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Step-by-step practice: Maths · Level 5Chapter 6 · Adding and subtracting fractions with the same denominator
Level 5 · Chapter 6

Adding and subtracting fractions with the same denominator

By the end of this chapter you will be able to

  • Compare and order fractions with the same denominator
  • Write equivalent fractions with a small denominator
  • Add two fractions with the same denominator, simplifying when needed
  • Subtract fractions with the same denominator
  • Write a total greater than one as a mixed number, and solve fraction problems with lengths and amounts

You should already know

  • Fractions of wholes and amounts (Level 4, chapter 7)
  • Halves and quarters of amounts (Level 4, chapter 7)

Where this leads

  • Fractions with different denominators (a later level)
  • Decimals to tenths and hundredths (chapter 7)

Step-by-step practice: maths (Level 5). Small steps, repeated until fluent. Content: Add and subtract fractions with the same denominator (Y5 paraphrase), Recognise and show, using diagrams, equivalent fractions with a small denominator (Y4 paraphrase), Compare and order fractions with the same denominator (Y5 paraphrase), Solve problems involving fractions of a length or an amount (Y5 paraphrase).

Exercise P  Do you remember?

Non-calculator
1
FWhat is 1/2 of 14?
2
FWrite 2 + 3/10 as a decimal.
3
FWhat is 6 × 7 + 3?
4
FWork out 5,000 - 1,250.
5
FWhich is greater: 0.6 or 0.5?
6.1

Comparing and ordering fractions, and equal fractions

Level 5
Compare and order fractions with the same denominator, Show equivalent fractions with small denominators, using diagrams
Video lesson: Comparing and ordering fractions, and equal fractionsComing soon

When fractions have the same bottom number, the parts are the same size. Compare the top numbers: 6/8 is more than 3/8.

Equal fractions have the same value but different names. Half of a bar is the same as 2 parts of 4: 1/2 = 2/4.

Key ideas

  • Same bottom number: compare the top numbers.
  • Multiply the top and bottom by the same number to find an equal fraction.
  • A fraction with the top and bottom equal is one whole.
Worked example 6.1.1
Which is greater: 3/8 or 6/8?

Method. With the same denominator, compare the numerators.

  1. Look at the bottom numbers. Both fractions have the same bottom number, so the parts are the same size.
    Both are 8ths
  2. Compare the top numbers. With the same size of part, the bigger top number is the bigger fraction.
    3 and 6
  3. Choose the greater fraction. The fraction with more parts is greater.
    6/8

Check. Check: 6 parts is more than 3 parts of the same size.

Watch out. Do not compare the bottom numbers. The bigger the bottom number, the smaller each part.

Worked example 6.1.2
Put these in order, smallest first: 5/6, 2/6, 4/6, 1/6.

Method. With the same denominator, the order is the order of the numerators.

  1. Look at the bottom numbers. They are all the same, so compare the top numbers only.
    All are 6ths
  2. Order the top numbers. Smallest top number first.
    1, 2, 4, 5
  3. Write the fractions in order. Put the fractions back in the same order.
    1/6, 2/6, 4/6, 5/6

Check. Check: the first fraction has the fewest parts, and the last has the most.

Watch out. The fractions are the same size if the top numbers are equal. Use the top numbers to order them.

Worked example 6.1.3
Write 1/2 as an equal fraction with denominator 8.

Method. Multiply the top and bottom by the same number. The fraction stays the same size.

  1. Find how many times the bottom number goes into 8. The denominator is multiplied by this number.
    2 × 4 = 8
  2. Multiply the top number by the same number. Do the same to the top and bottom, so the fraction stays equal.
    1 × 4 = 4
  3. Write the equal fraction. The new fraction has the same value.
    4/8

Check. Check: 4/8 simplifies to 1/2.

Watch out. Multiply the top and the bottom. Do not add the same number to each.

Watch out

Watch out. 3/8 is not greater than 3/4. The bigger bottom number means smaller parts.

Exam tip

Grown-up tip. Fold a strip of paper into 8 and shade 3 parts, then shade 6 parts on another strip. Compare them.

