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Primary Maths · Year 4Chapter 5 · Fractions
Year 4 · Chapter 5

Year 4 Fractions: worked examples and questions with answers

By the end of this chapter you will be able to

  • Name unit and non-unit fractions of shapes and sets, and know that the parts must be equal
  • Find a fraction of an amount: divide by the bottom number, then multiply by the top number
  • Show families of equivalent fractions with bars, fraction walls and number lines, and fill in missing numbers
  • Add and subtract fractions with the same denominator, including answers that make a whole
  • Compare and order fractions using the top numbers, a fraction wall, or equivalent fractions
  • Write improper fractions and mixed numbers from pictures and number lines, and change between them
  • Count up and down in tenths and hundredths, and link tenths to hundredths

You should already know

  • Halves, quarters and three quarters from earlier years
  • Times tables and division facts up to 12 × 12 (Chapter 3)
  • Dividing by one-digit numbers (Chapter 4)
  • Number lines and counting on and back (Chapter 1)

Where this leads

  • Decimals and money: tenths and hundredths as decimals (Chapter 6)
  • Fractions of measures: lengths, masses and capacities (Chapter 7)
  • Adding fractions with different denominators (Years 5 and 6)
  • Simplifying fractions and fractions of amounts in Year 5

National Curriculum for England, Key Stage 2 mathematics, Year 4. Content: Recognise and show, using diagrams, families of common equivalent fractions, Solve problems involving increasingly harder fractions to calculate quantities, and fractions to divide quantities, including non-unit fractions where the answer is a whole number, Add and subtract fractions with the same denominator, Count up and down in tenths and hundredths, Recognise fractions greater than 1 as mixed numbers and improper fractions (introduced).

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FWhat fraction of the bar is shaded?
2
FWork out 24 ÷ 4.
3
FWhat is half of 14?
4
FFind the missing number.   4 × ? = 36
5
RIs this rectangle cut into quarters? Give a reason.
5.1

Fractions of shapes and sets

Year 4
Unit and non-unit fractions of shapes and sets, Parts must be equal

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Fractions of shapes and setsComing soon

A fraction is a part of a whole. To make a fraction we cut a whole thing into equal parts. Equal means exactly the same size. Think of a pizza cut into 8 equal slices. One slice is one eighth of the pizza.

A fraction has two numbers. The bottom number is the denominator. It tells us how many equal parts there are altogether. The top number is the numerator. It tells us how many of those parts we are talking about.

This bar is cut into 8 equal parts. 3 parts are shaded. The shaded part is three eighths, written 3/8.

38

The top number is the numerator (3 parts shaded). The bottom number is the denominator (8 equal parts).

A fraction with 1 on the top is a unit fraction: 1/2, 1/3, 1/4, 1/5... A fraction with any other number on the top, like 3/4 or 5/8, is a non-unit fraction. Three quarters is just three unit fractions of one quarter.

   14141414
3/4 is made of three parts, and each part is 1/4.

Fractions of a set. A fraction can also describe a group of things. Here are 8 counters. 3 are red. The fraction of the counters that are red is 3 out of 8, which is 3/8. The bottom number is the whole set (all 8 counters).

RRRBBBBB
3 of the 8 counters are red: the fraction is 3/8.

The parts must be equal. If the parts are different sizes, we cannot name them with a fraction like this.

This rectangle has 4 parts, but they are NOT equal. The shaded part is not one quarter.

Key words. The denominator is the bottom number. The numerator is the top number. A unit fraction has 1 on the top. A non-unit fraction has a top number that is not 1.

Key ideas

  • The bottom number (denominator) = how many equal parts in the whole.
  • The top number (numerator) = how many parts we are talking about.
  • The parts MUST be equal in size.
  • A unit fraction has 1 on the top: 1/5. A non-unit fraction does not: 3/5.
  • For a set, the bottom number is the number of things in the whole set.
  • All the parts together make 1 whole: 8/8 = 1.
Worked example 5.1.1
What fraction of the bar is shaded?

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. A fraction describes equal parts of a whole. Count all the equal parts for the bottom number. Count the parts you want for the top number.

  1. Count ALL the equal parts in the whole. The bottom number of a fraction (the denominator) tells us how many equal parts there are altogether.
    There are 8 equal parts. The denominator is 8.
  2. Count the parts that are shaded. The top number (the numerator) tells us how many parts we are talking about.
    3 parts are shaded. The numerator is 3.
  3. Write the fraction. Put the numerator on top and the denominator on the bottom.
    38   (say “three eighths”)

Check. The parts that are shaded and the other parts together make all 8 parts: 3 + 5 = 8. ✓

Watch out. Count ALL the parts for the bottom number, not just the white ones.

Worked example 5.1.2
What fraction of the circle is shaded?

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. A fraction describes equal parts of a whole. Count all the equal parts for the bottom number. Count the parts you want for the top number.

  1. Count ALL the equal parts in the whole. The bottom number of a fraction (the denominator) tells us how many equal parts there are altogether.
    There are 8 equal parts. The denominator is 8.
  2. Count the parts that are shaded. The top number (the numerator) tells us how many parts we are talking about.
    5 parts are shaded. The numerator is 5.
  3. Write the fraction. Put the numerator on top and the denominator on the bottom.
    58   (say “five eighths”)

Check. The parts that are shaded and the other parts together make all 8 parts: 5 + 3 = 8. ✓

Watch out. Count ALL the parts for the bottom number, not just the white ones.

Worked example 5.1.3
What fraction of the hexagon is not shaded?

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. Count all the equal parts for the bottom number. Count the parts that are NOT shaded for the top number.

  1. Count ALL the equal parts. This is the bottom number (denominator).
    There are 6 equal parts altogether.
  2. Count the parts that are NOT shaded. These are the white parts.
    6 − 5 = 1. So 1 part is not shaded.
  3. Write the fraction. The white parts go on the top. The total number of parts goes on the bottom.
    16

Check. Shaded + not shaded: 56 + 16 = 66 = 1 whole. ✓

Watch out. The question asks for the parts that are not shaded, so count the white ones.

Worked example 5.1.4
A bag has 3 red counters, 4 blue counters and 2 green counters. What fraction of the counters are blue?
RRRBBBBGG

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. A fraction of a set compares the part with the whole set. The total goes on the bottom and the part you want goes on the top.

  1. Count ALL the counters. The total goes on the bottom of the fraction.
    red: 3
    blue: 4
    green: 2
    Total: 3 + 4 + 2 = 9
  2. Count the blue counters. This number goes on the top of the fraction.
    There are 4 blue counters.
  3. Write the fraction. 4 out of 9 is the fraction 4/9.
    49   (say “four ninths”)

Check. Add the fractions of every colour: the tops add up to 9, so the fractions together make 99 = 1 whole. ✓

Watch out. The bottom number is the whole set (9), not the number of the other colours.

Worked example 5.1.5
15 counters are put into 5 equal groups. 3 of the groups are coloured red. What fraction of the groups are coloured?

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. A fraction describes equal parts of a whole. Count all the equal parts for the bottom number. Count the parts you want for the top number.

  1. Count ALL the equal parts in the whole. The bottom number of a fraction (the denominator) tells us how many equal parts there are altogether.
    There are 5 equal groups. The denominator is 5.
  2. Count the groups that are coloured. The top number (the numerator) tells us how many parts we are talking about.
    3 groups are coloured. The numerator is 3.
  3. Write the fraction. Put the numerator on top and the denominator on the bottom.
    35   (say “three fifths”)

Check. The parts that are coloured and the other parts together make all 5 parts: 3 + 2 = 5. ✓

Watch out. Count ALL the parts for the bottom number, not just the white ones.

Worked example 5.1.6
Shade 5/6 of the shape. How many parts do you shade?

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. The bottom number tells us how many equal parts the whole is cut into. The top number tells us how many to shade.

  1. Look at the bottom number of 56. It is 6, so the whole must be cut into 6 equal parts.
    The picture has 6 equal parts. ✓
  2. Look at the top number. This is how many parts to shade.
    The top number is 5.
  3. Write the answer.
    Shade 5 of the 6 parts.

Check. Count again: 5 shaded and 1 white make 6 parts. ✓

Watch out. The bottom number is the total number of parts. Do not shade that many!

Worked example 5.1.7
Aisha says, “The shaded part is one quarter, because the rectangle has 4 parts.” Is she right?

Method. A fraction needs equal parts. Look at whether the parts are the same size.

  1. Count the parts. There are 4 parts.
    The rectangle has 4 parts.
  2. Check the sizes. The parts are not the same width. The shaded part is much smaller than the longest part.
  3. Decide. To be “one quarter” the whole must be cut into 4 EQUAL parts.
    No. The parts are not equal, so the shaded part is not 1/4.

Check. If the 4 parts were equal, each would be the same width as the shaded one. They are not. ✓

Watch out. Do not just count the parts. Check that they are equal.

Watch out

The parts must be equal. A shape cut into 4 parts is not always cut into quarters. Check that the parts are the same size. Also remember: the bottom number is the number of parts in the whole, not the number of parts you can see shaded or the number of white parts.

