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Primary Maths · Year 5Chapter 6 · Fractions
Year 5 · Equivalent Fractions and Mixed Numbers · Lesson 6.7

Year 5: Multiplying fractions by whole numbers and fractions of amounts

Part of the Year 5 equivalent fractions and mixed numbers chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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6.7

Multiplying fractions by whole numbers and fractions of amounts

Year 5
Multiply proper fractions and mixed numbers by whole numbers, supported by materials and diagrams, Fractions of amounts, Solve problems with bar models

Multiplying by a whole number is repeated addition. 3 × 25 means three lots of two fifths: 25 + 25 + 25 = 65 = 115. Look at the picture: 3 groups of 2 fifths make 6 fifths, which is one whole bar and 1 fifth. Multiply the numerator by the whole number and keep the denominator.

3 × 25 = 65 = 115. Each colour is one lot of 25.

For a mixed number, multiply the whole part and the fraction part separately, then add. 4 × 112: 4 × 1 = 4 and 4 × 12 = 42 = 2. Together 4 + 2 = 6. You can check by changing to an improper fraction: 112 = 32 and 4 × 32 = 122 = 6.

Fractions of amounts (a reminder). To find 34 of 36, divide by the denominator to find one quarter, then multiply by the numerator: 36 ÷ 4 = 9, and 3 × 9 = 27.

369999
34 of 36: each quarter is 9, and three quarters is 27.

Key ideas

  • Fraction × whole number: multiply the numerator, keep the denominator. 3 × 25 = 65. Then write it as a mixed number and simplify.
  • Mixed number × whole number: multiply the whole part and the fraction part separately, then add (or change to an improper fraction first).
  • Fraction of an amount: ÷ by the denominator, × by the numerator.
  • Finding the whole: if 35 of a number is 18, one fifth is 18 ÷ 3 = 6, so the whole is 5 × 6 = 30.
Worked example 6.7.1
Work out 3 × 25.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 3 × 25 as 3 lots of 25. Multiplying by a whole number is repeated addition.
    25 + 25 + 25
  2. Multiply the numerator by 3. Keep the denominator. We are counting fifths. 3 lots of 2 fifths is 6 fifths.
    3 × 25 = 3 × 25 = 65
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 65 means 6 ÷ 5.
    65 = 115
  4. Write the answer.
    3 × 25 = 115

Check. Divide back: 115 ÷ 3 = 25. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.2
Work out 4 × 38.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 4 × 38 as 4 lots of 38. Multiplying by a whole number is repeated addition.
    38 + 38 + 38 + 38
  2. Multiply the numerator by 4. Keep the denominator. We are counting eighths. 4 lots of 3 eighths is 12 eighths.
    4 × 38 = 4 × 38 = 128
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 128 means 12 ÷ 8.
    128 = 148
  4. Simplify the fraction part. 4 and 8 can both be divided by 4.
    48 = 4 ÷ 48 ÷ 4 = 12
    So the answer is 112.
  5. Write the answer.
    4 × 38 = 112

Check. Divide back: 112 ÷ 4 = 38. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.3
Work out 6 × 13.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 6 × 13 as 6 lots of 13. Multiplying by a whole number is repeated addition.
    13 + 13 + 13 + 13 + 13 + 13
  2. Multiply the numerator by 6. Keep the denominator. We are counting thirds. 6 lots of 1 thirds is 6 thirds.
    6 × 13 = 6 × 13 = 63
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 63 means 6 ÷ 3.
    63 = 2
  4. Write the answer.
    6 × 13 = 2

Check. Divide back: 2 ÷ 6 = 13. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.4
Five friends each drink 34 of a litre of water. How much water do they drink altogether?

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the numerator by the whole number and keep the denominator. If the answer is an improper fraction, change it to a mixed number or a whole number.

