PSAT Data — percentages, and which number sits underneath

Almost every percentage question on this test is really one question: which number is the percent taken of. Get that right and the arithmetic is a single multiplication; get it wrong and the answer will still be one of the four options, because the people who write these questions know exactly which base you are likely to use by mistake.

Twelve questions, arranged so that each group isolates one way of getting the base wrong.

Two are a plain percent of a number, where the wrong options are the percent itself read as a count, a rounder percent used in its place, and the complement — the eighty-eight percent who did not take the elective rather than the twelve percent who did.

Three are percent change, and all three turn on the same rule: the change goes over the ORIGINAL amount. Dividing by the new figure instead is the commonest error in the domain, and every one of these three offers it as an option. So does the ratio of the two amounts — one hundred fifteen percent, which is what the new rent is as a share of the old rather than by how much it rose.

Two ask for the original amount from a figure that already contains the change. A twenty percent rise to sixty dollars does not come back to fifty by taking twenty percent off sixty; that gives forty-eight, and the explanation says so in those words. Dividing by one point two is the operation, and it is worth doing once slowly.

Two are successive changes, where the whole point is that percents do not add. Ten percent up then ten percent down leaves ninety-nine, not a hundred; twenty down then twenty up leaves ninety-six. Both questions offer "no change" as an option, and both explanations show the compounding on a base of one hundred, which is the fastest way to settle any question of this kind.

Two are a share of a share — twenty percent of the sixty percent in the arts program — where the two percentages multiply rather than add. And one turns on what "forty percent more than" means: a multiplication by one point four, not an addition of forty.

Every situation here was invented for the exercise, all values are exact, and nothing is reproduced from any published test.

• Name the base before touching the arithmetic, and check it against the sentence that set the question
• Compute a percent change with the original amount underneath, not the new one
• Recognize the ratio of two amounts as a different thing from the change between them
• Recover an original amount by dividing by the factor, and see why taking the percent off the new figure does not do it
• Combine two successive percent changes by multiplying the factors rather than adding the percents
• Find a share of a share by multiplying the two percentages
• Read "forty percent more than" as a multiplication by 1.4

Every situation here was written for this exercise, and no two questions share one. Among them: twelve percent of four hundred fifty science students; thirty-five percent of an eight-hundred-kilogram harvest; a rent moving from twelve hundred dollars to thirteen hundred eighty; a workshop going from two hundred employees to a hundred seventy; a coat cut from a hundred fifty dollars to a hundred twenty; a membership fee of sixty dollars after a twenty percent rise, asked for the fee before it; a laptop at nine hundred dollars after a twenty-five percent cut; a population up ten percent and then down ten; a stock down twenty percent and then up twenty; twenty percent of the sixty percent in an arts program; twenty-five percent of the forty percent in a marketing department; and a movie theater selling forty percent more tickets on Saturday than the two hundred fifty it sold on Friday. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not an test center.