Exercise 5A  Fractions of shapes and sets

Non-calculator
1
FWhat fraction of each shape is shaded?
(a)
 
(b)
 
(c)
 
(d)
 
(e)
 
(f)
 
2
FWrite each of these as a fraction with numbers.
(a)three quarters =
(b)two fifths =
(c)five eighths =
(d)seven tenths =
(e)one sixth =
(f)two thirds =
3
FWrite each fraction in words.
(a)14 =
(b)23 =
(c)56 =
(d)38 =
(e)910 =
(f)45 =
4
FHow many parts must be shaded to show each fraction? The shapes are not shaded yet.
(a)
Shade 35. How many parts?
(b)
Shade 58. How many parts?
(c)
Shade 710. How many parts?
5
FWhat fraction of each shape is not shaded?
(a)
 
(b)
 
(c)
 
(d)
 
6
FWhat fraction of the counters are the colour in the question?
(a)
RRRBBBBB
What fraction of the counters are red?
(b)
GGGGGGGYYY
What fraction of the counters are yellow?
(c)
RRBBBBGGG
What fraction of the counters are blue?
(d)
YYYYYPPPPPP
What fraction of the counters are purple?
7
FThe counters are in equal groups. Some groups are coloured red. What fraction of the groups are red?
(a)
Fraction of groups that are red =
(b)
Fraction of groups that are red =
(c)
Fraction of groups that are red =
8
PSA pizza is cut into 8 equal slices. Noah eats 3 slices. Mei eats 1 slice.
(a)What fraction of the pizza does Noah eat?
(b)What fraction of the pizza does Mei eat?
(c)What fraction of the pizza is not eaten by Noah?
9
RWhich two pictures show one quarter shaded? Tick two boxes.
A
B
C
D
E
10
RWhich fraction is the odd one out?   12    13    34    15.   Explain.
11
RAlways, sometimes or never?   When a shape is cut into 4 parts, each part is one quarter.
12
RLeo says, “13 of the rectangle is shaded, because there are 3 parts and 1 is shaded.” Is Leo right? Explain.
13
RA rectangle is cut into 12 equal squares. 3 squares are blue and 4 squares are yellow. The rest are white. What fraction of the squares are white?
5.2

Fractions of amounts

Year 4
Fractions of quantities, Non-unit fractions where the answer is a whole number, Fractions to divide quantities

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Fractions of amountsComing soon

We can find a fraction of a number, of a sum of money, or of a measurement. It always works the same way. The bottom number says how many equal parts to split into. The top number says how many parts to take.

Unit fractions. To find 1/4 of 12, share 12 into 4 equal groups. Each group is one quarter.

12 counters in 4 equal groups. One group is 1/4 of 12. 12 ÷ 4 = 3.

So 1/4 of 12 = 12 ÷ 4 = 3. Finding a unit fraction is just dividing by the bottom number. 1/2 of a number: divide by 2. 1/5 of a number: divide by 5. 1/10 of a number: divide by 10.

Non-unit fractions. To find 3/4 of 20, first find 1/4 of 20, then take 3 of them. A bar model helps.

555520
20 ÷ 4 = 5. One quarter is 5. Three quarters is 3 × 5 = 15.

The steps are always the same: 1. Divide by the bottom number. 2. Multiply by the top number. This works with money (1/5 of £35) and with measures (3/4 of 20 cm). In this chapter the sums come out as whole numbers.

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts to take. “Of” in “3/4 of 20” means we find a fraction of the amount.

Key ideas

  • Step 1: divide the amount by the bottom number. This gives ONE part.
  • Step 2: multiply by the top number. This gives all the parts you need.
  • Unit fraction (top number 1): only step 1 is needed.
  • Draw a bar model: the whole on top, equal parts below.
  • To find what is left, take your answer away from the whole amount.
  • Put the units (£, p, cm...) in your answer.
Worked example 5.2.1
Work out 1/4 of 12.

Key words. The denominator (bottom) says how many equal parts. The numerator (top) says how many parts to take.

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

  1. Draw a bar and cut it into equal parts. The bottom number of 14 is 4, so we cut the whole amount into 4 equal parts.
    The whole amount is 12.
    ????12
  2. Divide 12 by 4 to find the size of ONE part. 12 is shared into 4 equal parts. One part is 14 of the amount.
    12 ÷ 4 = 3
    333312
  3. Write the answer. The top number is 1, so we only need one part.
    14 of 12 = 3

Check. Multiply back: 3 × 4 = 12. ✓

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

Worked example 5.2.2
Work out 1/5 of £35.

Key words. The denominator (bottom) says how many equal parts. The numerator (top) says how many parts to take.

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

  1. Draw a bar and cut it into equal parts. The bottom number of 15 is 5, so we cut the whole amount into 5 equal parts.
    The whole amount is £35.
    ?????£35
  2. Divide £35 by 5 to find the size of ONE part. £35 is shared into 5 equal parts. One part is 15 of the amount.
    £35 ÷ 5 = £7
    £7£7£7£7£7£35
  3. Write the answer. The top number is 1, so we only need one part.
    15 of £35 = £7

Check. Multiply back: £7 × 5 = £35. ✓

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

Worked example 5.2.3
Work out 3/4 of 20.

Key words. The denominator (bottom) says how many equal parts. The numerator (top) says how many parts to take.

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

  1. Draw a bar and cut it into equal parts. The bottom number of 34 is 4, so we cut the whole amount into 4 equal parts.
    The whole amount is 20.
    ????20
  2. Divide 20 by 4 to find the size of ONE part. 20 is shared into 4 equal parts. One part is 14 of the amount.
    20 ÷ 4 = 5
    555520
  3. Multiply by 3 to get 3 parts. The top number is 3, so we need 3 of the parts.
    5 × 3 = 15

Check. The other 1 part makes 5. 15 + 5 = 20. ✓

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

Worked example 5.2.4
A ribbon is 18 cm long. Amira cuts off 2/3 of it. How long is the piece she cuts off?

Key words. The denominator (bottom) says how many equal parts. The numerator (top) says how many parts to take.

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

  1. Draw a bar and cut it into equal parts. The bottom number of 23 is 3, so we cut the whole amount into 3 equal parts.
    The whole amount is 18 cm.
    ???18 cm
  2. Divide 18 cm by 3 to find the size of ONE part. 18 cm is shared into 3 equal parts. One part is 13 of the amount.
    18 cm ÷ 3 = 6 cm
    6 cm6 cm6 cm18 cm
  3. Multiply by 2 to get 2 parts. The top number is 2, so we need 2 of the parts.
    6 cm × 2 = 12 cm

Check. The other 1 part makes 6 cm. 12 cm + 6 cm = 18 cm. ✓

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

Worked example 5.2.5
Noah has £40. He spends 3/8 of his money on a book. How much money does he have left?

Key words. The denominator (bottom) says how many equal parts. The numerator (top) says how many parts to take.

Method. Find the fraction of the amount that is used. If you want what is left, take it away from the whole amount.

  1. Find one part: divide £40 by 8. 38 means 3 out of 8 equal parts, so we first find one part.
    £40 ÷ 8 = £5
    £5£5£5£5£5£5£5£5£40
  2. Find the amount for 3 parts: multiply by 3. The top number of 38 is 3, so we need 3 of the parts.
    £5 × 3 = £15
  3. Take it away from £40 to find what is left. What is left is the amount we started with, take away the amount used.
    £40 − £15 = £25

Check. The parts left are 5 out of 8. 5 × £5 = £25. ✓

Watch out. Read the question again: do you want the amount used, or the amount left?

Worked example 5.2.6
1/6 of a number is 7. What is the number?

Key words. The denominator (bottom) says how many equal parts. The numerator (top) says how many parts to take.

Method. We know some parts and want the whole. Find the size of one part. Then multiply by the number of parts in the whole.

  1. Draw a bar with the right number of equal parts. The bottom number is 6, so there are 6 equal parts. We shade 1 of them.
    1 part makes 7.
  2. Work out one part: divide 7 by 1. One part is the amount we know.
    7 ÷ 1 = 7
  3. Work out the whole: multiply one part by 6. The whole is all 6 parts.
    7 × 6 = 42
    7?

Check. 16 of 42 = 42 ÷ 6 × 1 = 7. ✓

Watch out. Do not divide the whole by the top number. The whole is what we are finding.

Worked example 5.2.7
Which is more: 3/4 of 20 or 2/3 of 24?

Key words. The denominator (bottom) says how many equal parts. The numerator (top) says how many parts to take.

Method. Work out both amounts, then compare the answers. Do not just compare the fractions: the wholes are different.

  1. Work out 34 of 20. Divide by 4, then multiply by 3.
    20 ÷ 4 = 5
    5 × 3 = 15
  2. Work out 23 of 24. Divide by 3, then multiply by 2.
    24 ÷ 3 = 8
    8 × 2 = 16
  3. Compare the two answers. The bigger number is the bigger amount.
    15 < 16
  4. Answer.
    23 of 24 is more.

Check. Check: 15 and 16 are different, and 16 is the bigger one. ✓

Watch out. A bigger fraction of a smaller amount can be less than a smaller fraction of a bigger amount.

Watch out

Divide by the bottom number first. For 3/4 of 20, divide by 4 (not by 3). A common mistake is to divide by the top number. Also remember the units: 3/4 of £20 is £15, not 15.

