Year 8 · Unit 1 Ratio and proportion · Lesson 3 of 6

Sharing an amount in a ratio

Cut a bar into equal parts, one for every unit of the ratio. The value of one part turns the total into the shares, and the shares always add back to the total.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Year 7, Unit 3, lesson 6) Solve 4x = 36.
    Answerx = 9, because 36 ÷ 4 = 9.
  2. (Lesson 2) Write 18 : 30 in its simplest form.
    Answer3 : 5, dividing both parts by 6.
  3. (Year 7, Unit 5, lesson 5) What is 2/5 of 35?
    Answer14, because 35 ÷ 5 = 7 and 7 × 2 = 14.
Words used in this lesson
share
the part of a total that one person gets. For example: Tunde's share of the takings.
total
the whole amount before it is shared. For example: 100 naira.
parts
the equal pieces the total is cut into. For example: 3 + 2 = 5 parts.
one part
the value of each equal piece. For example: 100 ÷ 5 = 20.
difference
how much more one share is than another. For example: 5 parts and 4 parts differ by 1 part.

One part, then the shares

Tunde and Bisi run a market stall and share the day's takings in the ratio 3 : 2, because Tunde works longer. They do not split it in half, and they do not take 3 and 2 naira. How should the money be shared?

Predict first: what fraction of the takings does Tunde get? And if the takings were 100 naira, how much does each person get? Then use the bar below: it builds the shares up from one equal part.

2
5
5
7parts in all
35amount shared
10 and 25the two shares

Ratio 2 : 5 has 2 + 5 = 7 parts. Each part is worth 5, so the amount shared is 7 × 5 = 35, and the shares are 10 and 25.

Show me the steps
  1. The ratio 2 : 5 has 7 parts. Each part is worth 5, so 35 is shared and the shares are 2 × 5 = 10 and 5 × 5 = 25.
  2. The ratio 1 : 1 has 2 parts worth 8 each: 16 is shared, and the shares are 8 and 8.
  3. The ratio 4 : 3 has 7 parts worth 6 each: 42 is shared, and the shares are 24 and 18.
  4. The ratio 1 : 5 has 6 parts worth 3 each: 18 is shared, and the shares are 3 and 15. The shares always add back to the amount shared.

To share an amount in a ratio: (1) add the ratio numbers to find the number of equal parts; (2) divide the amount by the number of parts to find one part; (3) multiply one part by each ratio number.

The shares always add back to the amount. If one share is known instead, divide it by its ratio number to get one part, then find the rest. The gap between two shares is the difference of their ratio numbers, in parts.

Back to the stall: 3 + 2 = 5 parts, so Tunde gets 3/5 of the takings. For 100 naira one part is 100 ÷ 5 = 20, so Tunde gets 3 × 20 = 60 and Bisi gets 2 × 20 = 40.

A common mistake: sharing 48 in the ratio 1 : 3 by working out 48 ÷ 1 = 48 and 48 ÷ 3 = 16. Those shares add to 64, which is more than the 48 there was to share. Add the parts first: 1 + 3 = 4, and 48 ÷ 4 = 12, so the shares are 12 and 36.

Worked example: from the total to the shares

Ife and Tola share 72 naira in the ratio 5 : 4. How much does each get?

  1. Parts: 5 + 4 = 9.
  2. One part: 72 ÷ 9 = 8.
  3. Ife: 5 × 8 = 40. Tola: 4 × 8 = 32.
  4. Check: 40 + 32 = 72. 40 and 32 naira.

Two shares are in the ratio 2 : 7, and the smaller share is 14. What is the total?

  1. The smaller share is 2 parts, so one part is 14 ÷ 2 = 7.
  2. The total is 2 + 7 = 9 parts: 9 × 7 = 63.
  3. Check: the larger share is 7 × 7 = 49, and 14 + 49 = 63. 63.

Your go. The first step is given. Divide 56 stickers between two children in the ratio 3 : 5.

  1. Parts: 3 + 5 = 8.
  2. Now find one part and both shares yourself, then check.
Check the answer

One part is 56 ÷ 8 = 7, so the shares are 3 × 7 = 21 and 5 × 7 = 35. Check: 21 + 35 = 56. 21 and 35 stickers.

Check before you go on

1. 150 sweets are shared in the ratio 2 : 3. How many sweets does the larger share get?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. 108 pencils are shared between two children in the ratio 4 : 5. How many pencils is one part worth?

3. 150 stickers are shared in the ratio 1 : 4. How many stickers is the larger share?

Level 2 · Building

4. Rukhsana and Tariq share a rent of 840 taka in the ratio 3 : 4. How many taka does Tariq pay?

5. A garden has roses, tulips and daisies in the ratio 2 : 3 : 5. There are 60 flowers in all. How many tulips are there?

6. Dhruv and Anya split 2.4 kg of mixed nuts in the ratio 1 : 3. How many grams does Anya get?

Level 3 · Stretch

7. A prize of 120 pesos is shared in the ratio 7 : 5. Then 8 pesos are moved from the first share to the second. What is the second share now?

8. Red and blue counters are in the ratio 4 : 5. There are 12 more blue counters than red counters. How many counters are there in all?

Something to think about

Share 50 in the ratio 2 : 3, then share 51 in the ratio 2 : 3. For which total are both shares whole numbers? What must be true of the total for whole-number shares?

One way to think about it

50 gives 5 parts of 10, so the shares are 20 and 30. 51 gives parts of 51 ÷ 5 = 10.2, so the shares are 20.4 and 30.6. The total must be a multiple of the number of parts, here 5.

Exit ticket

9. 135 is shared in the ratio 4 : 5. Which pair of shares is right?