Ratios

The ratio of $m$ to $n$ can be expressed in the following ways:

• $m$ to $n$
• $m:n$
• $\frac{m}{n}$

Last expression tells us, that ratio is, actually, a fraction.

As well as fraction, ratio can be expressed in simplest form (in other words it can be reduced).

Example. Suppose, for 2 servings of a fruit dessert we need 14 oz. of yogurt. Find amount of yogurt needed for 1 serving and for 4 servings. How many servings can we make, if we have 56 oz. of yogurt.

Ratio of amount of yogurt to number of servings is ${14}$ to ${2}$ or $\frac{{14}}{{2}}$.

We can reduce this fraction by 2: $\frac{{{7}\cdot{\color{red}{{{2}}}}}}{{{1}\cdot{\color{red}{{{2}}}}}}=\frac{{7}}{{1}}$.

So, we need 7 oz. of yogurt to make 1 serving.

Now, we multiple numerator and denominator of the reduced fraction by 4: $\frac{{{7}\cdot{\color{red}{{{4}}}}}}{{{1}\cdot{\color{red}{{{4}}}}}}=\frac{{28}}{{4}}$.

Finally, we multiple numerator and denominator of the reduced fraction by 8 (to get 56 oz.): $\frac{{{7}\cdot{\color{red}{{{8}}}}}}{{{1}\cdot{\color{red}{{{8}}}}}}=\frac{{56}}{{8}}$.

Thus, we can make 8 servings from 56 oz. of yogurt.

This means that we need 28 oz. of yogurt to make 4 servings.

Example. The scale of a map is 5 inches=27 miles. The distance between two towns is 38 inches. What is the actual distance?

Scale is $\frac{{27}}{{5}}$ miles per inch.

If distance is 38 inches, then actual distance is $\frac{{27}}{{5}}\cdot{38}=\frac{{1026}}{{5}}={205.2}$ miles.

Ratios can be used to compare more than two values.

Example. Ratio of flour, sugar and water is ${50}:{2}:{100}$. Find amount of flour and water for 40 g. of sugar.

We can reduce this ratio by 2: ${25}:{1}:{50}$.

Now, amount of flour is ${25}\cdot{40}={1000}$ g.

Amount of water is ${50}\cdot{40}={2000}$ g.