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SATs practice · Year 6Chapter 4 · Decimals and percentages
Year 6 · Chapter 4

Decimals and percentages

By the end of this chapter you will be able to

  • Write a decimal as a fraction and a fraction as a decimal
  • Find the value of a digit in a decimal to three places
  • Compare, order and round decimals
  • Multiply and divide decimals by whole numbers
  • Change between fractions, decimals and percentages, and find a percentage of an amount

You should already know

  • Place value and tenths, hundredths (Year 5 maths)
  • Fractions with denominators 2, 4, 5 and 10 (Chapter 3 of this book)

Where this leads

  • Ratio and proportion, including percentages as a proportion of a whole (Chapter 5 of this book)

KS2 SATs practice for arithmetic (Year 6). Original questions in the style of the national tests. Content: Read and write decimals as fractions, for example 0.71 = 71/100, and recognise thousandths (Year 5), Identify the value of each digit in numbers given to three decimal places; multiply and divide by 10, 100 and 1000 (Year 6), Multiply and divide numbers with up to two decimal places by one-digit and two-digit whole numbers (Year 5), Round to a specified degree of accuracy, including in context (Year 6), Recall and use equivalences between simple fractions, decimals and percentages (Year 5-6), Calculate percentages of amounts, such as 15% of 360, and use percentages for comparison (Year 5-6).

Exercise P  Do you remember?

Non-calculator
1
FWrite 1/4 as a decimal.
2
FWork out 3.6 + 2.5.
3
FWork out 0.4 × 10.
4
FWhat is 25% of 100?
5
FRound 4.6 to the nearest whole number.
4.1

Decimals as fractions and place value to three places

Year 6
Read and write decimals as fractions, for example 0.71 = 71/100, and recognise thousandths (Year 5), Identify the value of each digit in numbers given to three decimal places; multiply and divide by 10, 100 and 1000 (Year 6)
Video lesson: Decimals as fractions and place value to three placesComing soon
Practice material. Non-calculator practice. This book is practice in the style of the Year 6 arithmetic test. It is not the official test, and no official test item is used. Do every question without a calculator. Marks and timings are a practice guide only.
Story. A bottle of juice is 4.572 litres, measured to the nearest millilitre. The label writes each digit in a column, and the class is asked what the 7 is worth.

Rule

Each decimal place is a column: tenths (0.1), hundredths (0.01) and thousandths (0.001). A decimal can be written as a fraction over 10, 100 or 1000: 0.71 = 71/100.

Multiplying by 10, 100 or 1000 moves the digits to the left; dividing moves them to the right. For the full work on decimals, read Year 6 maths, chapter 4.

Key ideas

  • 0.71 = 71/100: the number of decimal places sets the bottom number.
  • Thousandths are three places after the point: 0.005 = 5/1000.
  • Multiply by 100 moves the digits two places to the left.
  • Divide by 1000 moves the digits three places to the right.
Worked example 4.1.1
Write 0.71 as a fraction.

Method. Count the decimal places, then use that many zeros on the bottom.

  1. Count the places. 0.71 has two decimal places.
    two places
  2. Write the digits over the matching power of 10. Two places means 100 on the bottom.
    71/100
  3. Check it cannot be simplified. 71 is a prime number, so the fraction is simplest.
    71/100
  4. Write the answer.
    71/100

Check. 71 ÷ 100 = 0.71. ✓

Worked example 4.1.2
What is the value of the 7 in 4.572?

Method. Find the column of the 7, then write its value.

  1. Find the column. The 7 is the second digit after the point, so it is in the hundredths column.
    hundredths
  2. Write the value. 7 hundredths is 7 ÷ 100.
    7 × 0.01 = 0.07
  3. Write the answer. The value of the 7 is 0.07.
    0.07
  4. Write the answer.
    0.07

Check. 4.572 = 4 + 0.5 + 0.07 + 0.002. ✓

Worked example 4.1.3
Write 0.005 as a fraction in its simplest form.

Method. Three places means the bottom is 1000.

