Chapter 7 of 8

All 8 chapters
  1. 1Equal fractions and simplest form, readunderstood
  2. 2Adding and taking away fractions, readunderstood
  3. 3Multiplying and dividing fractions, readunderstood
  4. 4Tenths, hundredths and thousandths, readunderstood
  5. 5Adding and taking away decimals, readunderstood
  6. 6Multiplying and dividing decimals, readunderstood
  7. 7Fractions and decimals, readunderstood
  8. 8Percentages, readunderstood

M3.3.7 Chapter 7

Fractions and decimals

A fraction is a division. So what decimal is 3/8?

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 3/8 into a decimal by dividing
  • Change 0.35 into a fraction in simplest form
  • Spot a decimal that repeats, like 1/3 = 0.333...

A fraction is a division in disguise. 3/8 means 3 ÷ 8. To change a fraction into a decimal, divide the top number by the bottom number.

Draw it first

Find 3/8 on a number line

  1. A bar for one whole, cut into 8 equal parts with 3 shaded. Under it is a number line from 0.00 to 1.00 with a tick at every eighth, labelled 0.00, 0.25, 0.50, 0.75 and 1.00. A point labelled 0.375 sits under the end of the shaded part.

    1 The bar is one whole, cut into 8 equal parts. 3 parts are shaded. That is 3/8.

  2. A bar for one whole, cut into 8 equal parts with 3 shaded. Under it is a number line from 0.00 to 1.00 with a tick at every eighth, labelled 0.00, 0.25, 0.50, 0.75 and 1.00. A point labelled 0.375 sits under the end of the shaded part.0.000.250.500.751.00

    2 Draw a number line under the bar, from 0 to 1. Mark a tick at every eighth. Each tick is 0.125 further on.

  3. A bar for one whole, cut into 8 equal parts with 3 shaded. Under it is a number line from 0.00 to 1.00 with a tick at every eighth, labelled 0.00, 0.25, 0.50, 0.75 and 1.00. A point labelled 0.375 sits under the end of the shaded part.0.000.250.500.751.000.375

    3 The shaded part ends at the third tick. That point is 0.375. So 3/8 = 0.375.

Worked example

Change 3/8 into a decimal.

  1. Work out 3 ÷ 8. Write 3 as 3.000, so there are tenths, hundredths and thousandths to divide.
  2. Ones: 3 ÷ 8 = 0, with 3 left over. 3 ones are 30 tenths.
  3. Tenths: 30 ÷ 8 = 3, with 6 left over.
  4. Hundredths: 60 ÷ 8 = 7, with 4 left over.
  5. Thousandths: 40 ÷ 8 = 5, with 0 left over.
  6. The digits of the answer are 3, 7 and 5. So 3/8 = 0.375.

Answer: 0.375

A decimal that ends, like 0.375, is a terminating decimal. Its division finishes with nothing left over.

To go the other way, read the place value. 0.35 is 35 hundredths, so 0.35 = 35/100. Then simplify the fraction.

See it three ways

With things

Take a hundred grid. Shade 35 squares. The whole grid is 100 squares.

As a picture

A square grid of 100 small squares. Three whole rows are shaded with lines, and 5 more squares are shaded with dots. The caption says 35 of 100 squares.35 of 100 squares

In symbols

0.35 = 35/100 = 7/20. Divide the top and the bottom by 5.

Worked example

Change 0.35 into a fraction in simplest form.

  1. The last digit, 5, is in the hundredths place. So 0.35 is 35 hundredths.
  2. Write it as a fraction: 35/100.
  3. Find the HCF of 35 and 100. They share one 5, so the HCF is 5.
  4. Divide the top and the bottom by 5. 35 ÷ 5 = 7 and 100 ÷ 5 = 20.
  5. So 0.35 = 7/20.

Answer: 7/20

A number line from 0.00 to 1.00 with a tick every 0.05, labelled 0.00, 0.25, 0.50, 0.75 and 1.00. Three points are marked: 1/4 at 0.25, 1/2 at 0.5 and 3/4 at 0.75.0.000.250.500.751.001/41/23/4
Three fractions worth knowing as decimals: 1/4 = 0.25, 1/2 = 0.5 and 3/4 = 0.75.

Some divisions never end. Try 1/3. Divide 1 by 3 and watch what is left over each time.

Draw it first

Find 1/3 on a number line

  1. A bar for one whole, cut into 3 equal parts with 1 shaded. Under it is a number line from 0.0 to 1.0 with a tick for each tenth, labelled 0.0, 0.5 and 1.0. A point labelled 0.333... sits under the end of the shaded part, just after the tick for 0.3.