Exercise 5B  Fractions of amounts

Non-calculator
1
FQuick-fire! Find each unit fraction of the number.
(a)12 of 18 =
(b)13 of 24 =
(c)14 of 32 =
(d)15 of 45 =
(e)110 of 70 =
(f)16 of 42 =
2
FUse the groups of counters to help you. Work out the fraction of the number.
(a)
13 of 12 =
(b)
34 of 20 =
(c)
25 of 15 =
3
FQuick-fire! Find each fraction of the number.
(a)23 of 12 =
(b)34 of 16 =
(c)45 of 25 =
(d)310 of 40 =
(e)56 of 18 =
(f)78 of 24 =
4
FThe bar shows 36 split into quarters.
????36
(a)What is 14 of 36?
(b)What is 34 of 36?
5
FFind the amounts of money.
(a)12 of £18 =
(b)35 of £45 =
(c)310 of £80 =
(d)25 of 60p =
6
FFind the amounts. Remember the units in your answers.
(a)15 of 40 cm =
(b)34 of 24 litres =
(c)23 of 30 minutes =
(d)58 of 64 g =
7
PSThere are 24 children in a class. 3/8 of them walk to school. How many children walk to school?
????????24
8
PSMei’s book has 60 pages. She has read 2/5 of the book. How many pages has she not read yet?
?????60
9
PSA school trip costs £48. A parent pays 1/4 of the cost in the first week. How much do they pay in the first week?
10
PSLeo has £60. He spends 1/3 of it on a book and 1/4 of it on lunch. How much money does Leo have left?
11
PSA baker makes 48 rolls. 1/4 of the rolls are chocolate. 1/3 of the rolls are plain. The rest are cheese. How many rolls are cheese?
12
RWhich is more: 3/5 of 20 or 1/2 of 30? Show how you know.
13
RJack works out 3/5 of 20. He says the answer is 20 ÷ 3. What mistake has he made? Work out the correct answer.
14
R1/5 of a number is 8. What is the number?
8?
15
RAlways, sometimes or never?   Half of an even number is a whole number.
16
R3/4 of a number is 18. What is the number?   (Hint: 3 quarters make 18. Find one quarter first.)
?18
5.3

Equivalent fractions

Year 4
Families of equivalent fractions, Fraction walls, bars and number lines, Tenths and fifths

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Equivalent fractionsComing soon

Equivalent fractions are fractions that look different but show the same amount. Look at these three bars. They are all the same length. The shaded part is the same in each, so 1/2 = 2/4 = 4/8.

122448
One half, two quarters and four eighths are all the same amount.

A fraction wall shows this clearly. Each row is the same length, cut into different numbers of equal bricks. Bricks that line up cover the same amount.

11/21/21/41/41/41/41/81/81/81/81/81/81/81/8
On the wall, 1/2, 2/4 and 4/8 line up. The whole is the row at the top.

Why does it work? Cut every part of 1/2 into 2 smaller equal parts. Now the whole has 4 parts, not 2, and the 1 shaded part has become 2 shaded parts. So 1/2 = 2/4. We multiplied the top and the bottom by 2. Cut every part into 4 pieces and we get 4/8. We multiplied the top and the bottom by 4.

12 = 1 × 22 × 2 = 24     12 = 1 × 42 × 4 = 48

The rule: multiply the top and the bottom by the same number. You can also go the other way: divide the top and the bottom by the same number, if they both divide exactly (this joins parts together).

011201240148
On a number line the equivalent fractions are at the same point.

Some families to know. 1/2 = 2/4 = 3/6 = 4/8 = 5/10 = 6/12. And 1/3 = 2/6 = 3/9 = 4/12. And fifths and tenths: 1/5 = 2/10, 2/5 = 4/10, 3/5 = 6/10, 4/5 = 8/10.

132641235610
Thirds, sixths and twelfths. Fifths and tenths.

Key words. Equivalent means the same amount. A fraction wall is a chart of bricks that shows fractions side by side. Multiply the top and bottom by the same number to make an equivalent fraction.

Key ideas

  • Equivalent fractions show the same amount: 1/2 = 2/4 = 4/8.
  • Multiply the top AND the bottom by the same number to get an equivalent fraction.
  • You can divide the top AND the bottom by the same number too (if both divide exactly).
  • Bricks that line up on a fraction wall are equivalent.
  • Never add or subtract the same number to the top and bottom.
  • Fifths and tenths: 1/5 = 2/10, 2/5 = 4/10, 3/5 = 6/10, 4/5 = 8/10, 1/2 = 5/10.
Worked example 5.3.1
Use the fraction wall to find two fractions that are equal to 1/2.
11/21/21/41/41/41/41/81/81/81/81/81/81/81/8

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Look for bricks on the fraction wall that line up with 1/2.

  1. Find the 1/2 brick on the wall. It is the shaded brick in the second row.
    11/21/21/41/41/41/41/81/81/81/81/81/81/81/8
  2. Look at the rows below it. Find the bricks that cover exactly the same length as 1/2.
    11/21/21/41/41/41/41/81/81/81/81/81/81/81/8
  3. Count the shaded bricks in each row. In the quarters row 2 bricks line up. In the eighths row 4 bricks line up.
    12 = 24 = 48

Check. Multiply: 1 × 2 = 2 and 2 × 2 = 4, so 12 = 24. ✓

Watch out. The bricks must line up exactly. Look at the edges.

Worked example 5.3.2
Fill in the missing number.   13 = ?6

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Equivalent fractions show the same amount. Multiply (or divide) the top and the bottom by the same number.

  1. Look at the bottom numbers. We need to go from 3 to 6. What do we multiply by?
    3 × 2 = 6
  2. Do the same to the top number. Every piece is cut into 2 smaller pieces, so the top number is multiplied too. The fraction stays the same size.
    1 × 2 = 2
  3. Write the equivalent fractions. Both fractions show the same amount.
    13 = 26
    1326

Check. Look at the bars: the shaded parts are the same length. ✓

Watch out. Never add or subtract the same number to the top and bottom. You must multiply or divide both.

Worked example 5.3.3
Fill in the missing number.   34 = ?8

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Equivalent fractions show the same amount. Multiply (or divide) the top and the bottom by the same number.

  1. Look at the bottom numbers. We need to go from 4 to 8. What do we multiply by?
    4 × 2 = 8
  2. Do the same to the top number. Every piece is cut into 2 smaller pieces, so the top number is multiplied too. The fraction stays the same size.
    3 × 2 = 6
  3. Write the equivalent fractions. Both fractions show the same amount.
    34 = 68
    3468

Check. Look at the bars: the shaded parts are the same length. ✓

Watch out. Never add or subtract the same number to the top and bottom. You must multiply or divide both.

Worked example 5.3.4
Fill in the missing number.   35 = 6?

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Equivalent fractions show the same amount. Multiply (or divide) the top and the bottom by the same number.

  1. Look at the top numbers. We need to go from 3 to 6. What do we multiply by?
    3 × 2 = 6
  2. Do the same to the bottom number. We must multiply the top and the bottom by the same number to keep the fraction the same size.
    5 × 2 = 10
  3. Write the equivalent fractions. Both fractions show the same amount.
    35 = 610
    35610

Check. Look at the bars: the shaded parts are the same length. ✓

Watch out. Never add or subtract the same number to the top and bottom. You must multiply or divide both.

Worked example 5.3.5
Fill in the missing number. Divide this time.   48 = ?2

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Equivalent fractions show the same amount. Multiply (or divide) the top and the bottom by the same number.

  1. Look at the bottom numbers. We need to go from 8 to 2. What do we divide by?
    8 ÷ 4 = 2
  2. Do the same to the top number. We join 4 pieces into one bigger piece, so the top number is divided too. The fraction stays the same size.
    4 ÷ 4 = 1
  3. Write the equivalent fractions. Both fractions show the same amount.
    48 = 12
    4812

Check. Look at the bars: the shaded parts are the same length. ✓

Watch out. Never add or subtract the same number to the top and bottom. You must multiply or divide both.

Worked example 5.3.6
Are 3/8 and 1/2 equivalent fractions?

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Make the bottom numbers the same by multiplying the top and bottom of one fraction by the same number. Then compare.

  1. Look at the bottom numbers. The bigger bottom number is 8. We rewrite the other fraction so that its bottom number is 8 too.
    2 × 4 = 8
  2. Multiply the top number by the same number. Then the two fractions have the same bottom number and we can compare the tops.
    12 = 48 (because 1 × 4 = 4)
  3. Compare. The tops are different, so the fractions show different amounts.
    48 ≠ 38
  4. Answer.
    No, 38 and 12 are not equivalent.

Check. Look at the bars: 38 and 12 cover different lengths.

3812
✓

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

Worked example 5.3.7
Write 3/5 as tenths.

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Count how many tenths are in each fifth, then use the same idea for 3 fifths.

  1. Compare fifths and tenths. Each fifth is cut into 2 equal parts to make tenths. So 1 fifth = 2 tenths.
    15 = 210
    15210
  2. Do the same for 3 fifths. Multiply the top and the bottom by 2.
    35 = 3 × 25 × 2 = 610
  3. Write the answer.
    610

Check. 3 fifths of the bar is the same length as 6 tenths. ✓

Watch out. To change fifths to tenths, multiply the top and bottom by 2.

Watch out

Multiply, do not add. To get an equivalent fraction, multiply (or divide) the top and the bottom by the same number. If you add 1 to the top and the bottom, you make a different fraction: 1/3 is not the same as 2/4.

