Chapter 3 of 8

All 8 chapters
  1. 1Ratio: comparing parts, readunderstood
  2. 2Ratios and fractions, readunderstood
  3. 3Sharing in a ratio, readunderstood
  4. 4Finding one first, readunderstood
  5. 5Direct proportion, readunderstood
  6. 6Rates and speed, readunderstood
  7. 7Maps and scale drawings, readunderstood
  8. 8Inverse proportion, readunderstood

M3.4.3 Chapter 3

Sharing in a ratio

Amina and Ravi share 35 seeds in the ratio 2 : 5. How many seeds does each one get?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Share 35 seeds in the ratio 2 : 5
  • Find the total when you know the size of one share

To share in a ratio, think of equal parts. The ratio 2 : 5 gives Amina 2 parts and Ravi 5 parts. Every part is the same size.

First add the parts to get the number of equal parts. Then share the total by that number to find one part. Last, multiply by each share.

Draw it first

Share 35 seeds in the ratio 2 : 5

  1. Step 1: a bar cut into 7 equal parts, labelled whole: 35 seeds. Step 2: 2 parts are shaded and labelled Amina 2. The other 5 parts are labelled Ravi 5. Step 3: each part is marked 5, and the bar is labelled each part: 5 seeds.whole: 35 seeds

    1 Add the parts: 2 + 5 = 7. Draw a bar of 7 equal parts for the 35 seeds.

  2. Step 1: a bar cut into 7 equal parts, labelled whole: 35 seeds. Step 2: 2 parts are shaded and labelled Amina 2. The other 5 parts are labelled Ravi 5. Step 3: each part is marked 5, and the bar is labelled each part: 5 seeds.Amina 2Ravi 5

    2 Amina gets 2 parts and Ravi gets 5 parts.

  3. Step 1: a bar cut into 7 equal parts, labelled whole: 35 seeds. Step 2: 2 parts are shaded and labelled Amina 2. The other 5 parts are labelled Ravi 5. Step 3: each part is marked 5, and the bar is labelled each part: 5 seeds.Amina 2Ravi 55555555each part: 5 seeds

    3 35 ÷ 7 = 5, so each part is 5 seeds.

Worked example

Share 35 seeds between Amina and Ravi in the ratio 2 : 5.

  1. Add the parts: 2 + 5 = 7.
  2. Find one part: 35 ÷ 7 = 5.
  3. Amina has 2 × 5 = 10 seeds. Ravi has 5 × 5 = 25 seeds.
  4. Check: 10 + 25 = 35.

Answer: Amina gets 10 seeds. Ravi gets 25 seeds.

Sometimes you know one share and need the total. Then work the other way. One share gives you one part. One part gives you the total.

Worked example

Leila and Tariq share dates in the ratio 3 : 4. Leila gets 12 dates. How many dates are there in all?

  1. Leila has 3 parts, and 3 parts are 12 dates.
  2. One part is 12 ÷ 3 = 4 dates.
  3. There are 3 + 4 = 7 parts in all.
  4. The total is 7 × 4 = 28 dates.

Answer: 28 dates

A bar of 7 equal parts. 3 parts are shaded, for Leila, and 4 are plain, for Tariq. The caption says Leila: 3 parts = 12 dates.Leila: 3 parts = 12 dates
Leila's 3 parts hold 12 dates, so each part holds 4 dates. The whole bar has 7 parts.

See it three ways

With things

Mei and Diego share 12 beans in the ratio 1 : 2. Deal the beans into 3 equal piles. Mei takes 1 pile and Diego takes 2 piles.

As a picture

Three piles of 4 beans. The first pile is filled in and labelled Mei. It is Mei's share. The other two piles are empty and labelled Diego. They are Diego's share.MeiDiegoDiego

In symbols

1 + 2 = 3 parts. 12 ÷ 3 = 4. Mei gets 1 × 4 = 4 beans. Diego gets 2 × 4 = 8 beans.

Try it

Share 24 stones in the ratio 1 : 3

You need: 24 small stones, or beans.

  1. Make a heap of 24 stones.
  2. The parts are 1 + 3 = 4. Deal the stones one at a time into 4 piles until none are left.
  3. Friend A takes 1 pile. Friend B takes the other 3 piles.
  4. Count the stones each friend has.

What you will see: Each pile has 6 stones. Friend A has 6 stones and friend B has 18. 6 + 18 = 24, so nothing is lost.

