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Primary Maths · Year 6Chapter 4 · Fractions
Year 6 · Chapter 4

Year 6 Fractions: worked examples and questions with answers

By the end of this chapter you will be able to

  • Simplify a fraction to its simplest form by dividing the top and bottom by their highest common factor
  • Write fractions with the same denominator, using the lowest common denominator (a common multiple)
  • Compare and order fractions, including fractions and mixed numbers greater than 1
  • Add and subtract fractions and mixed numbers with different denominators, and simplify the answer
  • Multiply two proper fractions and write the answer in its simplest form
  • Divide a proper fraction by a whole number
  • Change a fraction to a decimal with long division, and a decimal to a fraction
  • Solve problems with fractions and explain whether a statement is always, sometimes or never true

You should already know

  • Factors, common factors and the HCF; multiples and the LCM (Year 6, Chapter 3)
  • Equivalent fractions and adding fractions with the same denominator (Year 5, Chapter 6)
  • Times tables up to 12 × 12 and the related division facts

Where this leads

  • Decimals: place value to three decimal places and rounding (Chapter 5)
  • Percentages as fractions and decimals, such as 15% of 360 (Chapter 6)
  • Ratio and proportion with fractions of amounts (Chapter 7)

National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Use common factors to simplify fractions, Use common multiples to express fractions in the same denomination, Compare and order fractions, including fractions greater than 1, Add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions, Multiply simple pairs of proper fractions, writing the answer in its simplest form (for example 1/4 × 1/2 = 1/8), Divide proper fractions by whole numbers (for example 1/3 ÷ 2 = 1/6), Associate a fraction with division and calculate decimal fraction equivalents (for example 3/8 = 0.375).

Exercise P  Do you remember?

Non-calculator
1
FWrite all the factors of 16.
2
FFind the highest common factor (HCF) of 8 and 20.
3
FFind the lowest common multiple (LCM) of 4 and 6.
4
FWork out 34 of 20.
5
FWork out 58 + 28. Give your answer as a fraction.
4.1

Simplifying fractions

Year 6
Use common factors to simplify fractions, Write a fraction in its simplest form, Use the HCF to simplify, and check the answer
Video lesson: Simplifying fractionsComing soon

A fraction can look different and still have the same value. The bars below are the same length, so 68 and 34 are equal. The second bar is simpler to read: each part is bigger, and there are fewer parts.

A fraction is in its simplest form when the top and the bottom have no common factor except 1. To simplify a fraction, find the highest common factor (HCF) of the top and bottom. Then divide both numbers by it. The value stays the same, because the top and the bottom are divided by the same number.

6/83/4
Six of the eight parts are shaded. The same length is shaded in the four-part bar, which is 3 of 4 parts. So 6/8 = 3/4.

Key ideas

  • A fraction is in its simplest form when the HCF of the top and bottom is 1.
  • To simplify, divide the top and the bottom by the same number, which must divide into both exactly.
  • The HCF gives the quickest answer, because you divide once. Dividing in small steps also works.
  • Check the answer: if the top and bottom still share a factor, you can simplify again.
Worked example 4.1.1
Write 1218 in its simplest form.

Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.

Method. Find the HCF of the top and the bottom, then divide both numbers by it.

  1. Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.
    Factors of 12: 1, 2, 3, 4, 6, 12
    Factors of 18: 1, 2, 3, 6, 9, 18
    HCF = 6
  2. Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.
    12 ÷ 6 = 2 and 18 ÷ 6 = 3
  3. Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.
    Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.

Check. 2 × 6 = 12 and 3 × 6 = 18. ✓

Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.

Worked example 4.1.2
Write 3648 in its simplest form.

Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.

Method. Find the HCF of the top and the bottom, then divide both numbers by it.

  1. Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.
    Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
    Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
    HCF = 12
  2. Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.
    36 ÷ 12 = 3 and 48 ÷ 12 = 4
  3. Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.
    Factors of 3: 1, 3. Factors of 4: 1, 2, 4. Only 1 is shared, so the simplest form is 34.

Check. 3 × 12 = 36 and 4 × 12 = 48. ✓

Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.

Worked example 4.1.3
Simplify 2440 in two steps. Show each step.

Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.

Method. You can simplify in more than one step. Keep dividing by common factors until only 1 is shared.

  1. Divide the top and the bottom by 2. Both numbers divide by 2, so 2 is a common factor.
    2440 = 1220
  2. Divide the top and the bottom by 4. Both numbers still share a factor, so divide again.
    1220 = 35
  3. Check that the answer cannot be simplified. The top and the bottom have no shared factor except 1.
    1, 3 and 1, 5: only 1 is shared, so the simplest form is 35

Check. 3 × 8 = 24 and 5 × 8 = 40. ✓

Watch out. Two small steps are fine. The final answer must be the simplest form, not just a smaller fraction.

Watch out

Do not divide only the top or only the bottom. 1218 is not 618 in simplest form, and the value has changed. Also, a common factor is not always 2: check the HCF first.