Exercise 5C  Equivalent fractions

Non-calculator
1
FThe bars are all the same length. Fill in the missing numbers.
(a)
1224
12 = ?4
(b)
1326
13 = ?6
(c)
3468
34 = ?8
(d)
25410
25 = ?10
2
FUse the fraction wall to fill in the missing numbers.
11/21/21/31/31/31/41/41/41/41/61/61/61/61/61/61/81/81/81/81/81/81/81/81/121/121/121/121/121/121/121/121/121/121/121/12
(a)12 = ?4
(b)12 = ?8
(c)14 = ?8
(d)13 = ?6
(e)13 = ?12
(f)12 = ?6
3
FQuick-fire! Fill in the missing top numbers.
(a)12 = ?10
(b)14 = ?12
(c)23 = ?6
(d)34 = ?8
(e)15 = ?10
(f)35 = ?10
4
FQuick-fire! Fill in the missing bottom numbers.
(a)12 = 3?
(b)13 = 4?
(c)25 = 4?
(d)34 = 9?
(e)23 = 8?
(f)35 = 12?
5
FThis time divide the top and the bottom by the same number.
(a)48 = ?2
(b)612 = ?4
(c)610 = 3?
(d)812 = 2?
(e)410 = ?5
6
FComplete the family of 14.   14 = 2? = ?12 = 4?
(a)14 = 2?
(b)14 = ?12
(c)14 = 4?
7
FThe number line is split into tenths. Write each point as a fraction in fifths.
01102103104105106107108109101ABC
(a)Point A is at 210. Write A in fifths:   210 = ?5
(b)Point B is at 610. Write B in fifths:   610 = ?5
(c)Point C is at 810. Write C in fifths:   810 = ?5
8
FAre the two fractions equivalent? Write Yes or No.
(a)12 and 48
(b)23 and 812
(c)35 and 510
(d)14 and 38
(e)26 and 13
(f)25 and 310
9
RWhich fraction is the odd one out?   12    36    48    512
10
RWhich fractions are equal to 12? Tick all the boxes that are equal to a half.   A 36   B 410   C 510   D 612   E 38
11
PSA pizza is cut into 6 equal slices. Tom eats 2/3 of the pizza. Use equivalent fractions to find how many slices Tom eats.   23 = ?6
12
RNoah eats 2/4 of a pizza. Mei eats 4/8 of an identical pizza. Who eats more pizza? Explain.
13
RDev says, “13 = 24, because you add 1 to the top and 1 to the bottom.” Is Dev right? Explain.
14
RComplete these fractions that are equal to 25.
(a)25 = ?10
(b)25 = ?15
(c)25 = ?20
5.4

Adding and subtracting fractions

Year 4
Add and subtract fractions with the same denominator, Answers that make a whole

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Adding and subtracting fractionsComing soon

Imagine a chocolate bar cut into 7 equal pieces. Zara eats 2 pieces. Then she eats 3 more pieces. How much has she eaten altogether? She has eaten 2 sevenths and 3 sevenths. That is 5 pieces, so 5 sevenths.

2/7 (blue) + 3/7 (orange) = 5/7. The pieces are all sevenths, so we just count them.

27 + 37 = 57

The rule. When the bottom numbers are the same, the pieces are the same size. Add the top numbers and keep the bottom number the same. The same is true for subtraction: subtract the top numbers and keep the bottom number.

Why does the bottom number stay the same? The bottom number names the size of the pieces (sevenths). Adding more pieces does not change their size. 2 apples + 3 apples = 5 apples. 2 sevenths + 3 sevenths = 5 sevenths.

Making a whole. When all the pieces are there, the answer is 1 whole. For example 3 eighths + 5 eighths = 8 eighths = 1 whole. Any fraction with the same top and bottom number equals 1: 4/4 = 1, 10/10 = 1.

3/8 + 5/8 = 8/8 = 1 whole.

This helps with taking a fraction away from 1. Write 1 as a fraction with the same bottom number. For example 1 = 5/5, so 1 − 2/5 = 5/5 − 2/5 = 3/5.

Key words. The numerator is the top number. The denominator is the bottom number. Whole means 1 complete thing.

Key ideas

  • Same bottom number: the pieces are the same size.
  • Add or subtract the TOP numbers. Keep the bottom number the same.
  • When the top number equals the bottom number, the fraction is 1 whole: 6/6 = 1.
  • To take a fraction from 1, write 1 as (bottom number)/(bottom number). 1 = 5/5.
  • To find what is missing from a whole, subtract the top number from the bottom number.
  • Never add the bottom numbers.
Worked example 5.4.1
Work out 2/7 + 3/7.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

  1. Check the bottom numbers. They are all 7. That means every piece is the same size (sevenths), so we can add them.
    27 + 37
  2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
    2 + 3 = 5
  3. Keep the bottom number the same. The pieces are still sevenths. Adding does not change the size of the pieces.
    27 + 37 = 57

Check. Take away one of the fractions from the answer: 5 − 3 = 2. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay sevenths, so the bottom number stays 7.

Worked example 5.4.2
Work out 3/8 + 5/8.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

  1. Check the bottom numbers. They are all 8. That means every piece is the same size (eighths), so we can add them.
    38 + 58
  2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
    3 + 5 = 8
  3. Keep the bottom number the same. The pieces are still eighths. Adding does not change the size of the pieces.
    38 + 58 = 88
  4. Write it as a whole number. 8 eighths make 1 whole.
    88 = 1

Check. Take away one of the fractions from the answer: 8 − 5 = 3. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay eighths, so the bottom number stays 8.

Worked example 5.4.3
Work out 7/10 − 4/10.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. The pieces are the same size, so take away the top numbers and keep the bottom number the same.

  1. Check the bottom numbers. They are both 10, so the pieces are the same size (tenths) and we can subtract.
    710 − 410
  2. Subtract the top numbers. We start with 7 pieces and take away 4 pieces.
    7 − 4 = 3
  3. Keep the bottom number the same. The pieces are still tenths.
    710 − 410 = 310

Check. Add back: 3 + 4 = 7, so 310 + 410 = 710. ✓

Watch out. Do not subtract the bottom numbers. The pieces are still tenths.

Worked example 5.4.4
Work out 1 − 3/5.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Write 1 whole as 5/5. Then subtract.

  1. Write 1 whole as a fraction. 1 whole is 5 pieces of size 15, because the pieces are fifths.
    1 = 55
  2. Subtract the top numbers. We take away 3 pieces from 5 pieces.
    5 − 3 = 2
  3. Keep the bottom number the same. The pieces are still fifths.
    1 − 35  →  55 − 35 = 25

Check. Add: 35 + 25 = 55 = 1. ✓

Watch out. 1 whole is 5 fifths. It is not 1 fifth.

Worked example 5.4.5
Find the missing number.   310 + ?10 = 810

Key words. The numerator is the top number. The denominator is the bottom number.

Method. The bottom numbers are all the same, so we only need to work out the missing top number.

  1. Look at the bottom numbers. All are 10, so the pieces are the same size. The missing number is a top number.
    310 + ?10 = 810
  2. Use the opposite (inverse) operation. 8 is the total. If we take away 3 from the total, we get the missing top number.
    8 − 3 = 5
  3. Write the missing number.
    The missing number is 5.

Check. 3 + 5 = 8. ✓

Watch out. The missing number goes on top. The bottom number stays the same.

Worked example 5.4.6
A pizza is cut into 8 equal slices. Zainab eats 2/8 of the pizza and Leo eats 3/8 of the pizza. What fraction of the pizza do they eat altogether?

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

  1. Check the bottom numbers. They are all 8. That means every piece is the same size (eighths), so we can add them.
    28 + 38
  2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
    2 + 3 = 5
  3. Keep the bottom number the same. The pieces are still eighths. Adding does not change the size of the pieces.
    28 + 38 = 58

Check. Take away one of the fractions from the answer: 5 − 3 = 2. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay eighths, so the bottom number stays 8.

Worked example 5.4.7
Zainab and Leo have eaten 5/8 of the pizza. What fraction of the pizza is left?

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Write 1 whole as 8/8. Then subtract.

  1. Write 1 whole as a fraction. 1 whole is 8 pieces of size 18, because the pieces are eighths.
    1 = 88
  2. Subtract the top numbers. We take away 5 pieces from 8 pieces.
    8 − 5 = 3
  3. Keep the bottom number the same. The pieces are still eighths.
    1 − 58  →  88 − 58 = 38

Check. Add: 58 + 38 = 88 = 1. ✓

Watch out. 1 whole is 8 eighths. It is not 1 eighth.

Watch out

Do not add the bottom numbers. 2/5 + 1/5 is 3/5, not 3/10. The pieces are still fifths. To take a fraction away from 1, write 1 as a fraction with the same bottom number: 1 = 5/5 (not 1/1).

