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GCSE Maths · EdexcelChapter 4 · Decimals, rounding and estimation
GCSE Maths · Decimals, Rounding and Estimation · Higher · Lesson 4.5

GCSE Maths: Decimals and fractions (Higher)

Part of the GCSE Maths decimals, rounding and estimation chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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4.5

Decimals and fractions

Edexcel 1MA1: N10, N2
Grade 3-5

Decimals and fractions are two ways of writing the same numbers. The columns after the decimal point are fractions: tenths 110, hundredths 1100 and thousandths 11000. So 0.7 means 710, and 0.37 means 37100. A terminating decimal is a decimal that stops (like 0.375). Every terminating decimal can be written as a fraction.

Changing a decimal into a fraction

  1. Look at the last digit and say which column it is in (tenths, hundredths, thousandths…). That gives the denominator: 10, 100, 1000…
  2. Write the digits after the decimal point as the numerator.
  3. Simplify the fraction by dividing the top and bottom by their highest common factor (Chapter 3).
  4. If there is a whole number part (like the 2 in 2.35), keep it in front as a mixed number.
75 of the 100 squares are shaded, so 0.75 = 75100 = 34.

Changing a fraction into a decimal

A fraction is a division: 38 means 3 ÷ 8. So you can divide the top by the bottom using short division (section 4.4), adding zeros after the decimal point until the remainder is 0. If the denominator goes into 10, 100 or 1000, there is a quicker way: make an equivalent fraction with denominator 10, 100 or 1000. For example 720 = 35100 = 0.35.

It helps to remember these:

121434151101838120
0.50.250.750.20.10.1250.3750.05

Not every fraction gives a terminating decimal. 13 = 0.3333… goes on for ever (a recurring decimal). You will meet these in Chapter 3 (Higher). Here we only use fractions that give decimals that stop.

Comparing fractions and decimals

To order a mixture of fractions and decimals, change everything to decimals, then compare as in section 4.1.

Key ideas

  • The last decimal place gives the denominator: tenths → 10, hundredths → 100, thousandths → 1000.
  • Always simplify the fraction (divide top and bottom by the HCF).
  • Fraction → decimal: top ÷ bottom, or make the denominator 10, 100 or 1000.
  • To compare a fraction with a decimal, change both to decimals.
Worked example 4.5.1
Write 0.375 as a fraction in its simplest form.

Key words. A terminating decimal stops after a few digits (like 0.375). A mixed number has a whole number part and a fraction part.

Method. A decimal is a fraction with a denominator of 10, 100, 1000... The last decimal place tells you which. Write the fraction, then simplify.

  1. Say the decimal part in words to find the denominator. The last digit is in the thousandths column, so the denominator is 1000.
    0.375: last digit in the thousandths column, so denominator = 1000
  2. Write the digits after the decimal point over the denominator. The digits after the decimal point make the numerator.
    0.375 = 3751000
  3. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
    Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
    The highest common factor is 125.
  4. Divide the top and the bottom by 125. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    375 ÷ 125 = 3 and 1000 ÷ 125 = 8
    3751000 = 38

Check. Change back: 3 ÷ 8 = 0.375. ✓

Watch out. Always simplify. 0.25 = 25/100 is not finished until it is 1/4.

Worked example 4.5.2
Write 2.35 as a mixed number in its simplest form.

Key words. A terminating decimal stops after a few digits (like 0.375). A mixed number has a whole number part and a fraction part.

Method. A decimal is a fraction with a denominator of 10, 100, 1000... The last decimal place tells you which. Write the fraction, then simplify.

  1. Say the decimal part in words to find the denominator. The last digit is in the hundredths column, so the denominator is 100.
    2.35: last digit in the hundredths column, so denominator = 100
  2. Keep the whole number part to one side. Write the decimal part as a fraction. The digits after the decimal point make the numerator.
    2.35 = 2 and 35100
  3. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 35: 1, 5, 7, 35
    Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
    The highest common factor is 5.
  4. Divide the top and the bottom by 5. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    35 ÷ 5 = 7 and 100 ÷ 5 = 20
    35100 = 720
  5. Write the whole number and the fraction together. This gives a mixed number.
    2720

Check. Change back: 47 ÷ 20 = 2.35. ✓

Watch out. Always simplify. 0.25 = 25/100 is not finished until it is 1/4.

Worked example 4.5.3
Write 720 as a decimal.

