2 Percentages and Conversions

What a percent represents

A percent means “per hundred,” so $37\%$ represents $\frac{37}{100}$, or $37$ parts out of every $100$. Fractions, decimals, and percentages are different ways to represent the same quantity.

Convert fractions, decimals, and percentages

To convert between forms, use the relationship between a percentage and a fraction out of $100$, and between a fraction and division.

• Percent to decimal: divide by $100$, which moves the decimal point two places left, and remove the percent sign. For example, $42\%=0.42$.

• Decimal to percent: multiply by $100$, which moves the decimal point two places right, and add a percent sign. For example, $0.065=6.5\%$.

• Fraction to decimal: divide the numerator by the denominator. For example, $\frac{3}{8}=3\div 8=0.375$.

• Decimal to fraction: write the decimal over its place-value denominator, then simplify when possible. For example, $0.375=\frac{375}{1000}=\frac{3}{8}$.

• Fraction to percent: divide the numerator by the denominator, then multiply by $100$. For example, $\frac{3}{8}=0.375=37.5\%$.

Some fractions produce a repeating decimal. For example, $\frac{1}{3}=0.333\ldots=33.333\ldots\%$. If a rounded answer is needed, state the rounding; for example, $33.3\%$ to the nearest tenth of a percent.

Find a percentage of a quantity

To find a part when you know the whole and the percentage, convert the percentage to a decimal and multiply it by the whole:

For example, to find $15\%$ of $80$, calculate:

So, $15\%$ of $80$ is $12$.

To find what percentage one quantity is of another, divide the part by the whole and multiply by $100\%$:

For example, to find what percentage $18$ is of $60$:

Identify the whole carefully before calculating: it is the reference amount.

Percentage increase and decrease

Percentage change compares the change with the original amount:

A positive result indicates an increase; a negative result indicates a decrease. For a decrease, you can instead put the positive size of the decrease in the numerator.

For an increase from $\$50$ to $\$60$, the change is $\$10$:

The price increased by $20\%$. For a decrease from $80$ to $68$, the decrease is $12$:

The quantity decreased by $15\%$. The denominator is the original amount, $80$, not the new value, $68$.

For a direct increase or decrease, use a multiplier on the original amount:

• Increase by $r\%$: multiply by $1+\frac{r}{100}$.

• Decrease by $r\%$: multiply by $1-\frac{r}{100}$.

For example, increasing $\$50$ by $20\%$ gives $50\times1.20=\$60$. Decreasing $80$ by $15\%$ gives $80\times0.85=68$.