Exercise 5D  Adding and subtracting fractions

Non-calculator
1
FThe bars show two fractions. Add them.
(a)
2737
27 + 37 =
(b)
1949
19 + 49 =
(c)
310410
310 + 410 =
(d)
2525
25 + 25 =
2
FQuick-fire! Add the fractions.
(a)13 + 13 =
(b)14 + 24 =
(c)25 + 25 =
(d)16 + 46 =
(e)37 + 47 =
(f)512 + 612 =
3
FQuick-fire! Subtract the fractions.
(a)34 − 24 =
(b)45 − 15 =
(c)56 − 46 =
(d)78 − 28 =
(e)910 − 210 =
(f)1112 − 412 =
4
FWhat must be added to each fraction to make 1 whole?
(a)37 + ? = 1
(b)14 + ? = 1
(c)49 + ? = 1
(d)710 + ? = 1
(e)512 + ? = 1
5
FWork out each subtraction from 1.
(a)1 − 25 =
(b)1 − 13 =
(c)1 − 58 =
(d)1 − 310 =
(e)1 − 16 =
6
FFind the missing top numbers.
(a)?9 + 49 = 79
(b)310 + ?10 = 810
(c)78 − ?8 = 28
(d)?12 − 512 = 412
(e)26 + ?6 = 56
(f)910 − ?10 = 410
7
FAdd three fractions.
(a)19 + 29 + 29 =
(b)210 + 110 + 410 =
(c)15 + 15 + 35 =
8
PSA chocolate bar has 12 equal pieces. Amir eats 5/12 of the bar and Bea eats 2/12 of the bar.
(a)What fraction of the bar do they eat altogether?
(b)What fraction of the bar is left?
9
PSA water tank is full at the start of the week. On Monday 3/8 of the water is used. On Tuesday 2/8 of the water is used.
(a)What fraction of the water is used on the two days?
(b)What fraction of the tank is left?
10
PSPriya has 9/10 of a bottle of juice. She drinks 2/10 of the bottle. What fraction of the bottle is left?
11
RBen says, “25 + 15 = 310.” What mistake has Ben made? What is the correct answer?
12
RWhich two pairs of fractions add up to exactly 1 whole? Tick two boxes.    A 38 + 58   B 26 + 36   C 710 + 310   D 49 + 49
13
RAlways, sometimes or never?   When you add two fractions with the same bottom number, the answer is less than 1.
14
RTwo fractions each have a bottom number of 9. Both top numbers are whole numbers more than 0. The fractions add up to 79. Write all the possible pairs of top numbers. (Order does not matter.)
5.5

Comparing and ordering fractions

Year 4
Compare and order fractions with the same denominator, Use a fraction wall, Unit fractions with different denominators

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Comparing and ordering fractionsComing soon

To compare two fractions means to say which one is bigger. We use the signs < (less than) and > (greater than), and = (equal to).

Same bottom number. The pieces are the same size, so the fraction with more pieces is bigger. Compare the top numbers. 5/8 is bigger than 3/8, because 5 pieces are more than 3 pieces.

3858
3/8 < 5/8. With the same bottom number, the bigger top number is the bigger fraction.

Unit fractions with different bottom numbers. Cut a whole into 4 equal parts, and another whole into 8 equal parts. The more parts you cut, the smaller each part is. So 1/8 is smaller than 1/4. A fraction wall shows this well.

11/21/21/31/31/31/41/41/41/41/61/61/61/61/61/61/81/81/81/81/81/81/81/8
On the wall, 1/8 is a shorter brick than 1/4. So 1/8 < 1/4.

Different bottom numbers, harder fractions. Make the bottom numbers the same using equivalent fractions (Section 5.3). To compare 3/4 and 5/8, change 3/4 into eighths: 3/4 = 6/8. Now compare 6/8 and 5/8. 6 is more than 5, so 3/4 is more than 5/8.

346858
3/4 = 6/8, and 6/8 is longer than 5/8.

Ordering means putting fractions in a list from smallest to biggest (or biggest to smallest). Give them all the same bottom number, then order the top numbers. A handy trick is to compare with one half. Half of 12 is 6, so 7/12 is more than a half and 5/12 is less than a half.

Key words. To compare is to say which is bigger. To order is to put things in a list from smallest to biggest, or the other way round.

Key ideas

  • Same bottom number: the bigger top number is the bigger fraction.
  • Unit fractions: the bigger the bottom number, the SMALLER the fraction. (1/8 < 1/4)
  • Different bottom numbers: change one fraction into an equivalent fraction so the bottom numbers match.
  • A fraction wall shows the sizes: a longer brick is a bigger fraction.
  • Compare with 1/2: half of the bottom number is the top number of a half.
  • < means less than. > means greater than.
Worked example 5.5.1
Write < or > in the box.   38  [ ]   58

Key words. To compare is to say which is bigger. < means “is less than” and > means “is greater than”.

Method. When the bottom numbers are the same, the pieces are the same size. The fraction with the bigger top number is bigger.

  1. Look at the bottom numbers. They are both 8, so the pieces are the same size.
    38 and 58
    3858
  2. Compare the top numbers. The same size pieces: more pieces means a bigger fraction.
    3 < 5
  3. Write the answer.
    38 < 58

Check. Think of bars or a fraction wall: the longer shaded bar is the bigger fraction. ✓

Watch out. Only compare the top numbers when the bottom numbers are the same.

Worked example 5.5.2
Which is bigger, 1/5 or 1/8?

Key words. To compare is to say which is bigger. < means “is less than” and > means “is greater than”.

Method. For unit fractions (1 on the top), the bigger the bottom number, the smaller the piece. Use a fraction wall to see it.

  1. Look at the bottom numbers. They are different: 5 and 8. The pieces are not the same size.
    15 and 18
    11/51/51/51/51/51/81/81/81/81/81/81/81/8
  2. Think about the size of the pieces. When you cut a whole into MORE equal parts, each part is SMALLER.
    5 parts: each part is bigger. 8 parts: each part is smaller.
  3. Write the answer. The fraction with the bigger bottom number is the smaller fraction (for unit fractions).
    15 > 18

Check. Think of bars or a fraction wall: the longer shaded bar is the bigger fraction. ✓

Watch out. 1/8 is smaller than 1/4. A bigger bottom number does not mean a bigger fraction!

Worked example 5.5.3
Write these fractions in order, smallest first.   79,  29,  49,  19

Key words. To compare is to say which is bigger. < means “is less than” and > means “is greater than”.

Method. The bottom numbers are the same, so put the fractions in order using their top numbers, smallest first.

  1. Look at the bottom numbers. They are all 9, so the pieces are the same size and we can order the fractions by their top numbers.
    79,  29,  49,  19
  2. Put the top numbers in order. The bottom numbers are the same, so the bigger top number is the bigger fraction. We want smallest first.
    1, 2, 4, 7
  3. Write the ORIGINAL fractions in that order. The question uses the original fractions.
    19,  29,  49,  79

Check. Look at the picture: the bars get longer from left to right in the list. ✓

19294979

Watch out. Write the original fractions in your answer, not the rewritten ones.

Worked example 5.5.4
Write < or > in the box.   34  [ ]   58

Key words. To compare is to say which is bigger. < means “is less than” and > means “is greater than”.

Method. Change one fraction so both have the same bottom number. Then compare the top numbers.

  1. Look at the bottom numbers. They are different. But 8 is in the 4 times table, so we can change 34 into eighths.
    4 × 2 = 8
  2. Rewrite 34 as eighths. Multiply the top and the bottom by the same number.
    34 = 68
    346858
  3. Compare the top numbers. Now the bottom numbers are the same (8).
    6 and 5: 6 > 5
  4. Write the answer.
    34 > 58

Check. Think of bars or a fraction wall: the longer shaded bar is the bigger fraction. ✓

Watch out. Do not compare the bottom numbers or the top numbers on their own. Make the bottom numbers the same first.

Worked example 5.5.5
Write these fractions in order, smallest first.   12,  16,  13,  110

Key words. To compare is to say which is bigger. < means “is less than” and > means “is greater than”.

Method. For unit fractions, the bigger the bottom number, the smaller the piece. Use a fraction wall to see it.

  1. Look at the bottom numbers. These are all unit fractions (1 on the top). The bottom number tells us how many equal parts the whole is cut into.
    2, 6, 3, 10
  2. Think about the size of the pieces. More parts means smaller pieces. So the bigger the bottom number, the smaller the fraction.
    Biggest bottom number 10: smallest piece.
    Smallest bottom number 2: biggest piece.
    11/21/21/31/31/31/61/61/61/61/61/61/101/101/101/101/101/101/101/101/101/10
  3. Put them smallest first.
    110,  16,  13,  12

Check. On the wall the shaded bricks get shorter as we read down. ✓

Watch out. 5 is bigger than 3, but 1/5 is smaller than 1/3.

Worked example 5.5.6
Is 5/12 more than, less than or equal to 1/2?

Key words. To compare is to say which is bigger. < means “is less than” and > means “is greater than”.

Method. Half of the bottom number is the top number of a half. Compare the top number with it.

  1. Find half of the bottom number. The whole is 12 parts. Half of the whole is half of 12 parts.
    12 ÷ 2 = 6, so 612 = 12
  2. Compare the top number 5 with 6. If we have more than 6 parts we have more than a half.
    5 < 6
  3. Write the answer.
    512 is less than 12

Check. 12 = 612, and 512 is less than 612. ✓

Watch out. Compare the top number with HALF of the bottom number, not with the bottom number itself.

Worked example 5.5.7
Write these fractions in order, smallest first.   12,  38,  34

Key words. To compare is to say which is bigger. < means “is less than” and > means “is greater than”.

Method. Give the fractions the same bottom number. Then put the top numbers in order, smallest first.

  1. Find a bottom number that all the bottom numbers go into. The biggest bottom number is 8. The other bottom numbers (2, 4) all go into 8.
    Use eighths.
  2. Rewrite every fraction as eighths. Multiply the top and the bottom by the same number each time.
    12 = 48  (× 4)
    38 = 38
    34 = 68  (× 2)
  3. Put the top numbers in order. The bottom numbers are the same, so the bigger top number is the bigger fraction. We want smallest first.
    3, 4, 6
  4. Write the ORIGINAL fractions in that order. The question uses the original fractions.
    38,  12,  34

Check. Look at the picture: the bars get longer from left to right in the list. ✓

381234

Watch out. Write the original fractions in your answer, not the rewritten ones.