If you cannot try it, this is what happens.
Four piles of 6 stones. The first pile is filled in and labelled A. It is the share of friend A, 1 part. The other three piles are empty and each is labelled B. They are the share of friend B, 3 parts.ABBB

A wrong idea many people have

The ratio 2 : 5 means one person gets 2 seeds and the other gets 5 seeds.

Why it feels right: The numbers 2 and 5 are right there, and it is quick to use them as they are.

The test that breaks it: Give 2 seeds to Amina and 5 to Ravi. That uses 2 + 5 = 7 seeds. There were 35, so 28 seeds are still in the heap.

What is true: The numbers in a ratio count equal parts, not seeds. Find how many seeds one part holds. Here one part is 5 seeds.

Key words

total
the whole amount that is shared.
one part
the amount that each equal part of the ratio holds.

Back to your guess

The answer is C: Amina gets 10 seeds and Ravi gets 25 seeds.

The ratio 2 : 5 means 2 + 5 = 7 equal parts. 35 seeds in 7 parts is 5 seeds a part. Amina has 2 parts, which is 10 seeds. Ravi has 5 parts, which is 25 seeds.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Share 40 beans between Sipho and Lucía in the ratio 3 : 5. How many beans does each get?

    Tap your answer.

    1. Not this one. The numbers in a ratio are parts, not beans. There are 3 + 5 = 8 parts, and 40 beans to share out. Go to M3.4.3. Try another.

    2. Not this one. That is an equal share. The ratio 3 : 5 gives Lucía more than Sipho. Go to M3.4.3. Try another.

    3. Not this one. You treated 3 : 5 like the fractions 3/5 and 2/5. But the whole is 3 + 5 = 8 parts, not 5. Go to M3.4.3. Try another.

    4. Yes, that is right. 3 + 5 = 8 parts. 40 ÷ 8 = 5 beans a part. Sipho gets 3 × 5 = 15 and Lucía gets 5 × 5 = 25.

    See the answer

    15 and 25

    3 + 5 = 8 parts. 40 ÷ 8 = 5 beans a part. Sipho gets 3 × 5 = 15 and Lucía gets 5 × 5 = 25.

    • If you chose A: The numbers in a ratio are parts, not beans. There are 3 + 5 = 8 parts, and 40 beans to share out. Go to M3.4.3.
    • If you chose B: That is an equal share. The ratio 3 : 5 gives Lucía more than Sipho. Go to M3.4.3.
    • If you chose C: You treated 3 : 5 like the fractions 3/5 and 2/5. But the whole is 3 + 5 = 8 parts, not 5. Go to M3.4.3.

  2. Cut a rope 54 cm long into two pieces in the ratio 4 : 5. How long is each piece?

    Tap your answer.

    1. Not this one. That cuts the rope in half. The ratio 4 : 5 makes the pieces different lengths. Go to M3.4.3. Try another.

    2. Not this one. That is 4 × 9 and 5 × 9. 9 is the number of parts. You must divide 54 by 9 first to find one part. Go to M3.4.3. Try another.

    3. Yes, that is right. 4 + 5 = 9 parts. 54 ÷ 9 = 6 cm a part. The pieces are 4 × 6 = 24 cm and 5 × 6 = 30 cm.

    See the answer

    24 cm and 30 cm

    4 + 5 = 9 parts. 54 ÷ 9 = 6 cm a part. The pieces are 4 × 6 = 24 cm and 5 × 6 = 30 cm.

    • If you chose A: That cuts the rope in half. The ratio 4 : 5 makes the pieces different lengths. Go to M3.4.3.
    • If you chose B: That is 4 × 9 and 5 × 9. 9 is the number of parts. You must divide 54 by 9 first to find one part. Go to M3.4.3.

  3. Hana and Joseph share stones in the ratio 2 : 3. Hana gets 14 stones. How many stones are there in all?

    A bar of 5 equal parts. 2 parts are shaded and labelled Hana: 2 parts = 14 stones.Hana: 2 parts = 14 stones
    Tap your answer.

    1. Not this one. That is Joseph's share, 3 parts of 7. The question asks for all the stones, Hana's and Joseph's together. Go to M3.4.3. Try another.

    2. Not this one. That doubles Hana's 14, as if they shared equally. Joseph has 3 parts and Hana has only 2. Go to M3.4.3. Try another.

    3. Yes, that is right. Hana has 2 parts, which is 14 stones. One part is 14 ÷ 2 = 7. There are 2 + 3 = 5 parts, so 5 × 7 = 35 stones.

