Percentages

Digital SAT · Math · Problem-Solving and Data Analysis. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Percent means out of 100, so 35% = 35/100 = 0.35. Most questions become easy once you turn the percent into a multiplier. An increase of 12% means multiply by 1.12. A decrease of 12% means multiply by 0.88.

Percent change always compares the change with the original value, not the new one. To go back to an original value, divide by the multiplier.

For changes one after another, multiply the multipliers. Do not add the percents, because the second change acts on the new amount. The same idea gives growth over several years: multiply by the same factor once for each year.

Rules to remember

part = (percent ÷ 100) × whole, so whole = part ÷ (percent ÷ 100)
percent change = (new − original) ÷ original × 100
Increase by r%: multiply by (1 + r/100). Decrease by r%: multiply by (1 − r/100).
Original value = new value ÷ multiplier.
Changes in a row: multiply the multipliers, e.g. +20% then −10% is 1.20 × 0.90 = 1.08, a rise of 8%.
n periods at r% each: final = original × (1 + r/100)ⁿ
A fall from 8% to 6% is 2 percentage points, but a 25% decrease.

Common mistake

Reversing a percent change with the same percent. After a 12% rise you do not get back by taking 12% off; divide by 1.12 instead.

Worked example

After a 12% increase, the monthly rent for an apartment is $1,400. What was the rent before the increase?

An increase of 12% means the old rent was multiplied by 1.12.
So 1.12 × old rent = 1,400.
Old rent = 1,400 ÷ 1.12 = 1,250.
Check: 1,250 × 1.12 = 1,250 + 150 = 1,400.

Answer: $1,250

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