# Ratios and Rates as Percents

## Ratios and Percents

#### Ratio, Percent

We defined a ratio as a comparison, by division, of two pure numbers or two like denominate numbers. A most convenient number to compare numbers to is 100. Ratios in which one number is compared to 100 are called percents. The word percent comes from the Latin word "per centum". The word "per" means "for each" or "for every," and the word "centum" means "hundred". Thus, we have the following definition.

Percent means "for each hundred," or "for every hundred".

The symbol % is used to represent the word percent.

#### Sample Set A

The ratio 26 to 100 can be written as 26%. We read 26% as "twenty-six percent".

The ratio $\dfrac{165}{100}$ can be written as 165%.

We read 165% as "one hundred sixty-five percent".

The percent 38% can be written as the fraction $\dfrac{38}{100}$.

The percent 210% can be written as the fraction $\dfrac{210}{100}$ or the mixed number $2 \frac{10}{100}$ or 2.1.

Since one dollar is 100 cents, 25 cents is $\dfrac{25}{100}$ of a dollar. This implies that 25 cents is 25% of one dollar.

#### The Relationship Between Fractions, Decimals, and Percents – Making Conversions

Since a percent is a ratio, and a ratio can be written as a fraction, and a fraction can be written as a decimal, any of these forms can be converted to any other.

Let's summarize the conversion techniques.

To Convert a Fraction
To Convert a Decimal To Convert a Percent
To a decimal: Divide the numerator by the denominator
To a fraction: Read the decimal and reduce the resulting fraction To a decimal: Move the decimal point 2 places to the left and drop the % symbol
To a percent: Convert the fraction first to a decimal, then move the decimal point 2 places to the right and affix the % symbol.
To a percent: Move the decimal point 2 places to the right and affix the % symbol To a fraction: Drop the % sign and write the number "over" 100. Reduce, if possible.

Conversion Techniques – Fractions, Decimals, Percents

#### Sample Set B

##### Convert 12% to a decimal.

$12%=\dfrac{12}{100}=0.12$

Note that

The % symbol is dropped, and the decimal point moves 2 places to the left.

##### Convert 0.75 to a percent.
$0.75=\frac{75}{100}=75\%$

Note that

The % symbol is affixed, and the decimal point moves 2 units to the right.

##### Convert $\frac{3}{5}$ to a percent.

We can convert a decimal to a percent. We also know that we can convert a fraction to a decimal. Thus, we can see that if we first convert the fraction to a decimal, we can then convert the decimal to a percent.

$\begin{array}{r}.6 \\5 \text{⟌ } \overline{3 . 0} \\\frac{3 \; 0}{0}\end{array}$

or

$\frac{3}{5} =0.6 = \frac{6}{10} =\frac{60}{100}=60\%$

##### Convert 42% to a fraction.

$42\%= \frac{42}{100}=\frac{21}{50}$

or

$42\%=0.42=\frac{42}{100}=\frac{21}{50}$

Source: Rice University, https://cnx.org/contents/XeVIW7Iw@4.6:lqTF3_1U@2/Percent