    See the answer

    35 stones

    Hana has 2 parts, which is 14 stones. One part is 14 ÷ 2 = 7. There are 2 + 3 = 5 parts, so 5 × 7 = 35 stones.

    • If you chose A: That is Joseph's share, 3 parts of 7. The question asks for all the stones, Hana's and Joseph's together. Go to M3.4.3.
    • If you chose B: That doubles Hana's 14, as if they shared equally. Joseph has 3 parts and Hana has only 2. Go to M3.4.3.

  4. The bar shows 28 sweets shared in the ratio 3 : 4. How many sweets are in the share with 3 parts?

    A bar of 7 equal parts. 3 parts are shaded. The caption says 28 sweets in 7 equal parts.28 sweets in 7 equal parts
    Tap your answer.

    1. Not this one. That is the size of one part, 28 ÷ 7. The share has 3 parts, so multiply by 3. Go to M3.4.3. Try another.

    2. Not this one. That is the share with 4 parts. The question asks about the share with 3 parts. Go to M3.4.3. Try another.

    3. Not this one. That is 3 × 7. The 7 is the number of parts, not the size of a part. Divide 28 by 7 first. Go to M3.4.3. Try another.

    4. Yes, that is right. 3 + 4 = 7 parts. 28 ÷ 7 = 4 sweets a part. The share with 3 parts is 3 × 4 = 12 sweets.

    See the answer

    12 sweets

    3 + 4 = 7 parts. 28 ÷ 7 = 4 sweets a part. The share with 3 parts is 3 × 4 = 12 sweets.

    • If you chose A: That is the size of one part, 28 ÷ 7. The share has 3 parts, so multiply by 3. Go to M3.4.3.
    • If you chose B: That is the share with 4 parts. The question asks about the share with 3 parts. Go to M3.4.3.
    • If you chose C: That is 3 × 7. The 7 is the number of parts, not the size of a part. Divide 28 by 7 first. Go to M3.4.3.

  5. Red and white paint are mixed in the ratio 2 : 7 to make 36 litres. How much red paint is there?

    A bar of 9 equal parts. 2 parts are shaded, for red paint. The caption says 36 litres in 9 equal parts.36 litres in 9 equal parts
    Tap your answer.

    1. Yes, that is right. 2 + 7 = 9 parts. 36 ÷ 9 = 4 litres a part. Red has 2 parts: 2 × 4 = 8 litres.

    2. Not this one. That is the size of one part. Red has 2 parts, so multiply 4 by 2. Go to M3.4.3. Try another.

    3. Not this one. That is half of 36. Red and white are not shared equally. Red has 2 parts out of 9. Go to M3.4.3. Try another.

    4. Not this one. That is the white paint: 7 parts of 4 litres each. Red is the smaller share. Go to M3.4.3. Try another.

    See the answer

    8 litres

    2 + 7 = 9 parts. 36 ÷ 9 = 4 litres a part. Red has 2 parts: 2 × 4 = 8 litres.

    • If you chose B: That is the size of one part. Red has 2 parts, so multiply 4 by 2. Go to M3.4.3.
    • If you chose C: That is half of 36. Red and white are not shared equally. Red has 2 parts out of 9. Go to M3.4.3.
    • If you chose D: That is the white paint: 7 parts of 4 litres each. Red is the smaller share. Go to M3.4.3.

  6. Share 63 cm of ribbon between Ana and Joseph in the ratio 2 : 7. How long is each piece? Ana has the 2 parts.

    See the answer

    14 cm and 49 cm

    2 + 7 = 9 parts. 63 ÷ 9 = 7 cm a part. The pieces are 2 × 7 = 14 cm and 7 × 7 = 49 cm.

  7. Ana and Diego share seeds in the ratio 3 : 5. Ana gets 15 seeds. How many seeds are there in all?

    See the answer

    40 seeds

    Ana has 3 parts, which is 15 seeds. One part is 15 ÷ 3 = 5. There are 3 + 5 = 8 parts, so 8 × 5 = 40 seeds.

  8. From an earlier chapter

    Black and white beads are in the ratio 3 : 7. What fraction of the beads is black?

    See the answer

    3/10

    3 + 7 = 10 parts in all. Black is 3 of the 10 parts.

Keep it

Copy this into your exercise book.

Add the parts. Divide the total by the number of parts to find one part. Multiply one part by each share. Check that the shares add up to the total.

Draw: Draw a bar of 7 equal parts for 35 seeds. Shade 2 parts. Write 5 in every part.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (710 KB). It opens on any phone.