Exercise 4A  Simplifying fractions

Non-calculator
1
FWrite each fraction in its simplest form.
(a)68
(b)1015
(c)912
(d)1216
(e)1525
(f)2030
2
FWrite each fraction in its simplest form.
(a)1421
(b)1827
(c)1640
(d)2436
(e)3550
(f)4256
3
FWhich of these fractions is in its simplest form? Write it down.
68    34    912
4
PSMia has read 16 of the 24 pages in her book. Write the fraction of the book she has read, in its simplest form.
5
PSA bag has 35 sweets. 14 of them are red. What fraction of the sweets are red? Give your answer in its simplest form.
6
PSThere are 48 pupils in Year 6. 18 of them are in the choir. What fraction of Year 6 are in the choir? Give your answer in its simplest form.
7
PSKai practises the piano for 50 minutes. What fraction of an hour is this? Write the fraction in its simplest form.
8
RSara says that 721 cannot be simplified, because 7 is a prime number. Is Sara right? Explain your answer.
9
RAlways, sometimes or never?   If the top and the bottom of a fraction are both even, the fraction can be simplified.
10
RChallenge: how many fractions with a bottom number less than 50 are equal to 25? Include 25 itself.
4.2

Common multiples and the same denominator

Year 6
Use common multiples to express fractions in the same denomination, Find the lowest common denominator of two or more fractions
Video lesson: Common multiples and the same denominatorComing soon

To add, subtract or compare fractions, the bottom numbers (the denominators) must be the same. We say the fractions are in the same denomination. Look at the bars below. 14 and 56 are cut into different-sized parts, so it is hard to compare them. Cut both bars into twelfths, and the fractions are easy to see: 312 and 1012.

The new bottom number must be a common multiple of the bottoms: 12 is in the 4 times table and in the 6 times table. The lowest common denominator is the lowest common multiple (LCM), so it gives the smallest numbers to work with.

3/1210/12
Both bars are cut into twelfths. Three twelfths is 1/4, and ten twelfths is 5/6.

Key ideas

  • A common denominator is a bottom number that two or more fractions share.
  • The lowest common denominator is the LCM of the bottom numbers.
  • To change a fraction, multiply its top and bottom by the same number. The value stays the same.
  • Work out what the bottom must be multiplied by: 14 needs × 3 to make 12.
Worked example 4.2.1
Write 14 and 56 with the same denominator.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Find the lowest common denominator, then change each fraction to it.

  1. Find the lowest common denominator. It is the LCM of the bottom numbers.
    LCM of 4, 6 = 12
  2. Make every bottom number 12. Multiply the top and the bottom of each fraction by the same number.
    14 = 312 (top and bottom × 3)
    56 = 1012 (top and bottom × 2)
  3. Write the answer. All the fractions now have the same bottom number.
    312 and 1012

Check. Each new fraction still has the same value as the old one. ✓

Watch out. The lowest common denominator is the smallest common multiple of the bottoms, so the numbers stay as small as possible.

Worked example 4.2.2
Write 23 and 35 with the same denominator.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Find the lowest common denominator, then change each fraction to it.

  1. Find the lowest common denominator. It is the LCM of the bottom numbers.
    LCM of 3, 5 = 15
  2. Make every bottom number 15. Multiply the top and the bottom of each fraction by the same number.
    23 = 1015 (top and bottom × 5)
    35 = 915 (top and bottom × 3)
  3. Write the answer. All the fractions now have the same bottom number.
    1015 and 915

Check. Each new fraction still has the same value as the old one. ✓

Watch out. The lowest common denominator is the smallest common multiple of the bottoms, so the numbers stay as small as possible.

Worked example 4.2.3
Write 58 and 712 with the lowest common denominator.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Find the lowest common denominator, then change each fraction to it.

  1. Find the lowest common denominator. It is the LCM of the bottom numbers.
    LCM of 8, 12 = 24
  2. Make every bottom number 24. Multiply the top and the bottom of each fraction by the same number.
    58 = 1524 (top and bottom × 3)
    712 = 1424 (top and bottom × 2)
  3. Write the answer. All the fractions now have the same bottom number.
    1524 and 1424

Check. Each new fraction still has the same value as the old one. ✓

Watch out. The lowest common denominator is the smallest common multiple of the bottoms, so the numbers stay as small as possible.

Worked example 4.2.4
Write 12, 23 and 34 with the same denominator.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Find the lowest common denominator, then change each fraction to it.

  1. Find the lowest common denominator. It is the LCM of the bottom numbers.
    LCM of 2, 3, 4 = 12
  2. Make every bottom number 12. Multiply the top and the bottom of each fraction by the same number.
    12 = 612 (top and bottom × 6)
    23 = 812 (top and bottom × 4)
    34 = 912 (top and bottom × 3)
  3. Write the answer. All the fractions now have the same bottom number.
    612, 812 and 912

Check. Each new fraction still has the same value as the old one. ✓

Watch out. The lowest common denominator is the smallest common multiple of the bottoms, so the numbers stay as small as possible.

Watch out

Any common multiple works as a common denominator, but the lowest one gives the smallest numbers. 4 and 6 share 12 and 24, but 12 is quicker. Do not just multiply the two bottoms together (4 × 6 = 24) unless you have to.

