Ratios and Proportions

Introduction

A ratio compares two or more quantities. Recipes, maps, mixes, speeds and prices all use ratios. Proportional thinking helps you scale things up and down and compare options fairly.

Writing a ratio

If a drink mix uses 2 cups of juice for every 3 cups of water, the ratio of juice to water is 2 : 3. Order matters: the ratio of water to juice is 3 : 2.

Simplify by dividing every part by a common factor. 10 : 15 divides by 5 to give 2 : 3.

A ratio can also be written as a fraction: juice is 2 out of 5 total parts, so 2/5 or 40 percent of the drink.

Scaling

Because the ratio stays the same, you can multiply both parts by the same number.

If 2 cups of juice go with 3 of water, then 6 cups of juice need 9 cups of water (multiply both by 3).

Sharing in a ratio

Share 60 in the ratio 1 : 2 : 3.

1. Add the parts: 1 + 2 + 3 = 6.
2. Find one part: 60 / 6 = 10.
3. Multiply: 10, 20, 30.

Check: 10 + 20 + 30 = 60.

Direct proportion

Two quantities are in direct proportion if one doubles when the other doubles. If 4 notebooks cost 10, then 6 notebooks cost:

1. Find the cost of one: 10 / 4 = 2.5.
2. Multiply: 2.5 x 6 = 15.

This "unitary method" (find one, then scale) works for many problems.

Unit rates and best value

To compare prices, find the price per unit.

• 500 g for 2.40: 2.40 / 500 = 0.0048 per g, which is 0.48 per 100 g.
• 750 g for 3.45: 3.45 / 750 = 0.0046 per g, which is 0.46 per 100 g.

The larger pack is cheaper per gram. Remember to check the quality, whether you can use all of it before it spoils, and whether storage is practical.

Maps and scale

A scale of 1 : 50,000 means 1 cm on the map is 50,000 cm in reality, which is 500 m. So 4 cm on the map is 2,000 m, which is 2 km.

Try it yourself

A paint mix uses 3 parts blue to 2 parts white. How much of each for 20 litres? Which is the better buy: 1.5 litres for 3.60 or 2 litres for 4.50?

Summary

• A ratio compares quantities, and can be simplified or scaled by multiplying or dividing all parts.
• To share in a ratio, add the parts, find one part, and multiply.
• Compare prices with unit rates, and use the unitary method for proportion problems.