Exercise 6A  Practice: comparing, ordering and equal fractions

Non-calculator
1
FWrite 1/2 as an equal fraction with denominator 4.
1/2
2
FWrite 1/2 as an equal fraction with denominator 6.
1/2
3
FWrite 3/4 as an equal fraction with denominator 12.
3/4
4
FWrite 1/3 as an equal fraction with denominator 6.
1/3
5
FPut these in order, smallest first: 1/8, 6/8, 3/8, 8/8.
6
FPut these in order, smallest first: 8/10, 5/10, 7/10, 9/10.
7
FWrite 1/2 as an equal fraction with denominator 8.
1/2
8
FWhich is greater: 1/4 or 2/4?
9
FWrite 2/3 as an equal fraction with denominator 6.
2/3
10
FWrite 1/3 as an equal fraction with denominator 12.
1/3
11
FWhich is greater: 3/4 or 2/4?
12
FWrite 2/4 as an equal fraction with denominator 8.
2/4
13
FPut these in order, smallest first: 6/8, 3/8, 8/8, 4/8.
14
FWrite 2/4 as an equal fraction with denominator 12.
2/4
15
FWhich is greater: 1/4 or 3/4?
16
FPut these in order, smallest first: 6/6, 2/6, 4/6, 3/6.
6.2

Adding fractions with the same denominator

Level 5
Add two fractions with the same denominator, Simplify the answer when you can
Video lesson: Adding fractions with the same denominatorComing soon

Add the top numbers. Keep the bottom number the same, because the parts are the same size: 2/8 + 3/8 = 5/8.

If the answer can be simplified, divide the top and bottom by the same number: 2/8 + 2/8 = 4/8 = 1/2.

Key ideas

  • Add the top numbers; keep the bottom number.
  • Simplify by dividing the top and bottom by the same number.
  • Never add the bottom numbers.
Worked example 6.2.1
Work out 2/8 + 3/8. Write the answer in its simplest form.

Method. Add the top numbers and keep the bottom number. Simplify if you can.

  1. Add the top numbers. The bottom number stays the same, because the parts are the same size.
    2 + 3 = 5
  2. Keep the bottom number 8. The total is counted in the same parts.
    5/8
  3. Write the answer. This is the simplest form.
    5/8

Check. Check: 5/8 is equal to 5/8.

Watch out. Do not add the bottom numbers too. 2/8 + 3/8 is 5/8, not 5/16.

Worked example 6.2.2
Work out 2/8 + 2/8. Write the answer in its simplest form.

Method. Add the top numbers and keep the bottom number. Simplify if you can.

  1. Add the top numbers. The bottom number stays the same, because the parts are the same size.
    2 + 2 = 4
  2. Keep the bottom number 8. The total is counted in the same parts.
    4/8
  3. Simplify. Divide the top and bottom by the same number, which is 4.
    4/8 = 1/2
  4. Write the answer. This is the simplest form.
    1/2

Check. Check: 1/2 is equal to 4/8.

Watch out. Do not add the bottom numbers too. 2/8 + 3/8 is 5/8, not 5/16.

Worked example 6.2.3
Work out 3/10 + 4/10. Write the answer in its simplest form.

Method. Add the top numbers and keep the bottom number. Simplify if you can.

  1. Add the top numbers. The bottom number stays the same, because the parts are the same size.
    3 + 4 = 7
  2. Keep the bottom number 10. The total is counted in the same parts.
    7/10
  3. Write the answer. This is the simplest form.
    7/10

Check. Check: 7/10 is equal to 7/10.

Watch out. Do not add the bottom numbers too. 2/8 + 3/8 is 5/8, not 5/16.

Watch out

Watch out. 2/8 + 3/8 is 5/8, not 5/16. The bottom number does not change when you add.

Exam tip

Grown-up tip. Ask the child: “Are the pieces the same size? If so, what changes when you add?”