Exercise 4B  Common multiples and common denominators

Non-calculator
1
FList the first four common multiples of 3 and 4.
2
FWrite each pair of fractions with the lowest common denominator.
(a)12 and 13
(b)34 and 16
(c)25 and 310
(d)56 and 38
3
FWrite each fraction with the bottom number shown.
(a)35 with bottom number 20
(b)78 with bottom number 24
(c)56 with bottom number 18
4
FWhat is the lowest common denominator of each pair?
(a)14 and 16
(b)15 and 18
(c)13 and 110
(d)19 and 112
5
PSMia runs 34 km and Ravi runs 56 km. Who runs further? Write both distances with the same denominator first.
6
PSA cake recipe needs 14 of a cup of oats and 23 of a cup of flour. Write both amounts with the same denominator.
7
PSBen has 58 of a bag of marbles. Lee has 712 of a bag. Write both fractions with the lowest common denominator.
8
RBen says 15 is a common denominator for 13 and 15. Lily says 30 is also a common denominator. Who is right? Explain.
9
RWrite 14 and 38 with the lowest common denominator.
10
RChallenge: write 512 and 718 with the lowest common denominator.
4.3

Comparing and ordering fractions

Year 6
Compare and order fractions, including fractions greater than 1, Use a common denominator to compare fractions, Use mixed numbers and improper fractions
Video lesson: Comparing and ordering fractionsComing soon

To compare two fractions, make their bottom numbers the same. Then the bigger top number is the bigger fraction. Fractions can be greater than 1 too. On the number line below, 32 is one and a half, and 74 is one and three quarters, so 74 is the bigger one.

A mixed number such as 1 1/2 and an improper fraction such as 3/2 can be the same amount. Changing the mixed number to an improper fraction makes it easier to compare.

0123/27/4
Each small tick is a quarter. 7/4 is a little further to the right than 3/2.

Key ideas

  • Compare fractions by writing them with the same bottom number, then compare the top numbers.
  • When the bottoms are the same, the bigger top number is the bigger fraction.
  • A fraction bigger than 1 has a top number bigger than its bottom number. A mixed number like 2 1/3 is bigger than 2.
  • To order fractions, put them all over the same bottom number first.
Worked example 4.3.1
Which is bigger, 58 or 712?

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the same bottom number, then compare the top numbers.

  1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
    LCM of 8 and 12 = 24
  2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
    58 = 1524
    712 = 1424
  3. Compare the top numbers. When the bottoms are the same, the bigger top number is the bigger fraction.
    15 and 14: the bigger top number is 15
  4. Say which fraction is bigger. Write the smaller one, then the bigger one.
    712 < 58

Check. 1524 is the bigger value of the two, so 58 is the bigger fraction. ✓

Watch out. Do not just compare the bottom numbers. A bigger bottom number means smaller pieces.

Worked example 4.3.2
Put 23, 710, 56 and 35 in order, smallest first.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Make the bottom numbers the same, then compare the top numbers.

  1. Write them all with the same bottom number, 30. The lowest common denominator of the bottoms is 30. Change each fraction so that its bottom number is 30.
    23 = 2030
    710 = 2130
    56 = 2530
    35 = 1830
  2. Compare the top numbers. The bigger top number is the bigger fraction. Put them smallest first.
    18, 20, 21, 25 (out of 30)
  3. Write the fractions in order, using the fractions from the question.
    35, 23, 710, 56

Check. 35 is the smallest and 56 is the largest. ✓

Watch out. A bigger bottom number means smaller pieces. Compare the top numbers only after the bottoms are the same.

Worked example 4.3.3
Put 114, 52, 138 and 32 in order, smallest first. These are all bigger than 1.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Make the bottom numbers the same, then compare the top numbers.

  1. Write them all with the same bottom number, 8. The lowest common denominator of the bottoms is 8. Change each fraction so that its bottom number is 8.
    114 = 228
    52 = 208
    138 = 138
    32 = 128
  2. Compare the top numbers. The bigger top number is the bigger fraction. Put them smallest first.
    12, 13, 20, 22 (out of 8)
  3. Write the fractions in order, using the fractions from the question.
    32, 138, 52, 114

Check. 32 is the smallest and 114 is the largest. ✓

Watch out. A bigger bottom number means smaller pieces. Compare the top numbers only after the bottoms are the same.

Worked example 4.3.4
Which is bigger, 2 25 or 2 12?

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the same bottom number, then compare the top numbers.

  1. Write any mixed number as an improper fraction. Multiply the whole number by the bottom, then add the top.
    225: 2 × 5 + 2 = 12, so 125
    212: 2 × 2 + 1 = 5, so 52
  2. Find the lowest common denominator. Use the LCM of the two bottom numbers.
    LCM of 5 and 2 = 10
  3. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
    125 = 2410
    52 = 2510
  4. Compare the top numbers. When the bottoms are the same, the bigger top number is the bigger fraction.
    24 and 25: the bigger top number is 25
  5. Say which fraction is bigger. Write the smaller one, then the bigger one.
    225 < 212

Check. 2510 is the bigger value of the two, so 212 is the bigger fraction. ✓

Watch out. Do not just compare the bottom numbers. A bigger bottom number means smaller pieces.

Watch out

A bigger bottom number means smaller pieces, not a bigger fraction. 13 is bigger than 15, because thirds are bigger than fifths. Also, do not compare the top numbers before you make the bottoms the same.

