Percentages

This skill point covers percent change, successive percent changes, translating percent language into an equation, and reverse percentages — being handed the value after a change and asked for the value before it. It sits inside Problem-Solving and Data Analysis, which is the domain where the Digital SAT dresses arithmetic in sentences and the sentence is the difficulty.

The test asks it in four recognisable shapes. Compute a percent change: two values, "by what percent did it increase." Apply a chain of changes: a markup then a discount, a discount then a tax, a rate compounding over several years. Translate: "40% of," "40% more than," "40% less than," "40% off," "140% of" — each of which is a different multiplier and three of which get confused for each other. And run the chain backwards: a final figure plus the changes that produced it, asking for the original.

College Board publishes the domain weight, not the skill weight: Problem-Solving and Data Analysis is about 15% of the scored Math questions — roughly 5 to 7 of them — spread across seven separate skills, of which percentages is one. How many of those are percent items on any given form is not published. The prep-industry figure of one to three per test is an inference from released material, not an official number; treat it as a rough prior rather than something to plan around.

One structural fact governs everything below. A calculator is permitted on the entire Math section — Bluebook has Desmos built in, and you may bring your own approved one — so no percent question is difficult because of the multiplication. The difficulty is entirely in deciding which number is the base, which direction the change runs, and which of the several quantities floating around in your work the question actually asked for. That is where the points are, and it is where the distractors are aimed.

If you can say without pausing that a $60 item after a 35% discount costs 0.65 × 60 = $39, and that recovering the $60 from the $39 means dividing by 0.65, skip to the next block; nothing here will be new. If either half of that made you hesitate, this is the block that matters most on the page, and it assumes nothing.

Percent means per hundred. The symbol % is shorthand for "÷ 100." So 37% is 37/100 is 0.37, and converting is just moving the decimal point two places left: 5% → 0.05, 62.5% → 0.625, 140% → 1.40, 0.4% → 0.004. Do that conversion once, at the very start, and then work in decimals for the rest of the problem. Half the errors on this topic are decimal-place errors committed midway through, when there is no longer anything to check them against.

The word "of" means multiply, and the number that follows "of" is the base — the thing being called 100%. "18% of 250" is 0.18 × 250 = 45. This is the single most useful sentence in the lesson: find the word "of" (or "than"), and the quantity right after it is your denominator, your 100%, the number everything else is measured against.

Going the other way — "what percent of B is A" — is part over whole: A ÷ B, then multiply by 100 to express it as a percent. 45 is what percent of 250? 45 ÷ 250 = 0.18 = 18%. Note which number went underneath: the one that followed "of." Swapping them gives 250 ÷ 45 ≈ 5.56, or 556%, and the absurd size of that number is usually the first sign the fraction went in upside down.

Now the habit that makes the rest of the topic easy: work in multipliers, never in two-step add-ons. An increase of p% is a multiplication by (1 + p/100); a decrease of p% is a multiplication by (1 − p/100). Up 8% is ×1.08. Down 8% is ×0.92. Up 150% is ×2.50 — not ×1.50, which is the trap. Down 100% is ×0. A multiplier is one operation instead of two, it never loses the direction of the change, and, as the next two blocks show, it is the only form in which chains and reversals are straightforward.

Percent change has one formula and it is worth writing out longhand until it is automatic: percent change = (new value − old value) ÷ old value, then × 100. The numerator is the amount of the change. The denominator — the base — is always the value you started from, no matter which of the two numbers happens to be larger, and no matter which one the sentence mentioned first.

Finally, the reversal. If a known multiplier m turned an unknown original into a known result, then original × m = result, and the original is result ÷ m. That is ordinary algebra: you undo a multiplication by dividing. The reason it deserves its own name is that the intuitive move — take the same percent back off — is wrong, and wrong in a way that produces an answer close enough to look right.

Almost every percent item on the Digital SAT opens with one of five sentence shapes. Learning them as a table is worth more marks than any amount of extra arithmetic practice, because once the sentence is converted the question is over.

"A is 30% of B" becomes A = 0.30B. "A is 30% more than B" — and its synonyms "30% greater than," "exceeds B by 30%," "increased by 30% from B" — becomes A = 1.30B. "A is 30% less than B," or "30% off B," or "reduced by 30%," becomes A = 0.70B. "A is 130% of B" becomes A = 1.30B, which is the same relationship as "30% more than B" written a different way. And "A is 130% more than B" becomes A = 2.30B, which is not the same as any of the others and is the one that catches strong students.

The base is whatever follows "of" or "than." In "A is 30% more than B," the 30% is 30% of B, not of A and not of the difference. This is why the reverse questions work: when the sentence names B as the base but hands you A as the number, the unknown is sitting in the denominator, and the operation is division.

One more distinction, and it is the one that separates a careful reader from a fast one: percent versus percentage points. When the quantity that changed is itself a percentage — an unemployment rate, a recycling rate, a market share — the difference between the two figures is measured in percentage points, and the percent change is that difference divided by the starting percentage. A rate going from 24% to 30% has risen 6 percentage points and 25%. Both statements are true, they are different numbers, and answer choices in this family routinely offer you both.

