Solving Percentage Questions on the SAT

What this skill covers on the SAT

Percentages is one of seven skills in College Board’s Problem-Solving and Data Analysis domain, alongside ratios and rates, probability and four data skills. The domain is about 15% of SAT Math, or 5 to 7 questions, shared among all seven; there’s no official count for a single skill. Percent change also powers exponential growth and decay in nonlinear functions.

Math is two adaptive modules of 22 questions, 35 minutes each, with Desmos built in. About a quarter of the questions are student-produced response (SPR), with no choices to check against. The framework’s testing point for this skill reads:

“Use percentages to solve problems in a variety of contexts: including, but not limited to, discounts, interest, taxes, and tips … including those that involve percent increases and decreases for many different quantities. Understand and use the relationship between percent change and growth factor (5% and 1.05, for example); include percentages greater than or equal to 100%.”

College Board’s Skills Insight ties these statements about percentages to Math section score bands:

The ladder climbs by number of steps, not harder arithmetic: one percent of one number at the bottom, chained or undone changes at 610–670. College Board’s own sample question asks for a percent of a percent, with wrong choices from subtracting, adding and dividing the two percents. Section 06 shows why you multiply.

The one equation that solves every percent question

Nearly every percent question on the Digital SAT — find a percent, find a part, find a whole, or find a percent change — comes from the same relationship: Part = Percent × Whole, where the percent is written as a decimal. The skill is translating English into that equation correctly.

Translate word-by-word: “is” becomes an equals sign, “of” becomes multiplication, and “what percent” or “what number” becomes your unknown. “18 is what percent of 20” becomes 18 = (x/100)(20). Once it’s an equation, solving is ordinary algebra.

Walk through that exact question below, then try one yourself.

A shortcut: x% of y = y% of x, so 8% of 25 equals 25% of 8, which is 2. That translate-then-solve habit is the whole method. The rest of this guide applies it to change, reversal, repeated change and percent of a percent.

Percent increase and decrease

To apply a percent change forward, multiply the original value by (1 + rate) for an increase or (1 − rate) for a decrease. A $64 shirt marked up 25% costs 64 × 1.25 = $80. A $64 shirt marked down 25% costs 64 × 0.75 = $48. Building the multiplier (1.25 or 0.75) in one step avoids the common error of calculating the change and forgetting to add it back.

If you’re instead given the before-and-after values and asked for the rate, use Percent change = (New − Original) / Original × 100. The denominator is always the original value — using the new value there is a frequent trap.

The testing point also says to include percentages of 100% or more. “Increase by” adds 1 to the multiplier and “of” doesn’t: a 150% increase multiplies by 2.5, while 150% of a value multiplies by 1.5. A 100% increase doubles a value. Pick the growth factor for each change below.

Reverse percent: working backward from the result

A reverse-percent question gives you the value after a change and asks for the value before it. The trap is treating the final value as if you should undo the percent by simple addition or subtraction. Instead, write the same multiplier equation and solve for the original.

If a $56 jacket reflects a 20% discount, the original price satisfies 0.8 × Original = 56, so Original = 56 / 0.8. Dividing by the multiplier — not multiplying the final price by the discount rate — recovers the original.

With two or more changes, chain the multipliers first, then divide once. Step through a bill with tax and a tip:

Desmos tip: type the equation exactly as you built it, 1.08 · 1.2x = 77.76. Desmos solves a one-variable equation like this and shows x = 60, so you skip the long division.

Successive percent changes — why they don't add

When a value changes by one percent and then by another, resist the urge to add or subtract the percents. Instead, multiply the two multipliers together. A 20% increase followed by a 15% decrease is 1.20 × 0.85 = 1.02 — a net increase of only 2%, not the 5% you’d get by naively subtracting 20 − 15.

This is also why an equal percent increase followed by the same percent decrease never returns you to the start (unless the percent is 0%): the multipliers (1 + r) and (1 − r) multiply to 1 − r², which is always a little less than 1.

Classify each pair of changes below by its net effect before you compute anything precisely.

When the same change repeats, the multiplier gets an exponent. That’s how interest works, and it’s the link between percents and exponential functions:

Desmos tip: for “after how many years” questions, graph y = 2500 · 1.04^x and a line such as y = 3000, then click the crossing.

Percent of a percent

“A percent of a percent” is a product. Suppose 60% of a school’s seniors applied to college early, and 35% of those early applicants were admitted early. The admitted group is 35% of the 60%, so it’s 0.35 × 0.60 = 0.21, or 21% of all seniors. Subtracting (25%), adding (95%) or dividing (about 58%) are the classic wrong moves, the same three that College Board’s sample question uses as distractors.

A quick sanity check: the result must be smaller than both percents, because you’re taking part of a part. If you get something bigger than 35%, you didn’t multiply. When the question gives a count instead of a percent, the product is your multiplier for a reverse-percent step, as in the next question.

Spot the traps in the choices

Wrong choices on percent questions are built from a short list of mistakes. Name them and you can often cross a choice out on sight.

• The wrong base. Percent change divides by the original value. From 80 to 92 is 12/80 = 15%, not 12/92.
• Adding successive changes. Up 20%, then down 15%, is ×1.02, not +5%.
• Undoing by subtraction. Adding 20% back to a sale price doesn’t recover the original. Divide by the multiplier.
• “Of” versus “increase by.” 150% of a value is ×1.5; a 150% increase is ×2.5.
• Percent versus percentage points. A rate going from 20% to 25% rose 5 percentage points, which is a 25% increase.
• Answering with the multiplier. A factor of 1.02 means a 2% increase. If an SPR question asks for p in “p percent,” the answer is 2, not 1.02 or 0.02.

Practice until it's automatic

Translate first, then compute. Name the variation before you start calculating.

Once these feel quick, the next natural step is applying the same “part over whole” thinking to ratios and rates, where the denominator isn’t fixed at 100.

Practice questions

Here are three Percentages questions from our problem sets, from easier to harder, in the same format as the Digital SAT. Answer each one to see the full explanation.

Quick checks

Seven shorter questions. Write the multiplier before you compute anything.

Frequently asked questions