    1 The bar is one whole, cut into 3 equal parts. 1 part is shaded. That is 1/3.

  2. A bar for one whole, cut into 3 equal parts with 1 shaded. Under it is a number line from 0.0 to 1.0 with a tick for each tenth, labelled 0.0, 0.5 and 1.0. A point labelled 0.333... sits under the end of the shaded part, just after the tick for 0.3.0.00.51.0

    2 Draw a number line under the bar with a tick at every tenth. The end of the shaded part is a little after the tick for 0.3.

  3. A bar for one whole, cut into 3 equal parts with 1 shaded. Under it is a number line from 0.0 to 1.0 with a tick for each tenth, labelled 0.0, 0.5 and 1.0. A point labelled 0.333... sits under the end of the shaded part, just after the tick for 0.3.0.00.51.00.333...

    3 The point is at 0.333... The 3 goes on for ever.

Worked example

Change 1/3 into a decimal. Does the decimal stop?

  1. Work out 1 ÷ 3. Write 1 as 1.000. Ones: 1 ÷ 3 = 0, with 1 left over. 1 one is 10 tenths.
  2. Tenths: 10 ÷ 3 = 3, with 1 left over.
  3. Hundredths: the 1 left over is 10 hundredths. 10 ÷ 3 = 3, with 1 left over.
  4. Thousandths: it is the same sum again. 10 ÷ 3 = 3, with 1 left over.
  5. The same 1 comes back every time, so the digit 3 comes back every time.
  6. So 1/3 = 0.333... and the 3s never stop.

Answer: No, it repeats: 0.333...

A decimal whose digits repeat for ever is a recurring decimal. We write three dots after the repeating digits. Some fractions give a recurring decimal and some give a terminating one. A fraction in simplest form ends when its bottom number is made only of 2s and 5s.

Try it

Check 1/4 with a strip of 20 squares

You need: squared paper, a pencil.

  1. Draw a strip 20 squares long and 1 square high. It is one whole.
  2. Cut the strip into 4 equal parts. Each part is 5 squares.
  3. Shade one part. That is 1/4.
  4. Write the shaded squares as a fraction of 20. Then change it into hundredths.

What you will see: You shaded 5 of 20 squares. 5/20 = 25/100, which is 0.25. So 1/4 = 0.25.

If you cannot try it, this is what happens.
A bar cut into 20 equal parts with 5 shaded. The caption says 5 of 20 squares.5 of 20 squares

A wrong idea many people have

3/8 = 0.38, because you write the 3 and the 8 on either side of the point.

Why it feels right: The fraction has two numbers and a decimal has two parts. It looks like a match.

The test that breaks it: Try the rule on 1/2. It would give 0.12. But one half is 0.5. The rule fails.

What is true: A fraction is a division. 3/8 means 3 ÷ 8, which is 0.375.

Key words

terminating decimal
a decimal that ends, like 0.375.
recurring decimal
a decimal whose digits repeat for ever, like 0.333...

Back to your guess

The answer is A: 0.375. Divide 3 by 8.

3/8 means 3 ÷ 8, and 3 ÷ 8 = 0.375. The top number is divided by the bottom number. Writing the numbers either side of a point does not work: by that rule 1/2 would be 0.12.

Check yourself

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

  1. What is 3/5 as a decimal?

    Tap your answer.

    1. Not this one. You wrote the 3 and the 5 either side of a point. A fraction is a division. Work out 3 ÷ 5. Go to M3.3.7. Try another.

    2. Yes, that is right. 3/5 means 3 ÷ 5. 30 tenths ÷ 5 = 6 tenths, so 3 ÷ 5 = 0.6.

    3. Not this one. You divided 5 by 3, which gives 1.666... The top number is divided by the bottom number, so work out 3 ÷ 5. Go to M3.3.7. Try another.

    See the answer

    0.6

    3/5 means 3 ÷ 5. 30 tenths ÷ 5 = 6 tenths, so 3 ÷ 5 = 0.6.

    • If you chose A: You wrote the 3 and the 5 either side of a point. A fraction is a division. Work out 3 ÷ 5. Go to M3.3.7.
    • If you chose C: You divided 5 by 3, which gives 1.666... The top number is divided by the bottom number, so work out 3 ÷ 5. Go to M3.3.7.

  2. The bar is one whole. What decimal is the shaded part?

    A bar cut into 4 equal parts with 3 shaded. The caption says one whole.one whole
    Tap your answer.

    1. Not this one. You wrote the 3 and the 4 either side of a point. Divide instead: 3 ÷ 4. Go to M3.3.7. Try another.

    2. Not this one. That is the part that is not shaded, 1/4. Count the shaded parts, 3, and divide by the 4 parts. Go to M3.3.7. Try another.