Exercise 4C  Comparing and ordering fractions

Non-calculator
1
FWhich fraction is bigger? Write the bigger one.
(a)23 or 35?
(b)58 or 12?
(c)710 or 34?
(d)49 or 512?
2
FPut each set in order, smallest first.
(a)34, 56, 23
(b)12, 712, 58
(c)45, 34, 710
3
FWhich is bigger: 2 14 or 94?
4
FWhich is bigger: 1 34 or 53?
5
PSSam runs 74 km and Lee runs 1 23 km. Who runs further?
6
PSKofi has 56 of a litre of juice and Ada has 78 of a litre. Who has more juice?
7
RAli says 56 is bigger than 78, because 6 is smaller than 8. Is Ali right? Explain your answer.
8
RAlways, sometimes or never?   If two fractions have the same top number, the one with the smaller bottom number is bigger.
9
RWrite two fractions with a bottom number of 12 that are between 12 and 34.
10
RChallenge: put 23, 34 and 57 in order, largest first. Explain how you know.
4.4

Adding and subtracting fractions and mixed numbers

Year 6
Add and subtract fractions with different denominators and mixed numbers, Use equivalent fractions with a common denominator
Video lesson: Adding and subtracting fractions and mixed numbersComing soon

To add or subtract fractions with different bottom numbers, first make the bottoms the same. Then add or subtract the top numbers. Look at the bars: 13 is the same as 412, and 14 is the same as 312. So 13 + 14 = 412 + 312 = 712.

A mixed number has a whole part and a fraction part. You can add or subtract the whole parts and the fraction parts separately, or change each mixed number to an improper fraction first. This chapter uses the improper fraction method, because it always works. When you subtract, it also avoids having to borrow a whole number.

1/31/4total
Four twelfths plus three twelfths makes seven twelfths.

Key ideas

  • To add or subtract, make the bottom numbers the same first (use the lowest common denominator).
  • Only the top numbers are added or subtracted. The bottom number stays the same.
  • For mixed numbers, change each one to an improper fraction: 2 1/3 = 7/3 (2 × 3 + 1 = 7).
  • Give the answer in its simplest form, and change an improper answer back to a mixed number.
Worked example 4.4.1
Work out 13 + 14.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the LCM of their bottoms, then add the top numbers.

  1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
    LCM of 3 and 4 = 12
  2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
    13 = 412
    14 = 312
  3. Add the top numbers. The bottom number stays the same, because the pieces are the same size.
    412 + 312 = 712

Check. 712 − 14 = 13. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Worked example 4.4.2
Work out 56 − 38.

Key words. A common denominator is a bottom number that two or more fractions share. The lowest common denominator is the lowest common multiple (LCM) of the bottom numbers.

Method. Write both fractions with the LCM of their bottoms, then subtract the top numbers.

  1. Find the lowest common denominator. Use the LCM of the two bottom numbers.
    LCM of 6 and 8 = 24
  2. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
    56 = 2024
    38 = 924
  3. Subtract the top numbers. The bottom number stays the same, because the pieces are the same size.
    2024 − 924 = 1124

Check. 1124 + 38 = 56. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Worked example 4.4.3
Work out 2 13 + 1 34. Give your answer as a mixed number.

Key words. A mixed number has a whole part and a fraction part, such as 2 1/3. An improper fraction has a top number that is bigger than its bottom number, such as 7/3. The two forms are the same amount.

Method. Write both fractions with the LCM of their bottoms, then add the top numbers.

  1. Write each mixed number as an improper fraction. Multiply the whole number by the bottom, then add the top. The bottom stays the same.
    213: 2 × 3 + 1 = 7, so 73
    134: 1 × 4 + 3 = 7, so 74
  2. Find the lowest common denominator. Use the LCM of the two bottom numbers.
    LCM of 3 and 4 = 12
  3. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
    73 = 2812
    74 = 2112
  4. Add the top numbers. The bottom number stays the same, because the pieces are the same size.
    2812 + 2112 = 4912
  5. Write the answer as a mixed number. The top is bigger than the bottom, so take out the whole parts.
    4912 = 4112

Check. 4112 − 134 = 213. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Worked example 4.4.4
Work out 3 14 − 1 56. The top fraction is smaller, so you need to change the whole number first.

Key words. A mixed number has a whole part and a fraction part, such as 2 1/3. An improper fraction has a top number that is bigger than its bottom number, such as 7/3. The two forms are the same amount.

Method. Write both fractions with the LCM of their bottoms, then subtract the top numbers.

  1. Write each mixed number as an improper fraction. Multiply the whole number by the bottom, then add the top. The bottom stays the same.
    314: 3 × 4 + 1 = 13, so 134
    156: 1 × 6 + 5 = 11, so 116
  2. Find the lowest common denominator. Use the LCM of the two bottom numbers.
    LCM of 4 and 6 = 12
  3. Write both with the same bottom number. Multiply the top and bottom of each fraction by the same number.
    134 = 3912
    116 = 2212
  4. Subtract the top numbers. The bottom number stays the same, because the pieces are the same size.
    3912 − 2212 = 1712
  5. Write the answer as a mixed number. The top is bigger than the bottom, so take out the whole parts.
    1712 = 1512

Check. 1512 + 156 = 314. ✓

Watch out. Only the top numbers are added or subtracted. Do not add the bottom numbers.