Exercise 6B  Practice: adding fractions

Non-calculator
1
FWork out 5/8 + 1/8. Write the answer in its simplest form.
5/81/8
2
FWork out 1/6 + 2/6. Write the answer in its simplest form.
1/62/6
3
FWork out 2/5 + 2/5. Write the answer in its simplest form.
2/52/5
4
FWork out 3/6 + 1/6. Write the answer in its simplest form.
3/61/6
5
FWork out 5/10 + 4/10. Write the answer in its simplest form.
5/104/10
6
FWork out 1/8 + 2/8. Write the answer in its simplest form.
1/82/8
7
FWork out 4/8 + 2/8. Write the answer in its simplest form.
4/82/8
8
FWork out 2/4 + 1/4. Write the answer in its simplest form.
2/41/4
9
FWork out 1/4 + 1/4. Write the answer in its simplest form.
1/41/4
10
FWork out 3/5 + 1/5. Write the answer in its simplest form.
3/51/5
11
FWork out 2/5 + 1/5. Write the answer in its simplest form.
2/51/5
12
FWork out 1/5 + 3/5. Write the answer in its simplest form.
1/53/5
13
FWork out 1/4 + 2/4. Write the answer in its simplest form.
1/42/4
14
FWork out 4/6 + 1/6. Write the answer in its simplest form.
4/61/6
15
FWork out 2/6 + 2/6. Write the answer in its simplest form.
2/62/6
16
FWork out 6/10 + 1/10. Write the answer in its simplest form.
6/101/10
6.3

Subtracting fractions with the same denominator

Level 5
Subtract one fraction from another with the same denominator, Simplify the answer when you can
Video lesson: Subtracting fractions with the same denominatorComing soon

Subtract the top numbers and keep the bottom number: 7/8 - 2/8 = 5/8.

Simplify the answer if you can: 6/8 - 2/8 = 4/8 = 1/2.

Key ideas

  • Subtract the top numbers; keep the bottom number.
  • Simplify the answer if you can.
  • Check by adding back.
Worked example 6.3.1
Work out 7/8 - 2/8. Write the answer in its simplest form.

Method. Subtract the top numbers and keep the bottom number. Simplify if you can.

  1. Subtract the top numbers. The bottom number stays the same.
    7 - 2 = 5
  2. Keep the bottom number 8. The parts are the same size, so the bottom number does not change.
    5/8
  3. Write the answer. This is the simplest form.
    5/8

Check. Check: 5/8 + 2/8 gives 7/8.

Watch out. Do not subtract the bottom numbers. 7/8 - 3/8 is 4/8, which is 1/2.

Worked example 6.3.2
Work out 6/8 - 2/8. Write the answer in its simplest form.

Method. Subtract the top numbers and keep the bottom number. Simplify if you can.

  1. Subtract the top numbers. The bottom number stays the same.
    6 - 2 = 4
  2. Keep the bottom number 8. The parts are the same size, so the bottom number does not change.
    4/8
  3. Simplify. Divide the top and bottom by 4.
    4/8 = 1/2
  4. Write the answer. This is the simplest form.
    1/2

Check. Check: 1/2 + 2/8 gives 6/8.

Watch out. Do not subtract the bottom numbers. 7/8 - 3/8 is 4/8, which is 1/2.

Worked example 6.3.3
Work out 9/10 - 4/10. Write the answer in its simplest form.

Method. Subtract the top numbers and keep the bottom number. Simplify if you can.

  1. Subtract the top numbers. The bottom number stays the same.
    9 - 4 = 5
  2. Keep the bottom number 10. The parts are the same size, so the bottom number does not change.
    5/10
  3. Simplify. Divide the top and bottom by 5.
    5/10 = 1/2
  4. Write the answer. This is the simplest form.
    1/2

Check. Check: 1/2 + 4/10 gives 9/10.

Watch out. Do not subtract the bottom numbers. 7/8 - 3/8 is 4/8, which is 1/2.

Watch out

Watch out. 7/8 - 2/8 is 5/8. Do not write 5/6, and do not subtract the bottom numbers.

Exam tip

Grown-up tip. Use a bar cut into 8 parts. Take away 2 parts and count what is left.