  1. Think of 5 × 34 as 5 lots of 34. Multiplying by a whole number is repeated addition.
    34 + 34 + 34 + 34 + 34
  2. Multiply the numerator by 5. Keep the denominator. We are counting quarters. 5 lots of 3 quarters is 15 quarters.
    5 × 34 = 5 × 34 = 154
  3. The answer is 1 or more, so change it to a mixed number or a whole number. 154 means 15 ÷ 4.
    154 = 334
  4. Write the answer.
    5 × 34 = 334

Check. Divide back: 334 ÷ 5 = 34. ✓

Watch out. Multiply the top only. The bottom stays the same: 3 × 2/5 is 6/5, not 6/15.

Worked example 6.7.5
Work out 4 × 112.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the whole part and the fraction part separately, then add the answers. (Or change to an improper fraction first.)

  1. Split the mixed number into its whole part and its fraction part. We multiply each part by the whole number, then add the answers (the distributive law).
    112 = 1 + 12
  2. Multiply the whole part. The whole numbers multiply as usual.
    4 × 1 = 4
  3. Multiply the fraction part. Multiply the numerator by 4 and keep the denominator.
    4 × 12 = 42 = 2
  4. Add the two answers. Put the whole parts and the fraction parts back together.
    4 + 2 = 6
  5. Write the answer.
    4 × 112 = 6

Check. Other way: 112 = 32, and 4 × 32 = 122 = 6. ✓

Watch out. Do not forget to multiply the whole part as well as the fraction part.

Worked example 6.7.6
Work out 3 × 214.

Key words. Multiplying a fraction by a whole number is the same as adding the fraction that many times. An improper fraction has a top number bigger than its bottom number.

Method. Multiply the whole part and the fraction part separately, then add the answers. (Or change to an improper fraction first.)

  1. Split the mixed number into its whole part and its fraction part. We multiply each part by the whole number, then add the answers (the distributive law).
    214 = 2 + 14
  2. Multiply the whole part. The whole numbers multiply as usual.
    3 × 2 = 6
  3. Multiply the fraction part. Multiply the numerator by 3 and keep the denominator.
    3 × 14 = 34
  4. Add the two answers. Put the whole parts and the fraction parts back together.
    6 + 34 = 634
  5. Write the answer.
    3 × 214 = 634

Check. Other way: 214 = 94, and 3 × 94 = 274 = 634. ✓

Watch out. Do not forget to multiply the whole part as well as the fraction part.

Worked example 6.7.7
Work out 34 of 36.

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. To find a fraction of an amount, divide by the denominator to find one part. Then multiply by the numerator.

  1. Divide the amount by the denominator. The denominator 4 tells us to cut the amount into 4 equal parts. One part is 14 of the amount.
    36 ÷ 4 = 9
  2. Multiply by the numerator. The numerator 3 tells us how many of those parts we want.
    3 × 9 = 27
  3. Write the answer.
    34 of 36 = 27
    369999

Check. Other way: 3 × 36 = 108, then ÷ 4 = 27. ✓

Watch out. Divide by the bottom number first. Then multiply by the top number.

Worked example 6.7.8
Sofia has £48. She spends 58 of it. How much does she spend?

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. To find a fraction of an amount, divide by the denominator to find one part. Then multiply by the numerator.

  1. Divide the amount by the denominator. The denominator 8 tells us to cut the amount into 8 equal parts. One part is 18 of the amount.
    £48 ÷ 8 = £6
  2. Multiply by the numerator. The numerator 5 tells us how many of those parts we want.
    5 × £6 = £30
  3. Write the answer.
    58 of £48 = £30
    £48£6£6£6£6£6£6£6£6

Check. Other way: 5 × £48 = £240, then ÷ 8 = £30. ✓

Watch out. Divide by the bottom number first. Then multiply by the top number.

Worked example 6.7.9
35 of a number is 18. What is the number?

Key words. “Of” means multiply. To find a fraction of an amount, divide by the denominator, then multiply by the numerator.

Method. Use a bar model. The known part tells us the value of one small part. Then multiply to find the whole.