  1. Write over 1000. 0.005 has three decimal places, so the bottom is 1000.
    5/1000
  2. Simplify. Divide the top and bottom by 5.
    5/1000 = 1/200
  3. Check. 1/200 = 0.005.
    0.005 = 1/200
  4. Write the answer.
    1/200

Check. 200 × 0.005 = 1. ✓

Exercise 4A  Practice: decimals as fractions and place value

Non-calculator
1
FWrite 0.3 as a fraction in its simplest form.
2
FWrite 0.45 as a fraction in its simplest form.
3
FWrite 0.125 as a fraction in its simplest form.
4
FWrite 2.6 as a fraction in its simplest form.
5
FWhat is the value of the 6 in 3.961?
6
FWhat is the value of the 4 in 0.248?
7
FWhat is the value of the 8 in 0.058?
8
FMultiply 0.047 by 100.
9
FMultiply 0.056 by 1000.
10
FDivide 3.56 by 10.
11
FDivide 4.8 by 100.
12
PSWhich digit is in the thousandths column of 4.317?
13
PSA bag of sugar weighs 1.650 kg. How many grams is that? Write the number of grams.
14
RSam says 0.7 and 0.70 are different because 0.70 has two digits. Explain whether he is right.
15
RAli says 0.125 is the same as 1/125. Is he right? Write the fraction for 0.125 in its simplest form.
4.2

Comparing, ordering and rounding decimals

Year 6
Round to a specified degree of accuracy, including in context (Year 6), Identify the value of each digit in numbers given to three decimal places (Year 6)
Video lesson: Comparing, ordering and rounding decimalsComing soon
Story. At a sports day, the times are written to three decimal places. The organiser must put the runners in order, and then round each time to two places for the newspaper.

Rule

To compare decimals, line up the decimal points and compare the digits from the left. Add zeros at the end if you need to.

To round, look at the digit just after the place you keep. If it is 5 or more, round up. If it is less, round down.

Key ideas

  • Line up the decimal points, then compare from the left.
  • 0.68 is bigger than 0.608, because 0.680 is bigger than 0.608.
  • Round up when the next digit is 5 or more.
Worked example 4.2.1
Fill in the box: 0.608 ☐ 0.68, using <, > or =.

Method. Line up the decimal points, adding a zero to 0.68.

  1. Line up the decimal points. Give both numbers the same number of decimal places.
    0.608 and 0.680
  2. Compare from the left. The first place that differs decides which is bigger.
    digits 0 and 8
  3. Write the sign. The bigger number is the one with the bigger digit at the first place that differs.
    0.608 < 0.68
  4. Write the answer.
    <

Check. 0.608 is less than 0.680. ✓

Watch out. Do not think 608 is bigger than 68: the places are different.

Worked example 4.2.2
Put these in order, smallest first: 0.9, 0.59, 0.609, 0.6.

Method. Write each with three decimal places, then compare the digits from the left.

  1. Write with three places. 0.9 = 0.900, 0.59 = 0.590, 0.6 = 0.600.
    0.900, 0.590, 0.600, 0.609
  2. Compare the tenths. Tenths: 5 for 0.590, 6 for 0.600 and 0.609, and 9 for 0.900.
    5 < 6 < 9
  3. Compare the rest. 0.600 is less than 0.609.
    0.59, 0.6, 0.609, 0.9
  4. Write the answer.
    0.59, 0.6, 0.609, 0.9

Check. 0.9 is the largest because 0.900 is the biggest. ✓

Worked example 4.2.3
Round 3.4762 to two decimal places.

Method. Keep two decimal places, then look at the next digit.

  1. Find the place you are rounding to. Rounding to two decimal places means keeping 2 decimal places.
    keep 2 places
  2. Look at the next digit. The digit after the place is 6. If it is 5 or more, round up; if less, round down.
    digit 6
  3. Write the rounded number. 3.4762 rounds to 3.48.
    3.4762 → 3.48
  4. Write the answer.
    3.48

Check. 3.48 is nearer to 3.4762 than 3.47 is. ✓

Exercise 4B  Practice: comparing, ordering and rounding

Non-calculator
1
FFill in the box with <, > or =: 0.4 ☐ 0.35
2
FFill in the box with <, > or =: 2.07 ☐ 2.7
3
FFill in the box with <, > or =: 0.6 ☐ 0.59
4
FFill in the box with <, > or =: 5.001 ☐ 5.01
5
FFill in the box with <, > or =: 1.5 ☐ 1.45
6
FFill in the box with <, > or =: 0.08 ☐ 0.8
7
FPut these in order, smallest first: 0.45, 0.4, 0.405, 0.5.
8
FRound 6.837 to one decimal place.
9
FRound 5.6249 to two decimal places.
10
FRound 0.0715 to two decimal places.
11
PSAli has £4.376 in his purse. Round the amount to the nearest penny. Write the answer in pounds.
12
PSA runner's time is 12.4 seconds. Round it to the nearest whole second.
13
RRound 2.45 to one decimal place. Write the answer.
14
RA recipe says 2.45 kg of flour. Ali rounds this to 2.4 kg and says it is close enough. Is his rounding correct? Explain.
4.3

Multiplying decimals by whole numbers

Year 6
Multiply and divide numbers with up to two decimal places by one-digit and two-digit whole numbers (Year 5)
Video lesson: Multiplying decimals by whole numbersComing soon
Story. A pen costs £1.25. Ali wants to buy 14 pens for his class, and he needs the total cost before he goes to the shop.