Exercise 6C  Practice: subtracting fractions

Non-calculator
1
FWork out 2/5 - 1/5. Write the answer in its simplest form.
2/51/5
2
FWork out 2/6 - 1/6. Write the answer in its simplest form.
2/61/6
3
FWork out 4/5 - 2/5. Write the answer in its simplest form.
4/52/5
4
FWork out 3/4 - 1/4. Write the answer in its simplest form.
3/41/4
5
FWork out 8/10 - 4/10. Write the answer in its simplest form.
8/104/10
6
FWork out 6/8 - 2/8. Write the answer in its simplest form.
6/82/8
7
FWork out 9/10 - 7/10. Write the answer in its simplest form.
9/107/10
8
FWork out 2/4 - 1/4. Write the answer in its simplest form.
2/41/4
9
FWork out 5/6 - 2/6. Write the answer in its simplest form.
5/62/6
10
FWork out 3/4 - 2/4. Write the answer in its simplest form.
3/42/4
11
FWork out 9/10 - 4/10. Write the answer in its simplest form.
9/104/10
12
FWork out 3/6 - 1/6. Write the answer in its simplest form.
3/61/6
13
FWork out 6/8 - 3/8. Write the answer in its simplest form.
6/83/8
14
FWork out 4/5 - 1/5. Write the answer in its simplest form.
4/51/5
15
FWork out 5/6 - 4/6. Write the answer in its simplest form.
5/64/6
16
FWork out 4/6 - 3/6. Write the answer in its simplest form.
4/63/6
6.4

Totals greater than one

Level 5
Write a total greater than one as a mixed number, in its simplest form
Video lesson: Totals greater than oneComing soon

When the top number is bigger than the bottom number, the total is more than one whole. 5/4 is 4/4 and 1/4, so it is 1 1/4.

Write the whole ones first, then the fraction that is left.

Key ideas

  • A top number bigger than the bottom means more than one whole.
  • Write the whole ones, then the fraction left over.
  • Simplify the fraction part if you can.
Worked example 6.4.1
Work out 3/4 + 2/4. Write your answer as a mixed number if it is more than 1.

Method. More than one whole: add the top numbers, then write the whole ones out.

  1. Add the top numbers. Keep the bottom number the same.
    3 + 2 = 5
  2. Write the total. 5 parts of size 1/4.
    5/4
  3. Write as a mixed number. There are 1 whole one and 1/4 left over.
    5/4 = 1 1/4
  4. Write the answer. This is the answer in its simplest form.
    1 1/4

Check. Check: 1 1/4 is more than 1, as 5/4 is more than 4/4.

Watch out. A top number bigger than the bottom is more than one whole. Write it as a mixed number.

Worked example 6.4.2
Work out 3/5 + 4/5. Write your answer as a mixed number if it is more than 1.

Method. More than one whole: add the top numbers, then write the whole ones out.

  1. Add the top numbers. Keep the bottom number the same.
    3 + 4 = 7
  2. Write the total. 7 parts of size 1/5.
    7/5
  3. Write as a mixed number. There are 1 whole one and 2/5 left over.
    7/5 = 1 2/5
  4. Write the answer. This is the answer in its simplest form.
    1 2/5

Check. Check: 1 2/5 is more than 1, as 7/5 is more than 5/5.

Watch out. A top number bigger than the bottom is more than one whole. Write it as a mixed number.

Worked example 6.4.3
Work out 4/6 + 3/6. Write your answer as a mixed number if it is more than 1.

Method. More than one whole: add the top numbers, then write the whole ones out.

  1. Add the top numbers. Keep the bottom number the same.
    4 + 3 = 7
  2. Write the total. 7 parts of size 1/6.
    7/6
  3. Write as a mixed number. There are 1 whole one and 1/6 left over.
    7/6 = 1 1/6
  4. Write the answer. This is the answer in its simplest form.
    1 1/6

Check. Check: 1 1/6 is more than 1, as 7/6 is more than 6/6.

Watch out. A top number bigger than the bottom is more than one whole. Write it as a mixed number.

Watch out

Watch out. 5/4 is 1 1/4, not 5 and 4 as separate numbers. Count the whole ones first.

Exam tip

Grown-up tip. Use two bars cut into 4 parts. Ask: “How many full bars are shaded? How many parts are left?”