Watch out

A bigger bottom number does not mean a bigger fraction. 1/8 is smaller than 1/4, even though 8 is bigger than 4. The bottom number says how many parts the whole is cut into. More parts means smaller parts. Also, only compare the top numbers when the bottom numbers are the same.

Exercise 5E  Comparing and ordering fractions

Non-calculator
1
FWhich fraction is bigger? The bars will help you.
(a)
3858
38 or 58
(b)
710410
710 or 410
(c)
2515
25 or 15
2
FQuick-fire! Write <, > or = in each box.
(a)25  [ ]   45
(b)78  [ ]   38
(c)59  [ ]   59
(d)16  [ ]   56
(e)910  [ ]   710
(f)412  [ ]   1112
3
FQuick-fire! These are unit fractions. Write < or > in each box.
(a)14  [ ]   16
(b)13  [ ]   12
(c)110  [ ]   15
(d)18  [ ]   112
(e)12  [ ]   19
(f)17  [ ]   14
4
FWrite each set of fractions in order, smallest first.
(a)37,  67,  17,  57
(b)56,  26,  46
(c)810,  310,  910,  110
5
FThese are all unit fractions. Write them in order, biggest first.
(a)13,  18,  15
(b)110,  12,  14,  15
(c)112,  16,  13
6
FUse the fraction wall or equivalent fractions. Write < or > in each box.
11/21/21/31/31/31/41/41/41/41/51/51/51/51/51/61/61/61/61/61/61/81/81/81/81/81/81/81/81/101/101/101/101/101/101/101/101/101/10
(a)12  [ ]   38
(b)34  [ ]   58
(c)23  [ ]   56
(d)14  [ ]   38
(e)25  [ ]   710
7
FIs each fraction more than, less than or equal to a half?
(a)38 is ... 12
(b)712 is ... 12
(c)510 is ... 12
(d)46 is ... 12
(e)310 is ... 12
(f)612 is ... 12
8
FWrite each set of fractions in order. Change the fractions so they have the same bottom number first.
(a)Smallest first: 12,  58,  34
(b)Biggest first: 56,  23,  12,  16
9
PSAmira drinks 3/5 of a bottle of water. Jack drinks 7/10 of an identical bottle. Who drinks more water? Show how you know.
35710
10
PSPriya eats 3/8 of a bar of chocolate. Sam eats 1/2 of an identical bar. Leo eats 5/8 of another identical bar. Write the fractions in order, biggest first.
11
RTom says, “18 is bigger than 14, because 8 is bigger than 4.” Is Tom right? Explain.
12
RWhich fraction is the odd one out?   58   46   310   712. Explain.
13
RAlways, sometimes or never?   The fraction with the bigger bottom number is the smaller fraction.
14
RHow many fractions with a bottom number of 8 are bigger than 14 and smaller than 58? Write them down.
5.6

Improper fractions and mixed numbers

Year 4
Fractions greater than 1, Improper fractions and mixed numbers (introduced), Counting in fractions

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Improper fractions and mixed numbersComing soon

Sometimes we have more than one whole. Imagine 7 slices of pizza, where each pizza is cut into 4 equal slices (quarters). 4 slices make one whole pizza. We have one whole pizza and 3 more slices.

Seven quarters: one whole circle and 3 quarters of another.

We can write this in two ways.

74   =   134

7/4 is an improper fraction. Its top number is bigger than its bottom number. It means “7 quarters”. 1 3/4 is a mixed number. It is a whole number and a fraction together: 1 whole and 3 quarters. They are two names for the same amount.

Improper fraction to mixed number. Ask how many wholes there are. 4 quarters make a whole, so how many 4s fit into 7? One 4 fits (4 × 1 = 4). 7 − 4 = 3 quarters are left over. So 7/4 = 1 3/4. In other words: divide the top number by the bottom number. The answer is the whole number. The remainder goes on top of the fraction.

Mixed number to improper fraction. Each whole is 4 quarters. 1 whole is 4 quarters, and 3 more quarters make 7 quarters. So 1 3/4 = 7/4. In other words: multiply the whole number by the bottom number, add the top number, and keep the bottom number.

014243411141241342A
On a number line the point A is 7 quarters from 0. That is 1 whole and 3 quarters. After 3/4 comes 4/4 = 1, then 1 1/4, 1 2/4, 1 3/4, then 2.

Counting in fractions. When we count in quarters we go 1/4, 2/4, 3/4, then 1 whole (4/4), then 1 1/4, 1 2/4, 1 3/4, then 2. The jumps are always 1/4. The whole numbers appear when the top number reaches the bottom number.

Key words. An improper fraction has a top number that is bigger than (or the same as) the bottom number. A mixed number is a whole number and a fraction together.

Key ideas

  • An improper fraction has a top number bigger than (or equal to) the bottom number: 7/4.
  • A mixed number is a whole number and a fraction: 1 3/4.
  • Improper → mixed: divide the top by the bottom. The answer is the whole number. The remainder is the new top number.
  • Mixed → improper: whole number × bottom number + top number. Keep the bottom number.
  • 4 quarters make 1 whole, 3 thirds make 1 whole: the top number equals the bottom number.
  • Counting in fractions: after the last fraction comes a whole number (3/4, 1, 1 1/4).
Worked example 5.6.1
Write the shaded amount as an improper fraction and as a mixed number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count how many parts are in one whole for the bottom number. Count all the shaded parts for the top number. Count the full wholes for the mixed number.

  1. Count the equal parts in ONE whole. This is the bottom number. All the wholes are cut the same way.
    Each whole is cut into 4 equal parts, so the bottom number is 4.
  2. Count ALL the shaded parts. This is the top number. It can be bigger than the bottom number.
    7 parts are shaded, so the top number is 7.   74 is an improper fraction.
  3. Count the whole ones and the parts left over. 4 parts make 1 whole.
    1 whole one are full and 3 parts are shaded in the next one.
    74 = 134

Check. 1 × 4 + 3 = 7. ✓

Watch out. The top number counts ALL the shaded parts, in every picture.

Worked example 5.6.2
Write 17/5 as a mixed number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

  1. Ask: how many wholes are in it? 5 fifths make 1 whole. So we ask how many 5s fit into 17.
    5 × 3 = 15, which is not more than 17.
    5 × 4 = 20, which is more than 17.
    So there are 3 whole ones.
  2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
    17 − 15 = 2
  3. Write the mixed number. The whole number is 3. The left over parts make the fraction, still fifths.
    175 = 325

Check. Go back: 3 × 5 + 2 = 17 = 17. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Worked example 5.6.3
Write 2 3/5 as an improper fraction.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. A mixed number is whole ones and a fraction. Change the wholes into parts, add the extra parts, and keep the bottom number.

  1. Find how many parts are in the whole ones. Each whole is cut into 5 equal parts.
    2 × 5 = 10 parts
  2. Add the extra parts. The fraction part has 3 more parts.
    10 + 3 = 13
  3. Write the improper fraction. We count in fifths so the bottom number stays 5.
    235 = 135

Check. Go back: 13 ÷ 5 = 2 remainder 3, which is 235. ✓

Watch out. Do not forget to add the extra parts. And do not change the bottom number.

Worked example 5.6.4
What number does the arrow show? Write it as a mixed number.
012A

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Count the steps between 0 and 1 for the bottom number. Count the steps from 0 to the point for the top number.

  1. Count the equal steps between 0 and 1. This is the bottom number. All the steps are the same size.
    There are 4 steps from 0 to 1. Each step is 14.
  2. Count the steps from 0 to the point. This is the top number.
    The point is 7 steps from 0, so it is 74.
  3. Write the answer. 4 steps make 1 whole.
    74 = 134

Check. 7 steps of 14 is 7 quarters. ✓

Watch out. Count the spaces (steps), not the tick marks.

Worked example 5.6.5
Count on in thirds. Write the missing numbers.
13,  23,  1,  ?,  123,  ?,  213,  223

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Find the jump. Then count on in thirds. Remember that 3 thirds make 1 whole.

  1. Find the jump between two numbers you know. The numbers go up by the same amount each time.
    13 to 23: the jump is 13
  2. Count on in thirds. Remember that 3 thirds make 1 whole, so 33 is written as 1.
    13,  23,  1,  113,  123,  2,  213,  223
    01323111312322132233
  3. Write the missing numbers.
    113,  2

Check. Count on again from the first number and see that the jump is always 13. ✓

Watch out. When the top number reaches the bottom number you have a whole: 33 is written as 1. After the whole, the fractions carry on: 1 13.

Worked example 5.6.6
Work out 4/5 + 3/5. Write the answer as a mixed number.

Key words. The numerator is the top number. The denominator is the bottom number.

Method. Fractions with the same bottom number are made of equal pieces. Add the top numbers and keep the bottom number the same.

  1. Check the bottom numbers. They are all 5. That means every piece is the same size (fifths), so we can add them.
    45 + 35
  2. Add the top numbers. The top numbers tell us how many pieces we have. We are counting pieces of the same size.
    4 + 3 = 7
  3. Keep the bottom number the same. The pieces are still fifths. Adding does not change the size of the pieces.
    45 + 35 = 75
  4. Write the answer as a mixed number. 7 fifths is more than 1 whole. 5 fifths make 1 whole, so we have 1 whole and 2 fifths left over.
    75 = 125

Check. Take away one of the fractions from the answer: 7 − 3 = 4. That is the top of the other fraction. ✓

Watch out. Do not add the bottom numbers. The pieces stay fifths, so the bottom number stays 5.