  1. 3 parts are worth 18. Find the value of one part. The top number 3 is how many parts make 18, so divide by 3.
    18 ÷ 3 = 6
  2. Find the whole. The whole is 5 parts. The bottom number 5 is how many equal parts there are altogether.
    5 × 6 = 30
  3. Check with the fraction. Take the fraction of the whole you found.
    35 of 30 = 6 × 3 = 18 ✓
?66666

Check. 35 of 30 = 18. ✓

Watch out. Divide by the numerator first to find one part, then multiply by the denominator.

Watch out

Multiply the numerator only. 3 × 25 = 65, not 615. The denominator tells the size of the parts and it does not change. And in a mixed number, do not forget to multiply the whole part: 4 × 112 is not 1 + 2.

Exercise 6G  Multiplying by whole numbers and fractions of amounts

Non-calculator
1
FThe picture shows 3 lots of two fifths. Write the multiplication and work out the answer.
2
FWork out each multiplication.
(a)3 × 14
(b)4 × 16
(c)5 × 13
(d)6 × 18
3
FWork out each multiplication. Write each answer as a mixed number.
(a)4 × 37
(b)5 × 23
(c)3 × 58
(d)6 × 310
4
FWork out each multiplication. Give each answer in its simplest form.
(a)4 × 38
(b)6 × 512
(c)6 × 710
(d)8 × 34
5
FWork out each multiplication.
(a)2 × 114
(b)3 × 123
(c)4 × 212
(d)5 × 1310
6
FFind each missing fraction.
(a)3 × ? = 910
(b)3 × ? = 65
(c)5 × ? = 158
7
FWork out each fraction of an amount.
(a)34 of 36
(b)25 of 45
(c)58 of 48
(d)710 of 90
(e)23 of 27
(f)56 of 42
8
FWork out each answer.
(a)25 of £35
(b)38 of 64 kg
(c)712 of 60 minutes
(d)310 of 250 ml
9
PSSolve each problem.
(a)Five friends each drink 34 of a litre of water. How much do they drink altogether?
(b)Four bottles each hold 310 of a litre. How much do they hold altogether?
10
PSSolve each problem.
(a)A cake needs 212 cups of flour. How much flour is needed for 4 cakes?
(b)Each piece of rope is 113 m long. What is the total length of 3 pieces?
11
RTom says, “3 × 25 = 615.” What is wrong? What is the correct answer?
12
RAlways, sometimes or never?
(a)When you multiply a proper fraction by a whole number bigger than 1, the answer is bigger than the fraction.
(b)When you multiply a proper fraction by 3, the answer is greater than 1.
13
RWhich is the odd one out? Explain how you know.    3 × 23   6 × 13   4 × 12   2 × 113
14
RChallenge! I think of a fraction. I multiply it by 6 and the answer is 412. What fraction did I think of?

Exercise 6H  Fraction problems

Non-calculator
1
PSUse a bar model to solve each problem.
(a)35 of the stickers in a pack are shiny. There are 18 shiny stickers. How many stickers are in the pack?
(b)Leo spends 23 of his money on a football. The football costs £24. How much money did Leo have?
2
PSThere are 40 children in a club. 38 walk to school and 14 cycle. The rest come by bus. How many children come by bus?
walk 3/8cycle 1/4bus ?40
3
PSA chocolate bar has 12 squares. Ravi eats 14 of the bar and Priya eats 13 of the bar. How many squares are left?
4
RIs 34 of 20 more than 23 of 24? Show how you know.
5
PSDev reads three tenths of a book on Monday and two fifths of it on Tuesday.
(a)What fraction of the book has he read?
(b)What fraction of the book is left to read?
6
FWork out 12 + 14 + 18.
7
PSIn a class of 28 children, 37 are boys. How many girls are in the class?
8
PSAisha has £60. She spends 14 of it on a book and 25 of it on trainers. How much money does she have left?
9
PSA tank is 34 full. It holds 36 litres. How many litres does the tank hold when it is full?
10
RMei says, “3 × 25 and 2 × 35 have the same answer.” Is she right? Explain.
11
RChallenge! One third of a number is 7 less than half of the number. What is the number?

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