Exercise 6D  Practice: totals greater than one

Non-calculator
1
PSWork out 2/3 + 2/3. Write your answer as a mixed number if it is more than 1.
2/32/3
2
PSWork out 5/8 + 4/8. Write your answer as a mixed number if it is more than 1.
5/84/8
3
PSWork out 4/6 + 4/6. Write your answer as a mixed number if it is more than 1.
4/64/6
4
PSWork out 2/4 + 3/4. Write your answer as a mixed number if it is more than 1.
2/43/4
5
PSWork out 2/5 + 4/5. Write your answer as a mixed number if it is more than 1.
2/54/5
6
PSWork out 6/8 + 6/8. Write your answer as a mixed number if it is more than 1.
6/86/8
7
PSWork out 4/6 + 3/6. Write your answer as a mixed number if it is more than 1.
4/63/6
8
PSWork out 3/8 + 7/8. Write your answer as a mixed number if it is more than 1.
3/87/8
9
PSWork out 3/5 + 4/5. Write your answer as a mixed number if it is more than 1.
3/54/5
10
PSWork out 7/8 + 2/8. Write your answer as a mixed number if it is more than 1.
7/82/8
11
PSWork out 4/5 + 3/5. Write your answer as a mixed number if it is more than 1.
4/53/5
12
PSWork out 5/8 + 5/8. Write your answer as a mixed number if it is more than 1.
5/85/8
13
PSWork out 3/4 + 2/4. Write your answer as a mixed number if it is more than 1.
3/42/4
14
PSWork out 7/8 + 7/8. Write your answer as a mixed number if it is more than 1.
7/87/8
15
PSWork out 4/6 + 5/6. Write your answer as a mixed number if it is more than 1.
4/65/6
16
PSWork out 3/4 + 3/4. Write your answer as a mixed number if it is more than 1.
3/43/4
6.5

Fractions of lengths and amounts

Level 5
Solve problems involving fractions of a length or an amount
Video lesson: Fractions of lengths and amountsComing soon

To find a fraction of an amount, divide by the bottom number to find one part, then multiply by the top number.

For 3/5 of 60: 60 ÷ 5 = 12 for one fifth, and 3 × 12 = 36.

Key ideas

  • Divide by the bottom number to find one part.
  • Multiply by the top number to find the parts you want.
  • Keep the unit from the question.
Worked example 6.5.1
A ribbon is 60 cm long. Ayo uses 3/5 of it. How many cm does Ayo use?

Method. Divide by the bottom number to find one part, then multiply by the top number.

  1. Find one 5th of 60. Share 60 into 5 equal parts.
    60 ÷ 5 = 12
  2. Take 3 of those parts. The top number tells us how many parts.
    3 × 12 = 36
  3. Write the answer. Keep the unit from the question.
    36 cm

Check. Check: 12 × 5 = 60, and 3 parts is 36.

Watch out. Divide first, then multiply. Do not multiply by the bottom number.

Worked example 6.5.2
A ribbon is 48 cm long. Ayo uses 3/8 of it. How many cm does Ayo use?

Method. Divide by the bottom number to find one part, then multiply by the top number.

  1. Find one 8th of 48. Share 48 into 8 equal parts.
    48 ÷ 8 = 6
  2. Take 3 of those parts. The top number tells us how many parts.
    3 × 6 = 18
  3. Write the answer. Keep the unit from the question.
    18 cm

Check. Check: 6 × 8 = 48, and 3 parts is 18.

Watch out. Divide first, then multiply. Do not multiply by the bottom number.

Worked example 6.5.3
A ribbon is 35 cm long. Ayo uses 2/5 of it. How many cm does Ayo use?

Method. Divide by the bottom number to find one part, then multiply by the top number.

  1. Find one 5th of 35. Share 35 into 5 equal parts.
    35 ÷ 5 = 7
  2. Take 2 of those parts. The top number tells us how many parts.
    2 × 7 = 14
  3. Write the answer. Keep the unit from the question.
    14 cm

Check. Check: 7 × 5 = 35, and 2 parts is 14.

Watch out. Divide first, then multiply. Do not multiply by the bottom number.

Watch out

Watch out. Divide first, then multiply. Do not multiply by the bottom number.

Exam tip

Grown-up tip. Ask: “How many equal parts is the whole split into? How many of those parts do we want?”