Watch out

Do not add the bottom numbers. 13 + 14 is not 27. Also, when you subtract a mixed number, check whether the top fraction is smaller. If it is, use improper fractions or borrow one whole.

Exercise 4D  Adding and subtracting fractions

Non-calculator
1
FWork out each addition. Give each answer in its simplest form.
(a)12 + 13
(b)25 + 110
(c)34 + 16
(d)58 + 14
2
FWork out each subtraction. Give each answer in its simplest form.
(a)34 − 13
(b)56 − 12
(c)710 − 14
(d)45 − 38
3
FWork out each addition. Give your answers as mixed numbers.
(a)213 + 134
(b)125 + 212
(c)316 + 114
4
FWork out each subtraction. Give your answers as mixed numbers.
(a)314 − 156
(b)413 − 234
(c)518 − 212
5
PSAva has 34 of a litre of juice. She drinks 13 of a litre. How much juice is left? Give your answer in its simplest form.
1/3 drunkleft3/4 litre
6
PSSam walks 2 12 km to the park and then 1 34 km to the shop. How far does Sam walk altogether? Give your answer as a mixed number.
7
PSA jug holds 2 13 litres. Ella pours out 34 of a litre. How much is left in the jug?
8
RLily says that 12 + 13 = 25. What has Lily done wrong? Write the correct answer.
9
RFind the missing fraction: 14 + ? = 56. Explain how you found it.
10
RChallenge: find two different unit fractions (with a top number of 1) that add up to 12. Both bottom numbers must be less than 20.
4.5

Multiplying fractions

Year 6
Multiply simple pairs of proper fractions and write the answer in its simplest form, Find a fraction of a fraction and of an amount
Video lesson: Multiplying fractionsComing soon

To multiply two proper fractions, multiply the tops and multiply the bottoms. Then simplify. The picture shows 12 × 14 as an area. The square is cut into 2 columns (halves across) and 4 rows (quarters up). The blue columns show a half. The orange rows show a quarter. The green overlap is the answer: one of the eight small squares, so 12 × 14 = 18.

Finding a fraction of an amount means the same as multiplying. 12 of 14 is also 18.

1/2 across1/4 up
Blue: a half across. Orange: a quarter up. Green: the overlap, which is one eighth of the whole square.

Key ideas

  • To multiply fractions: tops × tops and bottoms × bottoms.
  • Multiplying two proper fractions gives a smaller answer than either fraction, because you take a part of a part.
  • Simplify the answer at the end. You can also simplify early by dividing a top and a bottom by a common factor.
  • A fraction of an amount is a multiplication: 23 of 12 is 23 × 12 = 8.
Worked example 4.5.1
Work out 14 × 12. Use the area picture above to check.

Key words. To multiply two fractions, multiply the tops and multiply the bottoms. A fraction of an amount means the same as multiplying by that fraction.

Method. Multiply the tops, multiply the bottoms, then simplify.

  1. Multiply the top numbers. The top of the answer is the product of the two tops.
    1 × 1 = 1
  2. Multiply the bottom numbers. The bottom of the answer is the product of the two bottoms.
    4 × 2 = 8
  3. Write the answer in its simplest form. The top and bottom have no shared factor except 1.
    18

Check. The answer is smaller than both 14 and 12, because we took a part of a part. ✓

Watch out. You may divide a top and a bottom by a common factor before you multiply, to keep the numbers small. The answer is the same.

Worked example 4.5.2
Work out 23 × 35. Give your answer in its simplest form.

Key words. To multiply two fractions, multiply the tops and multiply the bottoms. A fraction of an amount means the same as multiplying by that fraction.

Method. Multiply the tops, multiply the bottoms, then simplify.

  1. Multiply the top numbers. The top of the answer is the product of the two tops.
    2 × 3 = 6
  2. Multiply the bottom numbers. The bottom of the answer is the product of the two bottoms.
    3 × 5 = 15
  3. Write the answer as one fraction. Put the two results together.
    23 × 35 = 615
  4. Simplify. Divide the top and the bottom by their HCF, 3.
    615 = 25

Check. The answer is smaller than both 23 and 35, because we took a part of a part. ✓

Watch out. You may divide a top and a bottom by a common factor before you multiply, to keep the numbers small. The answer is the same.

Worked example 4.5.3
Work out 34 × 23.

Key words. To multiply two fractions, multiply the tops and multiply the bottoms. A fraction of an amount means the same as multiplying by that fraction.

Method. Multiply the tops, multiply the bottoms, then simplify.

  1. Multiply the top numbers. The top of the answer is the product of the two tops.
    3 × 2 = 6
  2. Multiply the bottom numbers. The bottom of the answer is the product of the two bottoms.
    4 × 3 = 12
  3. Write the answer as one fraction. Put the two results together.
    34 × 23 = 612
  4. Simplify. Divide the top and the bottom by their HCF, 6.
    612 = 12

Check. The answer is smaller than both 34 and 23, because we took a part of a part. ✓

Watch out. You may divide a top and a bottom by a common factor before you multiply, to keep the numbers small. The answer is the same.

Worked example 4.5.4
Work out 56 × 310.