Method. Make the denominator 10, 100 or 1000 with an equivalent fraction, then read off the decimal.

  1. Find what to multiply the denominator by to get 100. A fraction with denominator 10, 100 or 1000 is easy to write as a decimal. We use equivalent fractions: multiply the top and bottom by the same number.
    20 × 5 = 100
  2. Multiply the top by the same number. Top and bottom must be multiplied by the same number to keep the fraction the same size.
    720 = 35100
  3. Write as a decimal. The denominator 100 tells us the last digit is in the hundredths column.
    35100 = 0.35

Check. Divide the numerator by the denominator: 7 ÷ 20 = 0.35. ✓

Watch out. Whatever you multiply the bottom by, multiply the top by as well.

Worked example 4.5.4
Write 38 as a decimal.

Method. A fraction means top divided by bottom. Use short division and add zeros after the decimal point until there is no remainder.

  1. A fraction is a division: the top divided by the bottom. The fraction bar means divide, so 38 means 3 ÷ 8.
    3 ÷ 8
  2. Write 3 as 3.0 and use short division by 8. Add a decimal point and zeros after the number (this does not change its value) so that you can keep dividing.
    3 ÷ 8 = 0 remainder 3
    The number has no decimal point written, so imagine one after the last digit (3 is 3.0). Put the decimal point in the answer straight above it.
    30 ÷ 8 = 3 remainder 6 (an extra zero added after the decimal point)
    60 ÷ 8 = 7 remainder 4 (an extra zero added after the decimal point)
    40 ÷ 8 = 5 exactly (an extra zero added after the decimal point)
  3. Read the answer.
    3 ÷ 8 = 0.375

Check. Multiply back: 0.375 × 8 = 3. ✓

Watch out. Divide the top by the bottom, never the other way round.

Worked example 4.5.5
Write these in order, smallest first:   0.6,   58,   0.58,   34

Method. Change everything to decimals, then order them.

  1. Change every fraction into a decimal. Dividing the top by the bottom gives the decimal. Now all the numbers are in the same form and can be compared.
    0.6 is already a decimal
    58 = 5 ÷ 8 = 0.625
    0.58 is already a decimal
    34 = 3 ÷ 4 = 0.75
  2. Give every decimal the same number of decimal places. Add zeros on the end until each has 3 decimal places.
    0.6 = 0.600
    58 = 0.625
    0.58 = 0.580
    34 = 0.750
  3. Put them in order, smallest first. Compare column by column from the left.
    0.580, 0.600, 0.625, 0.750
  4. Write the original numbers in this order. The answer should use the numbers from the question.
    0.58,  0.6,  58,  34

Check. Each number in the answer is bigger than the one before it. ✓

Watch out. You can also change everything to fractions with a common denominator, but decimals are usually quicker when the fractions have denominators like 2, 4, 5, 8 or 10.

Watch out

The denominator comes from the last decimal place, not the first. 0.05 has its last digit in the hundredths column, so 0.05 = 5100 = 120 (not 510).

Exercise 4E  Decimals and fractions

Non-calculator
1
FWrite each decimal as a fraction in its simplest form.
(a)0.7
(b)0.25
(c)0.6
(d)0.35
(e)0.125
(f)0.04
2
FWrite each fraction as a decimal.
(a)12
(b)35
(c)920
(d)1325
(e)1750
(f)45
3
FWrite each fraction as a decimal.
(a)38
(b)58
(c)18
(d)740
4
FWrite each decimal as a mixed number in its simplest form.
(a)3.5
(b)1.25
(c)2.04
(d)6.8
5
FThe diagram is a 10 by 10 grid. Some of the squares are shaded. Write the shaded part as a decimal and as a fraction in its simplest form.
6
FWrite these in order, smallest first.
(a)0.4, 38, 0.35
(b)710, 0.65, 45, 0.7
(c)0.55, 12, 920, 0.5
7
PSWhich is bigger, 38 or 0.4? Show your working.
8
PSA bag of flour has a mass of 1.5 kg. Joe uses 35 of the flour. How many kilograms of flour does he use?
9
RExplain why 0.5 = 510 = 12.
10
RIs 0.3 the same as 3100? Give a reason.
11
RWrite a fraction that is between 0.3 and 0.4. Give your answer in the form ab where b is no more than 20.
12
FWrite 0.0125 as a fraction in its simplest form.
13
RShow that 316 = 0.1875 as a decimal, and write 3.1875 as a mixed number in its simplest form.

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