Percentages and Percent Change

Percent Means "Per Hundred"

A percent is a fraction with a denominator of $100$. So $37\%$ means $\tfrac{37}{100} = 0.37$. Every percent problem on the SAT ultimately reduces to translating that word into a decimal and then multiplying or dividing. The word "of" almost always signals multiplication: "$25\%$ of $80$" means $0.25 \times 80 = 20$.

The most useful fluency is moving among the three quantities in $\text{part} = \text{percent} \times \text{whole}$. Know any two and you can find the third. To find what percent one number is of another, divide and multiply by $100$: "$18$ is what percent of $40$?" gives $\tfrac{18}{40} = 0.45$, so $45\%$. To find the whole from a part and a percent, divide: if $30\%$ of a number is $21$, the number is $\tfrac{21}{0.30} = 70$.

Percent Increase and Decrease

A percent change measures growth or shrinkage relative to the starting value:

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

The denominator is always the original amount — the most common SAT trap is dividing by the new value. A positive result is an increase; a negative result is a decrease.

There is a faster way to apply a percent change: use a multiplier. Increasing by $15\%$ multiplies by $1.15$; decreasing by $15\%$ multiplies by $0.85$. In general a $p\%$ increase uses the factor $1 + \tfrac{p}{100}$ and a $p\%$ decrease uses $1 - \tfrac{p}{100}$. Multipliers are what make successive percent changes tractable.

Successive Percent Changes

When two percent changes happen in a row, multiply the multipliers — do not add the percents. A jacket that costs $80 is raised $20\%$, then the new price is cut $30\%$:

$\begin{aligned} 80 \times 1.20 &= 96 \\ 96 \times 0.70 &= 67.20 \end{aligned}$

The final price is $67.20. Chaining the multipliers directly, $1.20 \times 0.70 = 0.84$, tells you the price ended at $84\%$ of the original — an overall decrease of $16\%$, not the $-10\%$ you would get by subtracting. Because multiplication commutes, the order of the two changes does not affect the final amount: $0.70 \times 1.20 = 0.84$ as well. This "percents don't add" idea is the entire point of most SAT successive-change questions.

Reversing a Percent Change

If a price after a $25\%$ increase is $50, the original is not $50 \times 0.75$. You must divide by the multiplier: $\tfrac{50}{1.25} = 40$. Undoing a percent change means dividing by the factor you would have multiplied by. This "work backward" version appears whenever the SAT hands you the result and asks for the starting amount — and it is where subtracting the percent quietly gives the wrong answer.

Percent-of-a-Percent and Language Traps

Watch the exact wording. "$B$ is $20\%$ more than $A$" means $B = 1.20A$. "$A$ is $20\%$ of $B$" means $A = 0.20B$ — a completely different equation. And "$40\%$ of the $30\%$ who passed" is a percent of a percent: $0.40 \times 0.30 = 0.12$, i.e. $12\%$ of the whole group. Reading the sentence structure carefully is often more important than the arithmetic itself. Translate each phrase into an equation before you compute, and the numbers fall into place.