Key words. To multiply two fractions, multiply the tops and multiply the bottoms. A fraction of an amount means the same as multiplying by that fraction.

Method. Multiply the tops, multiply the bottoms, then simplify.

  1. Multiply the top numbers. The top of the answer is the product of the two tops.
    5 × 3 = 15
  2. Multiply the bottom numbers. The bottom of the answer is the product of the two bottoms.
    6 × 10 = 60
  3. Write the answer as one fraction. Put the two results together.
    56 × 310 = 1560
  4. Simplify. Divide the top and the bottom by their HCF, 15.
    1560 = 14

Check. The answer is smaller than both 56 and 310, because we took a part of a part. ✓

Watch out. You may divide a top and a bottom by a common factor before you multiply, to keep the numbers small. The answer is the same.

Watch out

Do not add the tops and bottoms. 12 × 13 is 16, not 25. Also, the answer is smaller than both fractions. If you get a bigger answer, check your working.

Exercise 4E  Multiplying fractions

Non-calculator
1
FMultiply each pair of fractions. Give each answer in its simplest form.
(a)12 × 12
(b)13 × 14
(c)25 × 12
(d)34 × 13
(e)23 × 56
(f)35 × 13
2
FWork out each fraction of a fraction. Give each answer in its simplest form.
(a)34 of 25
(b)23 of 910
(c)58 of 45
(d)47 of 34
3
FMultiply and simplify each pair. Write each answer in its simplest form.
(a)23 × 38
(b)45 × 512
(c)37 × 79
4
PSA rectangle is 12 m wide and 23 m long. What is its area? Give your answer in m².
5
PSPriya has 25 of a cake. She gives 12 of her share to Ben. What fraction of the whole cake does Ben get?
6
PSSam reads 34 of a book on Monday. On Tuesday he reads 23 of the part that is left. What fraction of the whole book does he read on Tuesday?
7
RIs 12 × 12 = 12? Explain your answer.
8
RAlways, sometimes or never?   When you multiply two proper fractions, the answer is smaller than both of them.
9
RChallenge: which proper fraction, with a bottom number less than 10, multiplied by 34 gives 320? Write it in its simplest form.
10
PSRavi has 56 of a jug of lemonade. He pours 35 of it into cups. What fraction of the whole jug goes into cups? Give your answer in its simplest form.
4.6

Dividing fractions by whole numbers

Year 6
Divide proper fractions by whole numbers, writing the answer in its simplest form
Video lesson: Dividing fractions by whole numbersComing soon

Sharing a fraction between people is a division. Look at the picture. 13 of the bar is shaded. If we share it equally between 2 people, each person gets half of that shaded part. The bottom bar is cut into 6 parts, and one part is each share: 16.

So when you divide a fraction by a whole number, multiply the bottom number by that whole number. The top number stays the same. Then simplify if you can.

Amount: 1/3One share: 1/6 (one of 2 equal shares)
Top: one third of the bar. Bottom: the bar is cut into 6 equal parts, so one share of 1/3 shared between 2 is 1/6.

Key ideas

  • Dividing by a whole number makes the fraction smaller.
  • To divide a fraction by a whole number, multiply the bottom number by it: 13 ÷ 2 = 16.
  • If the top number divides exactly by the whole number, you can divide the top instead: 68 ÷ 2 = 38.
  • Check your answer by multiplying it back by the whole number.
Worked example 4.6.1
Work out 13 ÷ 2.

Key words. Dividing by a whole number shares a fraction into equal parts. Each part is smaller, so the answer is a smaller fraction.

Method. Dividing by a whole number makes the bottom number bigger. Multiply the bottom, keep the top, then simplify.

  1. Write the division. We are sharing the fraction into 2 equal parts.
    13 ÷ 2
  2. Multiply the bottom number by the whole number. Each piece is cut into smaller parts, so the bottom gets bigger. The top stays the same.
    13 ÷ 2 = 16
  3. Check that it cannot be simplified. The top and the bottom have no common factor except 1.
    16 is in its simplest form

Check. 16 × 2 = 13. ✓

Watch out. When the top divides exactly by the whole number, you can divide the top instead.

Worked example 4.6.2
Work out 34 ÷ 3. Give your answer in its simplest form.

Key words. Dividing by a whole number shares a fraction into equal parts. Each part is smaller, so the answer is a smaller fraction.

Method. Dividing by a whole number makes the bottom number bigger. Multiply the bottom, keep the top, then simplify.

  1. Write the division. We are sharing the fraction into 3 equal parts.
    34 ÷ 3
  2. Multiply the bottom number by the whole number. Each piece is cut into smaller parts, so the bottom gets bigger. The top stays the same.
    34 ÷ 3 = 312
  3. Simplify. Divide the top and the bottom by their HCF, 3.
    312 = 14

Check. 14 × 3 = 34. ✓

Watch out. When the top divides exactly by the whole number, you can divide the top instead.

Worked example 4.6.3
Work out 25 ÷ 4.

Key words. Dividing by a whole number shares a fraction into equal parts. Each part is smaller, so the answer is a smaller fraction.

Method. Dividing by a whole number makes the bottom number bigger. Multiply the bottom, keep the top, then simplify.