    3. Yes, that is right. 3 of 4 parts are shaded, so the fraction is 3/4. 3 ÷ 4 = 0.75.

    See the answer

    0.75

    3 of 4 parts are shaded, so the fraction is 3/4. 3 ÷ 4 = 0.75.

    • If you chose A: You wrote the 3 and the 4 either side of a point. Divide instead: 3 ÷ 4. Go to M3.3.7.
    • If you chose B: That is the part that is not shaded, 1/4. Count the shaded parts, 3, and divide by the 4 parts. Go to M3.3.7.

  3. Write 0.6 as a fraction in simplest form.

    Tap your answer.

    1. Yes, that is right. 0.6 is 6 tenths, which is 6/10. The HCF of 6 and 10 is 2, so 6/10 = 3/5.

    2. Not this one. That is the right amount, but it is not in simplest form. 6 and 10 are both in the 2 times table, so divide both by 2. Go to M3.3.7. Try another.

    3. Not this one. You put the 6 on the bottom. 0.6 is 6 tenths, so the 6 goes on the top, over 10. Go to M3.3.7. Try another.

    See the answer

    3/5

    0.6 is 6 tenths, which is 6/10. The HCF of 6 and 10 is 2, so 6/10 = 3/5.

    • If you chose B: That is the right amount, but it is not in simplest form. 6 and 10 are both in the 2 times table, so divide both by 2. Go to M3.3.7.
    • If you chose C: You put the 6 on the bottom. 0.6 is 6 tenths, so the 6 goes on the top, over 10. Go to M3.3.7.

  4. Write 0.125 as a fraction in simplest form.

    Tap your answer.

    1. Not this one. That is the right amount, but it is not in simplest form. 125 and 1 000 share a factor of 125, so you can divide both by it. Go to M3.3.7. Try another.

    2. Yes, that is right. 0.125 is 125 thousandths, 125/1000. The HCF of 125 and 1 000 is 125. 125 ÷ 125 = 1 and 1 000 ÷ 125 = 8.

    3. Not this one. You put the 125 on the bottom. 0.125 is 125 thousandths, so 125 goes on the top, over 1 000. Go to M3.3.7. Try another.

    See the answer

    1/8

    0.125 is 125 thousandths, 125/1000. The HCF of 125 and 1 000 is 125. 125 ÷ 125 = 1 and 1 000 ÷ 125 = 8.

    • If you chose A: That is the right amount, but it is not in simplest form. 125 and 1 000 share a factor of 125, so you can divide both by it. Go to M3.3.7.
    • If you chose C: You put the 125 on the bottom. 0.125 is 125 thousandths, so 125 goes on the top, over 1 000. Go to M3.3.7.

  5. Which of these fractions gives a decimal that repeats for ever?

    Tap your answer.

    1. Not this one. It stops at 0.375, as in this chapter. Nothing is left over, so it does not repeat. Go to M3.3.7. Try another.

    2. Not this one. Divide 1 by 4. You get 0.25, and nothing is left over. The decimal stops, so it does not repeat. Go to M3.3.7. Try another.

    3. Yes, that is right. 1 ÷ 3 leaves 1 over every time, so the 3 repeats for ever: 0.333... The other two divisions end with nothing left over.

    See the answer

    1/3

    1 ÷ 3 leaves 1 over every time, so the 3 repeats for ever: 0.333... The other two divisions end with nothing left over.

    • If you chose A: It stops at 0.375, as in this chapter. Nothing is left over, so it does not repeat. Go to M3.3.7.
    • If you chose B: Divide 1 by 4. You get 0.25, and nothing is left over. The decimal stops, so it does not repeat. Go to M3.3.7.

  6. Change 5/8 into a decimal.

    See the answer

    0.625

    5 ÷ 8: 50 ÷ 8 = 6 left 2, 20 ÷ 8 = 2 left 4, 40 ÷ 8 = 5. So 0.625.

  7. Write 0.45 as a fraction in simplest form.

    See the answer

    9/20

    0.45 is 45 hundredths, 45/100. The HCF of 45 and 100 is 5, so 45/100 = 9/20.

  8. From an earlier chapter

    Work out 2.4 × 5.

    See the answer

    12

    24 × 5 = 120. 2.4 has 1 decimal place, so the answer is 12.0, which is 12.

Keep it

Copy this into your exercise book.

A fraction is a division. 3/8 means 3 ÷ 8 = 0.375. Read a decimal by its place value, then simplify. If the same remainder comes back, the decimal repeats.

Draw: A bar cut in 8 with 3 shaded. Under it a number line 0 to 1 with a point at 0.375.

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 (828 KB). It opens on any phone.