Exercise 6E  Practice: fractions of lengths and amounts

Non-calculator
1
PSA ribbon is 66 cm long. Ayo uses 5/6 of it. How many cm does Ayo use?
2
PSThere are 80 sweets in a jar. 6/8 of them are sour. How many sweets are sour?
3
PSThere are 10 sweets in a jar. 4/5 of them are sour. How many sweets are sour?
4
PSThere are 80 sweets in a jar. 4/8 of them are sour. How many sweets are sour?
5
PSThere are 35 sweets in a jar. 1/5 of them are sour. How many sweets are sour?
6
PSThere are 44 sweets in a jar. 3/4 of them are sour. How many sweets are sour?
7
PSA ribbon is 30 cm long. Ayo uses 2/6 of it. How many cm does Ayo use?
8
PSThere are 50 sweets in a jar. 8/10 of them are sour. How many sweets are sour?
9
PSThere are 20 sweets in a jar. 1/10 of them are sour. How many sweets are sour?
10
PSA ribbon is 30 cm long. Ayo uses 3/10 of it. How many cm does Ayo use?
11
PSThere are 16 sweets in a jar. 3/8 of them are sour. How many sweets are sour?
12
PSA ribbon is 36 cm long. Ayo uses 3/6 of it. How many cm does Ayo use?
13
PSA ribbon is 66 cm long. Ayo uses 1/6 of it. How many cm does Ayo use?
14
PSA ribbon is 45 cm long. Ayo uses 1/5 of it. How many cm does Ayo use?
15
PSThere are 20 sweets in a jar. 2/4 of them are sour. How many sweets are sour?
16
PSThere are 24 sweets in a jar. 2/3 of them are sour. How many sweets are sour?

Chapter summary

SkillMethod
Same denominatorCompare and order by the top numbers; the parts are the same size
Equal fractionsMultiply the top and bottom by the same number
Adding and subtractingAdd or subtract the top numbers; keep the bottom number; simplify
Totals over oneWrite the whole ones, then the fraction left: 5/4 = 1 1/4
Fractions of amountsDivide by the bottom number, then multiply by the top number

Can you…?

  • I can compare and order fractions with the same denominator
  • I can write an equal fraction with a small denominator, using a diagram
  • I can add and subtract fractions with the same denominator, and simplify
  • I can write a total greater than one as a mixed number
  • I can find a fraction of a length or an amount

Key terms

numerator
the top number of a fraction
denominator
the bottom number of a fraction
simplify
divide the top and bottom by the same number to make a fraction smaller
mixed number
a whole number and a fraction, such as 1 1/4

Exercise 6Z  Check-up

Non-calculator
1
Work out 2/6 + 3/6. Write the answer in its simplest form.
2/63/6
2
Work out 4/5 + 4/5. Write your answer as a mixed number if it is more than 1.
4/54/5
3
Work out 7/10 - 4/10. Write the answer in its simplest form.
7/104/10
4
Work out 1/5 + 3/5. Write the answer in its simplest form.
1/53/5
5
Work out 1/6 + 3/6. Write the answer in its simplest form.
1/63/6
6
Work out 7/8 + 7/8. Write your answer as a mixed number if it is more than 1.
7/87/8
7
Put these in order, smallest first: 6/10, 2/10, 10/10, 5/10.
8
There are 90 sweets in a jar. 9/10 of them are sour. How many sweets are sour?
9
Write 3/4 as an equal fraction with denominator 8.
3/4
10
Work out 7/8 + 4/8. Write your answer as a mixed number if it is more than 1.
7/84/8

Total 19 marks. Allow about 19 minutes.

Exercise 6X  Practice test questions

Non-calculator
1
FWork out 3/5 - 1/5. Write the answer in its simplest form.[1 mark]
3/51/5
2
PSA ribbon is 8 cm long. Ayo uses 3/4 of it. How many cm does Ayo use?[1 mark]
3
FWork out 2/4 + 1/4. Write the answer in its simplest form.[1 mark]
2/41/4
4
FWork out 4/8 - 2/8. Write the answer in its simplest form.[1 mark]
4/82/8
5
PSThere are 32 sweets in a jar. 1/8 of them are sour. How many sweets are sour?[2 marks]
6
FWork out 1/10 + 4/10. Write the answer in its simplest form.[2 marks]
1/104/10
7
PSWork out 7/8 + 7/8. Write your answer as a mixed number if it is more than 1.[2 marks]
7/87/8
8
FWork out 2/10 + 1/10. Write the answer in its simplest form.[3 marks]
2/101/10
9
PSWork out 3/8 + 7/8. Write your answer as a mixed number if it is more than 1.[3 marks]
3/87/8
10
FWork out 8/10 - 4/10. Write the answer in its simplest form.[3 marks]
8/104/10
← Chapter 5: Short division with remaindersChapter 7: Decimals to tenths and hundredths →

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