Worked example 5.6.7
Each pizza is cut into 8 equal slices. Sam eats 19 slices. How many pizzas does he eat? Write your answer as a mixed number.

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. An improper fraction is more than 1 whole. Find how many wholes it contains. The parts left over make the fraction part.

  1. Ask: how many wholes are in it? 8 eighths make 1 whole. So we ask how many 8s fit into 19.
    8 × 2 = 16, which is not more than 19.
    8 × 3 = 24, which is more than 19.
    So there are 2 whole ones.
  2. Find how many parts are left over. Take the parts in the whole ones away from all the parts.
    19 − 16 = 3
  3. Write the mixed number. The whole number is 2. The left over parts make the fraction, still eighths.
    198 = 238

Check. Go back: 2 × 8 + 3 = 19 = 19. ✓

Watch out. The bottom number stays the same. Only the top number and the whole number change.

Watch out

Keep the bottom number. 11/4 = 2 3/4, not 2 1/4 and not 2 3/11. The remainder (3) goes on top and the bottom number stays 4. And when you count in quarters, after 3/4 comes 1 whole. 4/4 is the same as 1, so we call it 1.

Exercise 5F  Improper fractions and mixed numbers

Non-calculator
1
FWrite each shaded amount as an improper fraction and as a mixed number.
(a)
Improper fraction and mixed number:
(b)
Improper fraction and mixed number:
(c)
Improper fraction and mixed number:
(d)
Improper fraction and mixed number:
2
FQuick-fire! Write each mixed number as an improper fraction.
(a)134 =
(b)215 =
(c)238 =
(d)314 =
(e)156 =
(f)427 =
3
FQuick-fire! Write each improper fraction as a mixed number.
(a)94 =
(b)73 =
(c)115 =
(d)136 =
(e)178 =
(f)103 =
4
FRead the number line. Write A and C as mixed numbers. Write B and D as improper fractions.
0123ABCD
(a)A =
(b)B =
(c)C =
(d)D =
5
FCount on or back. Write the missing numbers.
(a)34,  1,  ?,  124,  134,  ?,  214
(b)113,  ?,  2,  213,  ?,  3
(c)225,  2,  ?,  115,  ?,  25
6
FAdd. Write each answer as a mixed number.
(a)45 + 35 =
(b)57 + 47 =
(c)58 + 68 =
(d)79 + 69 =
7
FWrite <, > or = in each box.
(a)74  [ ]   112
(b)95  [ ]   135
(c)116  [ ]   123
8
FWrite the numbers in order.
(a)Smallest first: 94,  134,  54,  112
(b)Biggest first: 118,  134,  32,  118
9
PSKim fills 9 cups with water. Each cup holds 1/4 of a litre. How many litres of water is that? Write your answer as a mixed number.
10
PSZainab lays 13 ribbons end to end. Each ribbon is 1/5 of a metre long. How many metres long is the line of ribbons? Write your answer as a mixed number.
11
RJack says, “114 = 214.” What mistake has Jack made? Write the correct mixed number.
12
RAlways, sometimes or never?   If the top number of a fraction is bigger than its bottom number, the fraction is greater than 1.
13
RMei has 334 pizzas. She cuts them all into quarters. How many quarters does she have?
5.7

Tenths and hundredths

Year 4
Count up and down in tenths and hundredths, Link tenths and hundredths

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Tenths and hundredthsComing soon

When we cut a whole into 10 equal parts, each part is one tenth, written 1/10. Here is a bar cut into tenths. 10 tenths make 1 whole, so 10/10 = 1.

110110110110110110110110110110
A whole cut into tenths. 3 tenths are shaded: 3/10.

Counting in tenths. 1/10, 2/10, 3/10 ... 9/10, then 10/10 which is 1 whole, then 1 1/10, 1 2/10 ... and 2. We can count backwards too.

012A
The number line from 0 to 2 in tenths. A is 13 tenths = 1 3/10.

Hundredths. Now cut each tenth into 10 equal parts. The whole is now cut into 10 × 10 = 100 equal parts. Each part is one hundredth, written 1/100. A hundred square shows this. Each row is one tenth. Each small square is one hundredth.

1101100one tenth cut into 10 parts
A tenth cut into 10 equal parts. Each part is a hundredth. 1/10 ÷ 10 = 1/100.
37 small squares are shaded: 37/100. That is 3 full rows (3 tenths = 30 hundredths) and 7 more squares.

Tenths and hundredths are linked. 1 tenth is 10 hundredths: 1/10 = 10/100. So 3/10 = 30/100. To change tenths into hundredths, multiply the top and bottom by 10 (see Section 5.3). We count in hundredths in the same way: 1/100, 2/100 ... 99/100, then 100/100 = 1 whole. Later (Chapter 6) you will see that these fractions can also be written as decimals.

Key words. A tenth is one of 10 equal parts of a whole. A hundredth is one of 100 equal parts of a whole.

Key ideas

  • A tenth is 1 of 10 equal parts: 1/10. 10 tenths = 1 whole.
  • A hundredth is 1 of 100 equal parts: 1/100. 100 hundredths = 1 whole.
  • 1/10 ÷ 10 = 1/100. Cutting each tenth into 10 parts makes hundredths.
  • 1/10 = 10/100, so tenths × 10 = hundredths.
  • Counting in tenths: after 9/10 comes 1 whole, then 1 1/10, 1 2/10.
  • In a hundred square, each row is one tenth and each small square is one hundredth.
Worked example 5.7.1
What fraction of the bar is shaded?

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. Tenths are made by cutting a whole into 10 equal parts. Count the shaded parts for the top number.

  1. Count the equal parts. The whole is cut into 10 equal parts. Each part is one tenth.
    There are 10 parts, so the bottom number is 10.
  2. Count the shaded parts. This is the top number.
    7 parts are shaded, so the top number is 7.
  3. Write the fraction.
    710   (say “seven tenths”)

Check. 7 shaded and 3 white make 10 parts. ✓

Watch out. The bottom number is the number of parts in the whole (10), not the number shaded.

Worked example 5.7.2
What fraction of the hundred square is shaded?

Key words. The denominator (bottom number) is how many equal parts. The numerator (top number) is how many parts we have.

Method. A hundred square has 100 small squares. Each one is a hundredth. Count the shaded squares for the top number.

  1. Count the small squares in the whole. There are 10 rows of 10 small squares. Each small square is one hundredth.
    10 × 10 = 100 small squares, so the bottom number is 100.
  2. Count the shaded squares. Count whole rows first (10 squares each), then the extra squares.
    3 full rows = 3 × 10 = 30 squares
    30 + 7 = 37
  3. Write the fraction.
    37100   (say “37 hundredths”)

Check. 37 shaded and 63 white make 100. ✓

Watch out. Count in tens for the full rows to save time.

Worked example 5.7.3
Count on in tenths. Write the missing numbers.
610,  710,  ?,  910,  1,  ?,  1210,  1310

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Find the jump. Then count on in tenths. Remember that 10 tenths make 1 whole.

  1. Find the jump between two numbers you know. The numbers go up by the same amount each time.
    610 to 710: the jump is 110
  2. Count on in tenths. Remember that 10 tenths make 1 whole, so 1010 is written as 1.
    610,  710,  810,  910,  1,  1110,  1210,  1310
    012
  3. Write the missing numbers.
    810,  1110

Check. Count on again from the first number and see that the jump is always 110. ✓

Watch out. When the top number reaches the bottom number you have a whole: 1010 is written as 1. After the whole, the fractions carry on: 1 110.

Worked example 5.7.4
Count back in tenths. Write the missing numbers.
1310,  1210,  ?,  1,  ?,  810,  710

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Find the jump. Then count back in tenths. Remember that 10 tenths make 1 whole.

  1. Find the jump between two numbers you know. The numbers go down by the same amount each time.
    1310 to 1210: the jump is 110
  2. Count back in tenths. Remember that 1 whole is 1010.
    1310,  1210,  1110,  1,  910,  810,  710
    012
  3. Write the missing numbers.
    1110,  910

Check. Count back again from the first number and see that the jump is always 110. ✓

Watch out. Be careful when you pass a whole number going backwards: 1 is the same as 1010.

Worked example 5.7.5
The hundred square has 3 rows shaded. 3/10 of the square is shaded. How many hundredths is that?
310 = ?100

Key words. Equivalent fractions are different fractions that show the same amount, like 1/2 and 2/4.

Method. Equivalent fractions show the same amount. Multiply (or divide) the top and the bottom by the same number.

  1. Look at the bottom numbers. We need to go from 10 to 100. What do we multiply by?
    10 × 10 = 100
  2. Do the same to the top number. Every piece is cut into 10 smaller pieces, so the top number is multiplied too. The fraction stays the same size.
    3 × 10 = 30
  3. Write the equivalent fractions. Both fractions show the same amount.
    310 = 30100

Check. 310 and 30100 are the same size. ✓

Watch out. Never add or subtract the same number to the top and bottom. You must multiply or divide both.

Worked example 5.7.6
Count on in hundredths. Write the missing numbers.
96100,  97100,  ?,  99100,  ?,  11100,  12100

Key words. An improper fraction has a top number at least as big as the bottom, like 7/4. A mixed number is a whole number and a fraction, like 1 3/4.