Weak path on all of this: read the sentence, form a rough impression of "a bit more" or "a bit less," and start computing. Strong path: write the equation — A = 1.30B, with the letters defined — before touching the calculator. The equation costs five seconds and it makes the direction, the base, and the target of the question all explicit at once.

When several percent changes are applied one after another, convert each to a multiplier and multiply them together. A 25% markup followed by a 30% discount is 1.25 × 0.70 = 0.875. A 15% discount followed by a 6% tax is 0.85 × 1.06 = 0.901. Nothing else is required, and in particular nothing needs to be computed in dollars along the way.

Reading the net change off the combined multiplier is one subtraction: take the multiplier, subtract 1, and express the result as a percent. 0.875 − 1 = −0.125, a 12.5% decrease. 0.901 − 1 = −0.099, a 9.9% decrease. 1.19 − 1 = 0.19, a 19% increase. If your combined multiplier is above 1 the net change is an increase; below 1, a decrease. That check alone eliminates every distractor pointing the wrong way.

The same machinery covers repeated changes. "Increases by 6% per year for three years" is ×1.06 three times: 1.06³ = 1.191016, an increase of about 19.1% overall — not 18%, which is what adding three 6% changes would give. The gap widens fast as the percent or the number of periods grows.

Order is irrelevant, and knowing that is worth a question in itself. Because multiplication commutes, a 20% discount followed by an 8% tax gives exactly the same total as an 8% tax followed by a 20% discount: 0.80 × 1.08 = 1.08 × 0.80 = 0.864. When an item asks which sequence is cheaper, "neither, they are equal" is a live and frequently correct answer. The exception, which the harder items exploit: this holds only for pure percent multipliers. Put a flat $10 coupon anywhere in the chain and the order matters enormously, because a fixed amount is not a multiplier.

At 1500 the multiplier is not the problem. The points that remain are lost to four hard variants and a handful of execution slips, and the slips cost more than the variants do.

Hard variant one: a percent of a percent, and bases that are not the whole. "18% of the 45% of respondents who owned a car" is 0.18 × 0.45 = 8.1% of all respondents, not 18% and not 45%. The general rule is that percentages compose by multiplication only when each one is taken of the group named immediately before it, and the moment the groups differ in size you cannot average or add the rates at all. The strong path on every question of this shape is to invent a convenient total — the one the question gives you, or 100, or 1,000 — turn every percentage into a headcount, do integer arithmetic, and convert back at the very end. It is more writing and it removes the error entirely.

Hard variant two: language above 100%. "250% of x" is 2.5x. "250% greater than x" is 3.5x. "Increased by 150%" is ×2.5. "Reduced by 100%" is ×0, and there is no such thing as a decrease of more than 100% for a quantity that cannot go below zero — a choice claiming one is eliminable on sight. The asymmetry is worth holding: increases are unbounded, decreases are capped at 100%, so any answer of "decreased by 125%" is wrong before you check the arithmetic.

Hard variant three: the reverse percentage buried inside a multi-step problem, where nothing announces itself as a reversal. The signature is structural rather than verbal — the unknown is the base of the percent rather than the result of it. Combine every change into one multiplier, set original × multiplier = the given final value, and divide once. Doing it in stages works too, but each stage is another chance to divide when you meant to multiply.

Hard variant four: order, and when it stops being irrelevant. Pure percent multipliers commute, so discount-then-tax equals tax-then-discount exactly, and an item offering those two orderings usually has "the same" as the credited answer. But a flat fee, a fixed-dollar coupon, a "discount applied only to amounts above $50," or a tax charged on the pre-discount price breaks the commutativity, and then order changes the total. Check whether every element in the chain is a percentage before you assert that the sequence does not matter.

Now the execution slips, which is where a 1560 usually comes from. First and most expensive: answering the right question about the wrong quantity. The item asks for the final price and you report the discount; it asks by what percent the price fell and you report what percent of the original remains; it asks for the original and you report the intermediate value after undoing only one of two changes. In the second worked example above, $70, 12.5%, and 87.5% are all sitting in the same three lines of work, and all three appear as answer choices on items of that shape. Before selecting, re-read the final clause of the prompt and say aloud what the answer is a quantity of.

Second: rounding early. 425 ÷ 4,675 = 9.0909…%, and carrying it as "9%" into a further multiplication produces a figure that is visibly off by the end. Keep the full decimal in the calculator and round exactly once, at the last step, to whatever precision the question requests.

Third: the student-produced response box. If a question asks for a percent, the grid wants 25, not 0.25 — they are different entries and only one is scored. If it asks for a value in dollars, no percent sign and no comma. Read what unit the answer is supposed to be in, and enter it in that unit.

Fourth: direction on the answer line. When the choices distinguish "increased by" from "decreased by," a correct magnitude with the wrong direction scores zero, and the combined multiplier settles it instantly — above 1 is an increase, below 1 is a decrease. This is the same class of error as a sign slip in algebra, and it survives to test day for the same reason: the arithmetic feels finished before the direction has been checked.