85=1.021.20 \times 0.85 = 1.02; the true net is +2%+2\%.
  5. Percentage points read as percent — a rate going from 24% to 30% rose 6 points, which is a 25% increase in the rate. The two numbers are both correct answers to different questions.
  6. Answering the complement — computing the 35% who volunteered and reporting the 65% who did not, or the empty seats instead of the filled ones.
  7. Decimal-point slips — 35% entered as 0.035, or a rate over 100% (a 140% increase) shrunk to 0.4 by dropping the “1 +”.
  8. Percent of a percent — 20% of the 45% who take a language is 0.20×0.45=9%0.20 \times 0.45 = 9\% of the school, not 65% and not 20% of everyone.
  9. Comparing rates when the question asked about counts — a lower approval rate on a much larger pool can still be a rising approval count.

FAQ

Is “percent of” ever the same as “percent change”? Only by accident. “A is 80% of B” compares two amounts; “A decreased by 80%” describes a move from B to 0.2B0.2B. If the stem mentions a before and an after, you are in change territory and the old value is your denominator.

What if the answer comes out above 100%? That is fine and often intended. An increase from 20 to 50 is a 150% increase — the change (30) is bigger than the base (20). Only a part-of-a-whole percent is capped at 100%.

How do I enter a percent in a grid-in? Enter the number the stem asks for. If it asks “by what percent,” type 24, not 24% or 0.24; if it asks for a dollar amount, type the amount without the dollar sign. Read the last line of the stem twice — SPR items on this skill are where the wrong-quantity error is most expensive.

Do I need to memorise a percent formula for test day? No formula sheet lists them, and none is needed: part = rate × whole plus the multiplier idea covers every archetype above, including reverse and successive changes.

Related skills

Coming next in this domain: percent growth decay context, conditional probability tables.

Editorial team · last reviewed 2026-08-06