Method. Find the jump. Then count on in hundredths. Remember that 100 hundredths make 1 whole.

  1. Find the jump between two numbers you know. The numbers go up by the same amount each time.
    96100 to 97100: the jump is 1100
  2. Count on in hundredths. Remember that 100 hundredths make 1 whole, so 100100 is written as 1.
    96100,  97100,  98100,  99100,  1,  11100,  12100
  3. Write the missing numbers.
    98100,  1

Check. Count on again from the first number and see that the jump is always 1100. ✓

Watch out. When the top number reaches the bottom number you have a whole: 100100 is written as 1. After the whole, the fractions carry on: 1 1100.

Worked example 5.7.7
What is 1/10 divided by 10? Use a picture.

Method. Cut each tenth into 10 equal parts. Each new part is a hundredth.

  1. Start with one tenth. A whole cut into 10 equal parts gives tenths.
    1101100one tenth cut into 10 parts
  2. Cut that tenth into 10 equal parts. This is the same as dividing the tenth by 10.
    110 ÷ 10 = 1100
  3. Count the parts in the whole. There are 10 tenths, and each tenth is 10 hundredths.
    10 × 10 = 100 parts, so each part is 1100.

Check. 10 hundredths make 1 tenth: 10100 = 110. ✓

Watch out. Dividing a tenth by 10 makes a hundredth, which is a much smaller piece.

Watch out

Hundredths are much smaller than tenths. 1/100 is smaller than 1/10, because the whole is cut into more parts. And do not forget the whole number when you count: after 9/10 comes 1 (10/10), then 1 1/10. After 99/100 comes 1.

Exercise 5G  Tenths and hundredths

Non-calculator
1
FWhat fraction of each bar is shaded?
(a)
Fraction shaded =
(b)
Fraction shaded =
(c)
Fraction shaded =
2
FWhat fraction of each hundred square is shaded?
(a)
Fraction shaded =
(b)
Fraction shaded =
(c)
Fraction shaded =
3
FCount on in tenths. Write the missing numbers.
(a)310,  410,  ?,  610,  ?,  810,  910
(b)710,  810,  910,  ?,  1110,  ?,  1310
(c)410,  610,  810,  ?,  1210,  ?
4
FCount back in tenths. Write the missing numbers.
(a)910,  810,  ?,  610,  510,  ?,  310
(b)1410,  ?,  1210,  1110,  ?,  910,  810
(c)1610,  1410,  1210,  ?,  810,  ?
5
FCount in hundredths. Write the missing numbers.
(a)97100,  98100,  ?,  1,  ?,  12100
(b)6100,  ?,  4100,  ?,  2100,  1100
6
FThe number line goes from 0 to 2 in tenths. Write each point as a mixed number or a fraction.
012ABC
(a)A =
(b)B =
(c)C =
7
FQuick-fire! Change tenths into hundredths.
(a)110 = ?100
(b)310 = ?100
(c)710 = ?100
(d)910 = ?100
(e)510 = ?100
(f)210 = ?100
8
FChange these hundredths into tenths.
(a)40100 = ?10
(b)70100 = ?10
(c)20100 = ?10
(d)90100 = ?10
9
FFind the amounts.
(a)110 of 80 =
(b)310 of £50 =
(c)710 of 90 g =
(d)910 of 40 m =
10
PSA water tank holds 100 litres when it is full. It is 7/10 full. A box has 100 stickers. 3/10 of the stickers are gold.
(a)How many litres of water are in the tank?
(b)How many stickers are gold?
11
RAisha says, “1100 is bigger than 110, because 100 is bigger than 10.” Is Aisha right? Explain.
12
RCompare and order fractions with different bottom numbers.
(a)Which is bigger, 710 or 60100?
(b)Smallest first: 310,  25100,  410
13
RShow that 4/10 is the same as 40/100. Use a hundred square to help you.
14
RMei shades 310 of a hundred square blue and 12100 red. What fraction of the hundred square is not shaded?

Chapter summary

SkillMethod
Naming fractionsBottom number (denominator) = number of EQUAL parts in the whole. Top number (numerator) = parts we have. 3 of 8 equal parts = 3/8.
Fractions of a setWhole set on the bottom, the part you want on the top: 3 red counters out of 8 counters = 3/8.
Fraction of an amountDivide by the bottom number to find one part. Multiply by the top number. 3/4 of 20: 20 ÷ 4 = 5, 5 × 3 = 15.
Fraction leftFind the amount used, then take it away from the whole. Or find the other parts: (bottom − top) parts.
Equivalent fractionsMultiply (or divide) the top AND the bottom by the same number. 1/2 = 2/4 = 4/8. Fifths and tenths: 1/5 = 2/10. Never add.
Add, same denominatorAdd the top numbers. Keep the bottom number. 2/7 + 3/7 = 5/7. If the answer is 8/8, that is 1 whole.
Subtract, same denominatorSubtract the top numbers. Keep the bottom number. 1 whole = 5/5, so 1 − 2/5 = 3/5.
Compare, same denominatorThe bigger top number is the bigger fraction: 5/8 > 3/8.
Compare unit fractionsThe bigger the bottom number, the smaller the fraction: 1/8 < 1/4. A fraction wall shows it.
Compare / order, different denominatorsChange to equivalent fractions with the same bottom number (3/4 = 6/8), then compare the top numbers. Compare with 1/2 as a quick check.
Improper fractions and mixed numbers7/4 = 1 3/4. Improper → mixed: divide the top by the bottom, the remainder goes on top. Mixed → improper: whole × bottom + top.
Counting in fractionsCount in steps of the unit fraction. After the last fraction comes a whole: 3/4, 1, 1 1/4. Count back the same way.
Tenths and hundredths10 tenths = 1. 100 hundredths = 1. 1/10 = 10/100. 1/10 ÷ 10 = 1/100. In a hundred square, a row is a tenth and a square is a hundredth.

Can you…?

  • I can name a fraction of a shape or a set, and I know the parts must be equal.
  • I can find a unit fraction of an amount by dividing.
  • I can find a non-unit fraction of an amount: divide by the bottom number, then multiply by the top number.
  • I can use fractions of money and measures to solve problems.
  • I can show equivalent fractions with bars, a fraction wall and a number line, and fill in missing numbers.
  • I can add and subtract fractions with the same denominator, including making a whole.
  • I can compare and order fractions with the same denominator, and use a fraction wall or equivalent fractions when they differ.
  • I know that a bigger denominator does not mean a bigger fraction.
  • I can write improper fractions and mixed numbers from pictures and number lines, and change between them.
  • I can count up and down in tenths and hundredths, including past 1.
  • I know that 10 hundredths make 1 tenth.

Key terms

Fraction
A part of a whole, or of a set, cut into equal parts.
Equal parts
Parts that are exactly the same size.
Denominator
The bottom number of a fraction. It says how many equal parts there are altogether.
Numerator
The top number of a fraction. It says how many parts we have.
Unit fraction
A fraction with 1 on the top, like 1/5.
Non-unit fraction
A fraction with a top number that is not 1, like 3/5.
Equivalent fractions
Fractions that show the same amount, like 1/2 and 2/4.
Fraction wall
A chart of bricks that shows fractions of one whole in rows.
Whole
One complete thing. 4/4 and 10/10 both equal 1 whole.
Compare
To say which of two numbers or fractions is bigger.
Order
To put numbers or fractions in a list from smallest to biggest, or biggest to smallest.
Improper fraction
A fraction whose top number is the same as or bigger than the bottom number, like 7/4.
Mixed number
A whole number and a fraction together, like 1 3/4.
Tenth
One of 10 equal parts of a whole: 1/10.
Hundredth
One of 100 equal parts of a whole: 1/100.
Bar model
A drawing of a bar cut into equal parts that shows a fraction of an amount.

Exercise 5Z  Check-up (non-calculator)

Non-calculator
1
What fraction of the circle is shaded?
2
Work out 1/5 of 40.
3
Work out 3/8 of 24.
4
Write the missing number.   23 = ?9
5
Work out 39 + 49.
6
Work out 1 − 27.
7
Write < or > in the box.   35  [ ]   710
8
Write 145 as a mixed number.
9
Count on in tenths. Write the missing numbers.
810,  910,  ?,  ?,  1210,  1310
10
Mia has £35. She spends 3/5 of her money. How much money does she have left?
?????£35

Total 15 marks. Allow about 15 minutes.

Exercise 5X  SATs-style questions

Non-calculator
1
FWhat fraction of the rectangle is shaded?[1 mark]
2
FWork out 3/4 of 28.[1 mark]
3
FWrite the missing number.   35 = ?10[1 mark]
4
FWork out 59 + 39.[1 mark]
5
FWrite 174 as a mixed number.[1 mark]
6
FWrite the missing numbers in the counting sequence.
24,  34,  ?,  114,  ?,  134[1 mark]
7
RWhich two fractions are equal to 12? Tick two boxes.   A 24   B 38   C 510   D 46[2 marks]
8
FWrite these fractions in order, smallest first.   34,  12,  58
Show your working.[2 marks]
9
RBen says, “25 + 25 = 410.” Is Ben correct? Explain how you know.[2 marks]
10
PSAmir has £48. He spends 1/4 of his money on a game. He then spends 1/3 of the money he has left on a book. How much money does Amir have now?   Show your working.[3 marks]
????£48
← Chapter 4: Multiplying and dividingChapter 6: Decimals and money →

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