Rule

Multiply as if there were no decimal point, then put the point back so the answer has the same number of decimal places as the decimal.

For a two-digit whole number, use two partial products: multiply by the ones digit, then by the tens digit, then add.

Key ideas

  • Ignore the decimal point, multiply, then put it back.
  • The answer has the same number of decimal places as the decimal number.
  • Estimate first to check the size of the answer.
Worked example 4.3.1
Work out 1.25 × 14.

Method. Use two partial products, then put the decimal point back.

  1. Multiply by the ones digit. Multiply the digits 125 by 4.
    125 × 4 = 500
  2. Multiply by the tens digit. Multiply the digits 125 by 10, which is 1 tens.
    125 × 10 = 1250
  3. Add the two parts. Add the partial products.
    500 + 1250 = 1750
  4. Put the decimal point back. 1.25 has 2 decimal places, so the answer has 2.
    1750 → 17.50
  5. Write the answer.
    17.50

Check. About 1 × 14 = 14, and 17.5 is about right. ✓

Watch out. Keep the two decimal places in the answer: 17.50.

Worked example 4.3.2
Work out 3.7 × 6.

Method. Multiply 37 by 6, then put the point back.

  1. Ignore the decimal point for now. Multiply the digits 37 by 6, as whole numbers.
    37 × 6 = 222
  2. Put the decimal point back. 3.7 has 1 decimal place, so the answer has 1.
    222 → 22.2
  3. Write the answer.
    22.2

Check. About 4 × 6 = 24, close to 22.2. ✓

Worked example 4.3.3
Work out 0.46 × 8.

Method. Multiply 46 by 8, then put the point back.

  1. Ignore the decimal point for now. Multiply the digits 46 by 8, as whole numbers.
    46 × 8 = 368
  2. Put the decimal point back. 0.46 has 2 decimal places, so the answer has 2.
    368 → 3.68
  3. Write the answer.
    3.68

Check. 0.5 × 8 = 4, close to 3.68. ✓

Exercise 4C  Practice: multiplying decimals by whole numbers

Non-calculator
1
FWork out 3.4 × 5.
2
FWork out 2.36 × 4.
3
FWork out 1.85 × 6.
4
FWork out 0.46 × 8.
5
FWork out 6.25 × 12.
6
FWork out 4.07 × 23.
7
FWork out 0.85 × 14.
8
FWork out 7.6 × 9.
9
FWork out 12.5 × 4.
10
FWork out 0.39 × 16.
11
PSAli buys 14 pens at £1.25 each. How much do they cost altogether?
12
PSA packet of rice weighs 0.75 kg. How many kilograms are in 12 packets?
13
REstimate 6.8 × 19 by rounding 6.8 to 7 and 19 to 20. Then say whether 129.2 is close to your estimate.
14
FWork out 4.05 × 3 and check your answer.
4.4

Dividing decimals by whole numbers

Year 6
Multiply and divide numbers with up to two decimal places by one-digit and two-digit whole numbers (Year 5), Multiply and divide by 10, 100 and 1000 (Year 6)
Video lesson: Dividing decimals by whole numbersComing soon
Story. A ribbon 8.4 metres long is cut into 4 equal pieces. The teacher needs to know how long each piece is, so she divides the length by 4.

Rule

Divide the whole number first, then keep dividing into the decimal places, placing the decimal point in the answer above the point in the number.

Add zeros after the point if you need more places. The answer should end exactly, with no remainder.

Key ideas

  • Place the decimal point in the answer straight above the point in the number.
  • Add zeros after the point to keep dividing.
  • Check by multiplying the answer by the whole number.
Worked example 4.4.1
Work out 6.84 ÷ 4.

Method. Divide the digits as a whole number, then put the point in the answer.

  1. Divide the digits as a whole number. 684 ÷ 4 = 171 remainder 0.
    684 ÷ 4 = 171 r 0
  2. Place the decimal point. Put the point in the answer directly above the point in 6.84, and count the places.
    6.84 ÷ 4 = 1.71
  3. Write the answer.
    1.71

Check. 1.71 × 4 = 6.84. ✓

Worked example 4.4.2
Work out 3.6 ÷ 8.

