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.