SAT Math — percentages, and which figure is the base

Almost every wrong answer to a percentage question comes from the same place: the percentage was taken as a share of the wrong thing. A percentage is never a number on its own, it is always a share of something, and identifying that something is the whole skill. This set is arranged so that the base changes from question to question and the arithmetic barely does.

Three questions ask for a percentage change, one a rise, one a fall, one from a before and an after figure. A percentage change is measured against what the quantity WAS, and the tempting wrong option in each divides by what it became instead. The useful phrasing, repeated in the explanations, is to ask which of the two figures came first.

Two questions apply one percentage and then another, and they exist to show that the two do not cancel. A price that rises by a tenth and then falls by a tenth does not come back to where it began, because the fall is a tenth of a larger amount than the rise was — the same percentage is worth more dollars on the way down. The wrong option in each is the answer a learner gets by combining the two percentages directly, which is the intuition this pair is meant to dislodge.

Two questions work backwards from a figure that has already changed: a price after a discount, a price after a rise. These are harder than they look, because the instinct is to apply the percentage to the number in front of you, and the number in front of you is the result rather than the base. Both explanations solve it as an equation and then work forwards to confirm that the original figure returns.

Two questions take a share of a share — a proportion of a group that is itself a proportion of a larger one — where the second percentage applies to the group the first produced, not to the whole. One of them is checked on a concrete population of a thousand, which is often faster than the algebra and is proof against the error rather than merely a way round it.

The rest cover finding a percentage of a quantity, finding what share one number is of another, and recovering a total from a count and the percentage it represents. Even here the explanations point at the base: in the phrase "what percentage of eighty is twenty", the number after the word "of" is the whole.

Every store, town and company is invented, sums of money are written in words, and nothing is drawn from any official publication.

• Identify which figure is the base of any percentage, and in particular that a change is measured against the original
• Explain why a rise and a fall of the same percentage do not cancel, and predict the direction of the shortfall
• Work backwards from a price after a discount or a rise by solving for the original rather than applying the percentage again
• Take a share of a share by applying the second percentage to the group the first produced
• Recover a total from a count and the percentage it represents, and check the answer by working forwards
• Read "what percentage of A is B" correctly, with A as the whole

Written for this catalogue, with no source document. The content is the percentage material named in the College Board's own public description of the Problem Solving and Data Analysis domain — percentages, percent change and proportional reasoning; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test also asks some questions in a student-produced response format, which a multiple-choice bank cannot reproduce. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.