Ratios: learn and practice

Who this lesson is for

This ratios lesson is for grades 7–8. Students share a whole amount in a given ratio by finding the value of one part. The practice link opens that skill at grade 7. The page has one address, and both grade hubs link here.

Add the parts, then find one part

A ratio 3:5 means 3 parts for the first share and 5 parts for the second. The parts are equal in size. Together there are 3 + 5 = 8 parts. If the whole amount is 40, one part is 40 ÷ 8 = 5. The smaller share is 3 × 5 = 15, and the larger share is 5 × 5 = 25. Check: 15 + 25 = 40, and 15:25 simplifies to 3:5.

The ratio is not the amount

The numbers in the ratio are counts of parts, not the final amounts, unless the whole happens to equal the sum of the ratio numbers. 3:5 of 40 is not 3 and 5. Those two numbers add to 8, not to 40. Multiplying the ratio numbers by the value of one part is what turns the ratio into amounts. Skipping that step leaves you with the labels instead of the shares.

Worked example 1: 40 shared as 3:5

Share 40 in the ratio 3:5. How much is the smaller share? There are 3 + 5 = 8 equal parts. One part is 40 ÷ 8 = 5. The smaller share has 3 parts, so it is 3 × 5 = 15. The larger share is 5 × 5 = 25. Add the shares: 15 + 25 = 40. The ratio of the shares is 15:25, and dividing both numbers by 5 returns 3:5. The smaller share is 15, which is the catalog answer for this problem.

Worked example 2: 20 shared as 2:3

Share 20 in the ratio 2:3. The number of parts is 2 + 3 = 5. One part is 20 ÷ 5 = 4. The first share is 2 × 4 = 8. The second share is 3 × 4 = 12. Check: 8 + 12 = 20, and 8:12 = 2:3. Giving 2 and 3 themselves would total 5, not 20. The ratio told you how to cut the 20. It was not already the list of amounts.

Worked example 3: 18 shared as 1:1

Share 18 in the ratio 1:1. There are 1 + 1 = 2 equal parts. One part is 18 ÷ 2 = 9. Each share is 1 × 9 = 9. A ratio of 1:1 means the two shares match. Check: 9 + 9 = 18. This is the same arithmetic as splitting in half, written in ratio language so it sits beside the uneven ratios above.

Checks before you leave the problem

Add the numbers in the ratio to count the parts. Divide the whole by that count. Multiply by the part of the ratio you were asked for. Add the shares and match the original whole. Then practice the ratio skill on the board.

Common questions about ratios

Is the smaller share in 3:5 always 3?

Only when the whole amount is 8, one for each part. For a whole of 40 the smaller share is 15. The 3 counts parts. It is not yet the amount.

What if the ratio has three numbers?

Add all of them to count the parts, then continue the same way. This page practices two-number ratios, which is what the catalog skill asks for.

How do I know which share is smaller?

The smaller number in the ratio marks the smaller share, as long as both numbers count the same size of part. In 3:5 the 3-part share is smaller.