  1. Write the division. We are sharing the fraction into 4 equal parts.
    25 ÷ 4
  2. Multiply the bottom number by the whole number. Each piece is cut into smaller parts, so the bottom gets bigger. The top stays the same.
    25 ÷ 4 = 220
  3. Simplify. Divide the top and the bottom by their HCF, 2.
    220 = 110

Check. 110 × 4 = 25. ✓

Watch out. When the top divides exactly by the whole number, you can divide the top instead.

Watch out

Do not multiply the top number when you divide. 13 ÷ 2 is 16, not 23. Dividing makes the answer smaller, so it must be less than the fraction you started with.

Exercise 4F  Dividing fractions by whole numbers

Non-calculator
1
FWork out each division. Give each answer in its simplest form.
(a)13 ÷ 2
(b)34 ÷ 3
(c)25 ÷ 4
(d)56 ÷ 5
(e)38 ÷ 3
(f)49 ÷ 2
2
FWork out each division. Give each answer in its simplest form.
(a)23 ÷ 4
(b)58 ÷ 5
(c)67 ÷ 2
(d)310 ÷ 6
3
FWork out each division. Give each answer in its simplest form.
(a)12 ÷ 3
(b)35 ÷ 2
(c)47 ÷ 4
4
PSElla shares 34 of a pizza equally between 3 friends. How much pizza does each friend get?
5
PSA 23 litre bottle of juice is poured equally into 4 cups. How much juice goes into each cup? Give your answer in litres.
6
PSThree children share 25 of a cake equally. What fraction of the whole cake is each child’s share?
7
RSam says that 68 ÷ 2 = 38, because he halved the top number. Is Sam right? Explain.
8
RBen says 12 ÷ 2 = 14. Lily says 12 ÷ 2 is the same as 12 × 12. Are they both right? Explain.
9
RAlways, sometimes or never?   A proper fraction divided by a whole number is smaller than the fraction.
10
RChallenge: a fraction divided by 4 gives 316. Write the original fraction in its simplest form.
4.7

Fractions as division and decimals

Year 6
Associate a fraction with division and calculate decimal fraction equivalents (3/8 = 0.375), Convert between fractions and decimals, and compare them
Video lesson: Fractions as division and decimalsComing soon

A fraction is a division: 38 means 3 ÷ 8. Long division gives the answer 0.375. The number line below is cut into eighths. The orange dot is 38, which is exactly the same place as 0.375.

Decimals are fractions with bottoms of 10, 100, 1000 and so on. So 0.6 = 610 = 35 in its simplest form.

013/8 = 0.375
Each tick is an eighth. The point 3/8 is the same as the decimal 0.375.

Key ideas

  • A fraction is a division: the top number ÷ the bottom number.
  • Use long division to change a fraction to a decimal. Add zeros after the point until the remainder is 0.
  • A decimal stops only when the simplest bottom number has no factors except 2 and 5. So 13 never stops, but 38 does.
  • Change a decimal to a fraction by writing it over 10, 100 or 1000, then simplify.
Worked example 4.7.1
Write 38 as a decimal.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.

Method. Use long division. Keep adding zeros until the remainder is 0.

  1. Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.
    3 becomes 3.000
  2. Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.
    30 ÷ 8 = 3, remainder 6
    60 ÷ 8 = 7, remainder 4
    40 ÷ 8 = 5, remainder 0
  3. Write the digits after the point. Stop when the remainder is 0. The decimal stops.
    38 = 0.375

Check. 0.375 × 8 = 3. ✓

Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.

Worked example 4.7.2
Write 0.6 as a fraction in its simplest form.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.

Method. Write the decimal as a fraction over a power of ten, then simplify.

  1. Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 1 digit after the point, so the bottom number is 10.
    0.6 = 610
  2. Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.
    HCF of 6 and 10 = 2
  3. Simplify. Divide the top and the bottom by the HCF.
    6 ÷ 2 = 3 and 10 ÷ 2 = 5, so 35

Check. 3 ÷ 5 = 0.6, and 0.6 is the same. ✓

Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.

Worked example 4.7.3
Write 720 as a decimal.

Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.

Method. Use long division. Keep adding zeros until the remainder is 0.

  1. Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.
    7 becomes 7.000
  2. Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.
    70 ÷ 20 = 3, remainder 10
    100 ÷ 20 = 5, remainder 0
  3. Write the digits after the point. Stop when the remainder is 0. The decimal stops.
    720 = 0.35

Check. 0.35 × 20 = 7. ✓

Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.

Worked example 4.7.4
Which is bigger, 38 or 0.4?

Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.

Method. Write the fraction as a decimal, then compare the decimals digit by digit.

  1. Write the fraction as a decimal. Use long division. The answer is a decimal that stops.
    38 = 0.375
  2. Write both decimals with the same number of places. Add zeros at the end so the decimal points line up.
    0.375 and 0.400
  3. Compare the digits from the left. The first digit that is different tells you which one is bigger.
    0.375 and 0.4: the first difference is at the same place value
  4. Say which one is bigger. Write the bigger decimal.
    0.4 is bigger

Check. 0.375 is 38 exactly, and 0.4 is the bigger value. ✓

Watch out. Compare decimals from the left, one place at a time. 0.4 is bigger than 0.375, because 0.400 is bigger than 0.375.

