Chapter 2 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

M3.4.2 Chapter 2

Ratios and fractions

Ana mixes juice and water in the ratio 2 : 3. Is the juice two thirds of the drink?

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

  • Change a ratio of 2 : 3 into fractions of the whole
  • Tell a part to part ratio from a part to whole fraction

In a ratio like 2 : 3, each number counts parts. The juice has 2 parts. The water has 3 parts. All the parts are the same size.

A ratio compares one part with another part. We call that part to part. A fraction compares one part with the whole. We call that part to whole.

Draw it first

Find the whole, then the fraction

  1. Step 1: a bar cut into 5 equal parts, labelled whole: 5 parts. Step 2: 2 parts at the left are shaded. Step 3: the shaded parts are labelled juice 2, and the other parts are labelled water 3.whole: 5 parts

    1 Draw a bar with 5 equal parts. 2 + 3 = 5, so the bar is the whole drink.

  2. Step 1: a bar cut into 5 equal parts, labelled whole: 5 parts. Step 2: 2 parts at the left are shaded. Step 3: the shaded parts are labelled juice 2, and the other parts are labelled water 3.whole: 5 parts

    2 Shade 2 parts for the juice.

  3. Step 1: a bar cut into 5 equal parts, labelled whole: 5 parts. Step 2: 2 parts at the left are shaded. Step 3: the shaded parts are labelled juice 2, and the other parts are labelled water 3.juice 2water 3

    3 The juice is 2 of the 5 parts. The water is the other 3.

Two bars, each with 5 equal parts and 2 shaded. The top bar is labelled part to part: 2 : 3. The bottom bar is labelled part to whole: 2 of 5.part to part: 2 : 3part to whole: 2 of 5
Same bar, two ways to say it. The ratio looks at the plain part too. The fraction looks at the whole bar.

So the juice is 2/5 of the drink. The water is 3/5. The bottom number of each fraction is the total of the ratio.

Worked example

Juice and water are in the ratio 2 : 3. What fraction of the drink is juice?

  1. 2 + 3 = 5, so the whole drink is 5 parts.
  2. The juice is 2 of the 5 parts.
  3. The fraction of juice is 2/5.

Answer: 2/5

Sometimes the fraction can be made simpler. Simplify it as you did before, by dividing the top and the bottom by the same number.

Worked example

Red and black seeds are in the ratio 4 : 6. What fraction of the seeds is red? Give it in simplest form.

  1. 4 + 6 = 10, so there are 10 parts.
  2. Red is 4 of the 10 parts, which is 4/10.
  3. Divide the top and the bottom by 2. This gives 2/5.

Answer: 2/5

Two bars. The top bar has 10 equal parts and 4 are shaded, labelled 4 of 10 parts. The bottom bar has 5 equal parts and 2 are shaded, labelled 2 of 5 parts. The shaded part is the same share of each bar.4 of 10 parts2 of 5 parts
The two shaded parts end at the same place. 4/10 and 2/5 are the same share.

See it three ways

With things

Yusuf has 2 red beans and 3 black beans. Put them in one line.

As a picture

A row of 5 beans. 2 are filled and 3 are empty.

In symbols

2 : 3 compares part with part. 2 + 3 = 5. Red is 2/5 of the beans. Black is 3/5.

Try it

Colour a strip of squares

You need: squared paper, a pencil.

  1. Draw a strip of 5 squares. Colour 2 of them. The ratio of coloured to plain is 2 : 3.
  2. Count all the squares. The coloured squares are 2 of 5.
  3. Now draw a strip of 10 squares. Colour 4 of them.
  4. Count the plain squares. Write the ratio of coloured to plain, and the fraction that is coloured.

What you will see: The second strip has 4 coloured and 6 plain squares. The ratio is 4 : 6 and the fraction is 4 out of 10. Both simplify to the first strip: 2 : 3 and 2 out of 5.

If you cannot try it, this is what happens.
A bar of 10 equal parts. 4 parts are shaded, labelled shaded 4, plain 6.shaded 4, plain 6

A wrong idea many people have

The ratio 2 : 3 means the juice is two thirds of the drink.

Why it feels right: You can see a 2 and a 3. Two thirds is also written with a 2 and a 3.

The test that breaks it: Draw 2 squares for juice and 3 squares for water in one strip. Count the whole strip. It has 5 squares, not 3. Two thirds of the strip would be more than 3 squares.

What is true: In a ratio, 3 is the water only. The whole drink is 2 + 3 = 5 parts. The juice is two fifths.

Key words

parts
the equal pieces that the numbers of a ratio count.
part to part
a ratio, which compares one part with another part.
part to whole
a fraction, which compares one part with all the parts.

Back to your guess

The answer is B: No. The juice is 2 of the 5 parts, so it is two fifths.

The ratio 2 : 3 compares juice with water. The whole drink is juice and water together: 2 + 3 = 5 parts. The juice is 2 of the 5 parts, which is two fifths.

Check yourself

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

  1. The ratio of goats to sheep is 1 : 4. What fraction of the animals are goats?

    A bar of 5 equal parts. 1 part is shaded and labelled goats 1, sheep 4.goats 1, sheep 4
    Tap your answer.

    1. Not this one. That compares goats with sheep only. The whole flock is goats and sheep together: 1 + 4 = 5 parts. Go to M3.4.2. Try another.