Method. Divide 36 by 8 and place the decimal point.

  1. Divide the digits as a whole number. 36 ÷ 8 = 4 remainder 4.
    36 ÷ 8 = 4 r 4
  2. Bring down a zero and keep dividing. The remainder becomes 40, and 8 goes in 5 times.
    40 ÷ 8 = 5 r 0
  3. Place the decimal point. Put the point in the answer directly above the point in 3.6, and count the places.
    3.6 ÷ 8 = 0.45
  4. Write the answer.
    0.45

Check. 0.45 × 8 = 3.6. ✓

Watch out. Put the decimal point in the answer above the one in the number: 0.45, not 4.5.

Worked example 4.4.3
Work out 0.9 ÷ 6.

Method. Divide 9 by 6, with zeros after the point if needed.

  1. Divide the digits as a whole number. 9 ÷ 6 = 1 remainder 3.
    9 ÷ 6 = 1 r 3
  2. Bring down a zero and keep dividing. The remainder becomes 30, and 6 goes in 5 times.
    30 ÷ 6 = 5 r 0
  3. Place the decimal point. Put the point in the answer directly above the point in 0.9, and count the places.
    0.9 ÷ 6 = 0.15
  4. Write the answer.
    0.15

Check. 0.15 × 6 = 0.9. ✓

Exercise 4D  Practice: dividing decimals by whole numbers

Non-calculator
1
FWork out 6.84 ÷ 4.
2
FWork out 3.6 ÷ 8.
3
FWork out 0.9 ÷ 6.
4
FWork out 7.5 ÷ 6.
5
FWork out 9.6 ÷ 12.
6
FWork out 0.48 ÷ 6.
7
FWork out 12.6 ÷ 4.
8
FWork out 5.4 ÷ 9.
9
FWork out 4.2 ÷ 6.
10
FWork out 8.4 ÷ 4.
11
PSSix children share £5.40 equally. How much does each child get?
12
PSA ribbon 8.4 m long is cut into 4 equal pieces. How long is each piece?
13
RSam says 3.6 ÷ 2 = 1.8, so 3.6 ÷ 20 is 0.18. Is he right? Explain how the answer changes.
14
FWork out 2.7 ÷ 9 and check your answer.
15
FWork out 10.8 ÷ 6.
16
RWork out 7.2 ÷ 8.
4.5

Fractions, decimals, percentages and amounts

Year 6
Recall and use equivalences between simple fractions, decimals and percentages (Year 5-6), Calculate percentages of amounts, such as 15% of 360, and use percentages for comparison (Year 5-6)
Video lesson: Fractions, decimals, percentages and amountsComing soon
Story. A school has 360 pupils. The head teacher says that 15% of the pupils are in the choir. The class wants to know how many that is, and whether more pupils walk to school than take the bus.

Rule

A percentage means "out of 100". 25% = 25/100 = 0.25 = 1/4.

To find a percentage of an amount, find 1% first (divide by 100), then multiply by the percentage.

Common equivalents: 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25%, 3/4 = 0.75 = 75%, 1/5 = 0.2 = 20%, 1/10 = 0.1 = 10%.

Key ideas

  • A percentage is out of 100: 25% = 25/100.
  • 1% of an amount is the amount divided by 100.
  • Use fractions, decimals and percentages to compare values.
Worked example 4.5.1
Work out 15% of 360.

Method. Find 1% first, then multiply by 15.

  1. Find 1%. 1% is one hundredth, so divide the amount by 100.
    360 ÷ 100 = 3.6
  2. Multiply by 15. 15% is 15 lots of 1%.
    3.6 × 15 = 54
  3. Write the answer.
    54

Check. 10% of 360 is 36, and 5% is 18, so 15% is 54. ✓

Worked example 4.5.2
Work out 20% of 85.

Method. Find 1%, then multiply by 20.

  1. Find 1%. 1% is one hundredth, so divide the amount by 100.
    85 ÷ 100 = 0.85
  2. Multiply by 20. 20% is 20 lots of 1%.
    0.85 × 20 = 17
  3. Write the answer.
    17

Check. 20% is one fifth of 85, and 85 ÷ 5 = 17. ✓

Worked example 4.5.3
Which is greater: 40% of 80, or 1/3 of 90?

Method. Work out both amounts, then compare.

