# Ratios

A **ratio** is a comparison of two numbers by division.

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.

A ratio called **scale** is used when making a model or drawing of something that is too large or too small to be conveniently drawn at actual size.

**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.

**Exercise 1.** Ratio of cows to goat on the farm is $$${15}:{6}$$$. Express this ratio in the simplest form and find number of cows if there are 12 goats. Find number of goats, if there are 45 cows.

**Answer:** Ratio is $$$\frac{{5}}{{2}}$$$. If there are 12 goats, then number of cows is 30 (multiple reduced fraction by 6). If there are 45 cows, then number of goats is 18 (multiple numerator and deominator of the reduced fraction by 9)

**Exercise 2.** Scale of the map is 2 inches=15 miles. If the actual distance is 90 miles, find distance between towns on the map.

**Answer**: ratio is $$$\frac{{2}}{{15}}$$$ inches per mile. Distance required is $$$\frac{{2}}{{15}}\cdot{90}={12}$$$ inches.

**Exercise 3.** Average fuel consumption of BMW X5 is 204 miles to 12 gallons. Express this ratio in simplest term and find how many miles can you drive, if you have 18 gallons of fuel.

**Answer**: $$$\frac{{{204}\ \text{miles}}}{{{12}\ \text{gallons}}}=\frac{{{17}\ \text{miles}}}{{{1}\ \text{gallon}}}$$$ or 17 miles per gallon. Now, multiple numerator and denominator by 18 to get $$$\frac{{{306}\ \text{miles}}}{{{18}\ \text{gallons}}}$$$. Thus, you can ride 306 miles.

**Exercise 4.** Ratio of height of Ann, Bob and John is $$${1.4}:{1.2}:{1.5}$$$. Find height of Ann and Bob, if height of John is 75 inches.

**Answer**: height of Ann is 70 inches, height of Bob is 60 inches.