    2. Yes, that is right. 1 + 4 = 5 parts in all. Goats are 1 of the 5 parts.

    3. Not this one. That is the fraction of sheep. The goats are 1 of the 5 parts. Go to M3.4.2. Try another.

    See the answer

    1/5

    1 + 4 = 5 parts in all. Goats are 1 of the 5 parts.

    • If you chose A: That compares goats with sheep only. The whole flock is goats and sheep together: 1 + 4 = 5 parts. Go to M3.4.2.
    • If you chose C: That is the fraction of sheep. The goats are 1 of the 5 parts. Go to M3.4.2.

  2. The ratio of dogs to cats is 5 : 3. What fraction of the animals are cats?

    Tap your answer.

    1. Not this one. That compares cats with dogs. The whole is dogs and cats together: 5 + 3 = 8 parts. Go to M3.4.2. Try another.

    2. Not this one. That is the fraction of dogs. The question asks about cats, which are 3 of the 8 parts. Go to M3.4.2. Try another.

    3. Yes, that is right. 5 + 3 = 8 parts in all. Cats are 3 of the 8 parts.

    See the answer

    3/8

    5 + 3 = 8 parts in all. Cats are 3 of the 8 parts.

    • If you chose A: That compares cats with dogs. The whole is dogs and cats together: 5 + 3 = 8 parts. Go to M3.4.2.
    • If you chose B: That is the fraction of dogs. The question asks about cats, which are 3 of the 8 parts. Go to M3.4.2.

  3. 2/7 of a class walk to school. The others come by bus. What is the ratio of walkers to bus riders?

    Tap your answer.

    1. Not this one. The 7 is the whole class. Bus riders are the parts that are left: 7 − 2 = 5. Go to M3.4.2. Try another.

    2. Not this one. The numbers are right but the order is flipped. Walkers come first. Go to M3.4.2. Try another.

    3. Yes, that is right. Out of 7 parts, 2 are walkers. The other 7 − 2 = 5 parts come by bus. The ratio is 2 : 5.

    See the answer

    2 : 5

    Out of 7 parts, 2 are walkers. The other 7 − 2 = 5 parts come by bus. The ratio is 2 : 5.

    • If you chose A: The 7 is the whole class. Bus riders are the parts that are left: 7 − 2 = 5. Go to M3.4.2.
    • If you chose B: The numbers are right but the order is flipped. Walkers come first. Go to M3.4.2.

  4. Look at the bar. What is the ratio of shaded parts to plain parts?

    A bar of 8 equal parts. 3 parts are shaded and 5 are plain. The caption says 8 equal parts.8 equal parts
    Tap your answer.

    1. Not this one. You counted all the parts in the bar. That makes a fraction, 3 out of 8. A ratio compares shaded with plain. Go to M3.4.2. Try another.

    2. Yes, that is right. 3 parts are shaded and 5 are plain. Shaded comes first, so the ratio is 3 : 5.

    3. Not this one. The order is flipped. Shaded comes first, so write the shaded count first. Go to M3.4.2. Try another.

    See the answer

    3 : 5

    3 parts are shaded and 5 are plain. Shaded comes first, so the ratio is 3 : 5.

    • If you chose A: You counted all the parts in the bar. That makes a fraction, 3 out of 8. A ratio compares shaded with plain. Go to M3.4.2.
    • If you chose C: The order is flipped. Shaded comes first, so write the shaded count first. Go to M3.4.2.

  5. Juice and water are in the ratio 2 : 3. Which sentence compares the juice with the whole drink?

    Tap your answer.

    1. Not this one. That uses the 3 from the ratio, but 3 is the water only. The whole drink is 2 + 3 = 5 parts. Go to M3.4.2. Try another.

    2. Not this one. This is true, but it compares part with part, juice with water. The question asks for part to whole. Go to M3.4.2. Try another.

    3. Yes, that is right. A fraction compares a part with the whole. The whole drink is 2 + 3 = 5 parts, and juice is 2 of them.

    See the answer

    Juice is 2/5 of the drink.

    A fraction compares a part with the whole. The whole drink is 2 + 3 = 5 parts, and juice is 2 of them.

    • If you chose A: That uses the 3 from the ratio, but 3 is the water only. The whole drink is 2 + 3 = 5 parts. Go to M3.4.2.
    • If you chose B: This is true, but it compares part with part, juice with water. The question asks for part to whole. Go to M3.4.2.

  6. Red and blue paint are mixed in the ratio 3 : 4. What fraction of the mix is red?

    See the answer

    3/7

    3 + 4 = 7 parts in all. Red is 3 of the 7 parts.

  7. 3/8 of a class walk to school. The others come by bus. What is the ratio of walkers to bus riders?

    See the answer

    3 : 5

    Out of 8 parts, 3 are walkers. The other 8 − 3 = 5 parts come by bus. The ratio is 3 : 5.

  8. From an earlier chapter

    Write the ratio 18 : 12 in simplest form.

    See the answer

    3 : 2

    The highest common factor of 18 and 12 is 6. 18 ÷ 6 = 3 and 12 ÷ 6 = 2.

Keep it

Copy this into your exercise book.

A ratio compares part with part. A fraction compares a part with the whole. Add the parts to find the whole. In 2 : 3 the whole is 5 parts, and the first part is 2 out of 5.

Draw: Draw a bar of 5 equal squares. Shade 2. Write 2 : 3 under it, and 2 out of 5 above it.

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.