Watch out

Do not stop long division too early. 38 is 0.375, not 0.3 or 0.38. The decimal stops when the remainder is 0. Also, 0.38 is a rounded answer, so it is not exactly 38.

Exercise 4G  Fractions and decimals

Non-calculator
1
FWrite each fraction as a decimal.
(a)14
(b)35
(c)58
(d)720
(e)925
(f)18
2
FWrite each decimal as a fraction in its simplest form.
(a)0.5
(b)0.75
(c)0.2
(d)0.45
(e)0.64
(f)0.125
3
FPut 14, 0.3 and 38 in order, smallest first.
4
PSOut of 20 pupils, 11 have a dog. What fraction of the pupils have a dog? Give your answer as a decimal.
5
RTom says that 38 of a metre is 0.38 of a metre. Is he right? Explain your answer.
6
RAlways, sometimes or never?   Every fraction with a bottom number of 10 can be written as a decimal with one decimal place.
7
RWhich of these fractions can be written as a decimal that stops? Write the ones that stop: 13, 38, 27 and 520.
8
PSWhich is bigger: 38 or 0.4? Write the bigger one.
9
PSA recipe needs 0.8 of a kilogram of flour. Write this as a fraction in its simplest form.
10
RChallenge: write 0.15 as a fraction in its simplest form. Multiply it by 8. Give your answer as a decimal.

Chapter summary

SkillMethod
SimplifyingFind the HCF of the top and bottom. Divide both by it. Check that only 1 is shared.
Same bottom numberFind the lowest common denominator (the LCM of the bottoms). Multiply the top and bottom of each fraction by the same number.
Comparing and orderingWrite them with the same bottom number, then compare the top numbers. Change mixed numbers to improper fractions first.
Adding and subtractingUse the LCM of the bottoms, add or subtract the top numbers, then simplify. Write improper answers as mixed numbers.
MultiplyingMultiply the tops, multiply the bottoms, then simplify. The answer is smaller than both fractions.
Dividing by a whole numberMultiply the bottom number by the whole number, then simplify.
DecimalsA fraction is a division: use long division. A decimal stops only when the simplest bottom number has factors 2 and 5 only.

Can you…?

  • I can simplify a fraction by dividing the top and bottom by their highest common factor.
  • I can find the lowest common denominator and write fractions with the same denominator.
  • I can compare and order fractions, including fractions and mixed numbers greater than 1.
  • I can add and subtract fractions with different denominators, and mixed numbers.
  • I can multiply two proper fractions and write the answer in its simplest form.
  • I can divide a proper fraction by a whole number.
  • I can change a fraction to a decimal with long division, and a decimal to a fraction.
  • I can say whether a statement is always, sometimes or never true, and explain why.

Key terms

Simplest form
A fraction is in its simplest form when the top and bottom have no common factor except 1.
Highest common factor (HCF)
The biggest number that divides exactly into two or more numbers.
Lowest common multiple (LCM)
The smallest number that is in the times tables of two or more numbers.
Common denominator
A bottom number that two or more fractions share.
Equivalent fraction
A fraction that has the same value as another, such as 1/2 = 2/4.
Improper fraction
A fraction whose top number is bigger than its bottom number, such as 7/4.
Mixed number
A whole number and a proper fraction together, such as 1 3/4.
Unit fraction
A fraction with a top number of 1, such as 1/5.
Terminating decimal
A decimal that stops, such as 0.375. Its fraction has a bottom number with only the factors 2 and 5 in its simplest form.
Long division
A method for dividing one number by another, one digit at a time.

Exercise 4Z  Check-up (non-calculator)

Non-calculator
1
Write 2436 in its simplest form.
2
Write 58 as an equivalent fraction with a bottom number of 24.
3
Which is bigger, 712 or 59?
4
Put 34, 56 and 23 in order, smallest first.
5
Work out 34 + 29.
6
Work out 5 13 − 2 56. Give your answer as a mixed number.
7
Work out 45 × 58.
8
Work out 512 ÷ 5.
9
Write 0.45 as a fraction in its simplest form.
10
Write 78 as a decimal.

Total 23 marks. Allow about 23 minutes.

Exercise 4X  SATs-style questions

Non-calculator
1
FWrite 1827 in its simplest form.[1 mark]
2
FWork out 34 + 18.[2 marks]
3
FWhich is bigger, 47 or 59? Show your working.[2 marks]
4
FWrite 35, 23 and 710 in order, smallest first.[2 marks]
5
FWork out 2 14 + 1 56. Give your answer as a mixed number.[2 marks]
6
FWork out 4 16 − 2 34. Show your working.[3 marks]
7
FWork out 23 × 910. Give your answer in its simplest form.[2 marks]
8
PSElla has 56 of a litre of water. She pours out 38 of a litre. How much water is left, in litres? Show your working.[3 marks]
9
RAlex says that 12 of 23 is bigger than 13. Is Alex right? Explain.[2 marks]
10
PSBen has 3 litres of paint. He uses 34 of a litre for a sign and 1 56 litres for a fence. How much paint is left, in litres? Show your working.[4 marks]
← Chapter 3: Common factors, multiples and primesChapter 5: Decimals →

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