Math Core

Discounts, tax, "a third of what was left," and percent increases show up on every MATHCOUNTS test and nearly every AMC 8. The fastest way through all of them is to stop thinking of a percent as something you add or subtract, and start thinking of it as something you multiply by.

Percents are multipliers

"Percent" means "per hundred," so $35\% = 0.35$ . Taking $35\%$ of a number means multiplying by $0.35$ .

A change is a multiplication too:

Increasing by $p\%$ multiplies by $1 + \dfrac{p}{100}$ . A $20\%$ raise multiplies by $1.2$ .
Decreasing by $p\%$ multiplies by $1 - \dfrac{p}{100}$ . A $30\%$ discount multiplies by $0.7$ .

Successive changes multiply

Several percent changes in a row combine by multiplying their multipliers. A $30\%$ discount followed by another $10\%$ off the sale price multiplies the price by

$0.7 \times 0.9 = 0.63,$

which is a total discount of $37\%$ , not $40\%$ .

The percent change from an old value to a new value is

$\frac{\text{new} - \text{old}}{\text{old}} \times 100\%.$

Always divide by the original amount.

Common mistake

Percent changes don't cancel. Raising a price by $20\%$ and then lowering the new price by $20\%$ gives $1.2 \times 0.8 = 0.96$ of the original, a $4\%$ loss. The second $20\%$ is taken from a bigger number.

Fractions of what's left

In "she spent $\dfrac{1}{3}$ of her money, then $\dfrac{1}{4}$ of the rest," the second fraction applies to what remains, not to the original. Work with the fraction remaining: after spending $\dfrac{1}{3}$ , she keeps $\dfrac{2}{3}$ ; after spending $\dfrac{1}{4}$ of that, she keeps $\dfrac{3}{4}$ of it. Remaining fractions multiply, just like percent multipliers.

Repeating decimals

A repeating block of $k$ digits sits over $k$ nines:

$0.\overline{7} = \frac{7}{9}, \qquad 0.\overline{36} = \frac{36}{99} = \frac{4}{11}, \qquad 0.\overline{123} = \frac{123}{999} = \frac{41}{333}.$

This works because, for example, $x = 0.\overline{36}$ gives $100x = 36.\overline{36}$ , and subtracting leaves $99x = 36$ .

If the repeat starts later, shift first. For $x = 0.1\overline{6}$ : $10x = 1.\overline{6} = 1 + \dfrac{6}{9} = \dfrac{5}{3}$ , so $x = \dfrac{1}{6}$ .

Worked example: Fraction of the rest

Maria spent $\dfrac{1}{3}$ of her money on a book and then $\dfrac{1}{4}$ of what was left on lunch. She had $24 left. How much did she start with?

After the book she had $\dfrac{2}{3}$ of her money. After lunch she had $\dfrac{3}{4}$ of that:

$\frac{3}{4} \cdot \frac{2}{3} = \frac{1}{2}.$

Half her money is $24, so she started with $48.

Worked example: Up and down

A price is raised $25\%$ and then the new price is cut by $20\%$ . What is the overall change?

The multipliers are $1.25$ and $0.8$ , and $1.25 \times 0.8 = 1$ . The price ends exactly where it started: no change.

Worked example: A percent of a percent

$A$ is $40\%$ of $B$ , and $B$ is $150\%$ of $C$ . What percent of $C$ is $A$ ?

$A = 0.4B = 0.4(1.5C) = 0.6C$ . So $A$ is $60\%$ of $C$ .

Worked example: Mixing groups

In a school, $40\%$ of the students play a sport. Of those, $25\%$ also play an instrument. Of the students who don't play a sport, $50\%$ play an instrument. What percent of all students play an instrument?

Think of $100$ students. There are $40$ athletes, and $25\%$ of them, $10$ students, play an instrument. There are $60$ non-athletes, and $50\%$ of them, $30$ students, play an instrument. That's $40$ out of $100$ : $40\%$ .

Tip

When no total is given, pick a friendly one: $100 for percent problems, or a common multiple of the denominators for fraction problems. The answer as a percent or fraction won't depend on your choice.