  1. Work out 40% of 80. 1% of 80 is 0.8, so 40% is 32.
    32
  2. Work out 1/3 of 90. 90 ÷ 3 = 30.
    30
  3. Compare. 32 is greater than 30.
    40% of 80 is greater
  4. Write the answer.
    40% of 80

Check. 40% of 80 is 32 and 1/3 of 90 is 30. ✓

Watch out. Compare the amounts you have worked out, not the percentage and the fraction.

Exercise 4E  Practice: fractions, decimals, percentages and amounts

Non-calculator
1
FWrite 3/5 as a decimal.
2
FWrite 1/8 as a decimal.
3
FWrite 7/10 as a decimal.
4
FWrite 3/4 as a decimal.
5
FWrite 9/20 as a decimal.
6
FWrite 3/5 as a percentage.
7
FWrite 1/4 as a percentage.
8
FWrite 9/20 as a percentage.
9
FWrite 7/10 as a percentage.
10
FWork out 15% of 360.
11
FWork out 20% of 85.
12
FWork out 35% of 240.
13
FWork out 5% of 60.
14
FWork out 12% of 250.
15
FWork out 7% of 300.
16
PS240 pupils are in a school. 35% walk to school. How many pupils walk?
17
PSWhich is more: 40% of 80, or 1/3 of 90? Write the greater amount.
18
RSam says 50% of 30 and 25% of 60 are both 15, so they are the same. Is he right? Write Yes or No.
19
PSA class has 80 pupils and 45% of them have a pet. How many pupils have a pet?
20
RAli scores 18 out of 20 and Priya scores 45 out of 50. Who scored the higher percentage? Explain.

Chapter summary

SkillMethod
Decimal as fraction0.71 = 71/100: as many zeros as decimal places.
Place valueTenths, hundredths and thousandths: 0.007 is 7 thousandths.
Compare decimalsLine up the decimal points and compare from the left.
RoundLook at the next digit: 5 or more rounds up, 4 or less rounds down.
Multiply by wholeMultiply as whole numbers, then keep the same number of decimal places.
Divide by wholeDivide, put the point in the answer above the point, and add zeros if needed.
Fractions, decimals, %3/4 = 0.75 = 75%; 1/5 = 0.2 = 20%.
Percentage of amountFind 1% (divide by 100), then multiply by the percentage: 15% of 360 = 54.

Can you…?

  • I can write a decimal as a fraction, and recognise thousandths
  • I can find the value of a digit to three decimal places
  • I can compare, order and round decimals
  • I can multiply and divide decimals by whole numbers
  • I can change between fractions, decimals and percentages
  • I can find a percentage of an amount and compare percentages

Key terms

decimal place
a position after the decimal point: tenths, hundredths, thousandths
thousandths
the third place after the decimal point, worth 0.001 each
percentage
a number out of 100, shown with the % sign
round
to change a number to a simpler value of a given accuracy
equivalent
the same value written in a different form, such as 1/2 = 0.5 = 50%

Exercise 4Z  Check-up

Non-calculator
1
Write 0.37 as a fraction.
2
What is the value of the 4 in 6.248?
3
Multiply 2.75 by 100.
4
Which is greater: 0.608 or 0.68? Choose one: 0.608 or 0.68.
5
Round 5.6249 to two decimal places.
6
Work out 2.36 × 4.
7
Work out 4.8 ÷ 6.
8
Work out 25% of 360.
9
Write 3/8 as a decimal, then as a percentage. Write the decimal, then the percentage, separated by a comma.
10
Explain why 2.5 × 4 and 2.50 × 4 are the same.

Total 17 marks. Allow about 17 minutes.

Exercise 4X  Practice test questions

Non-calculator
1
FWrite 0.8 as a fraction in its simplest form.[1 mark]
2
FWhat is the value of the 5 in 3.054?[1 mark]
3
FWork out 6.25 × 12.[2 marks]
4
FWork out 9.6 ÷ 12.[2 marks]
5
FRound 7.8359 to two decimal places.[1 mark]
6
FWork out 12% of 250.[2 marks]
7
PSWhich is greater: 35% of 200, or 3/4 of 120? Write the greater amount.[2 marks]
8
PSA shop sells 0.6 of its 120 biscuits by Friday. How many biscuits are left?[2 marks]
9
FWrite 0.45 as a percentage.[1 mark]
10
RA test has 40 marks. Ali gets 26 marks. Explain how to find his percentage score.[3 marks]
← Chapter 3: FractionsChapter 5: Ratio and proportion →

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