Year 5 · Chapter 7

Decimals and percentages

By the end of this chapter you will be able to

Write decimals with tenths, hundredths and thousandths as fractions in their simplest form, and write fractions as decimals
Understand place value to three decimal places, and use zeros as place holders
Compare and order decimals with up to three decimal places
Round a decimal with two decimal places to the nearest whole number and to one decimal place
Add and subtract decimals with up to three decimal places by lining up the decimal points, and solve problems with them
Understand that per cent means out of 100, and change between percentages, fractions with denominator 100 and decimals
Know the percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 and fractions with denominators that are multiples of 10 or 25
Explain your reasoning about decimals and percentages, and spot the mistakes in other people’s working

You should already know

Tenths and hundredths as fractions, and equivalent fractions (Year 5 Chapter 6)
Dividing and multiplying whole numbers and decimals by 10, 100 and 1,000 (Chapters 4 and 5)
Place value of numbers up to 1,000,000 (Chapter 1)
Simplifying fractions and finding the highest common factor (Chapter 6)

Where this leads

Percentages of amounts, and comparing with percentages (Year 6)
Decimals with more places, and rounding to a given accuracy (Year 6)
Ratio and proportion, and conversion between fractions, decimals and percentages in Year 6

National Curriculum for England, Key Stage 2 mathematics, Year 5. Content: Read and write decimal numbers as fractions; recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents, Round decimals with two decimal places to the nearest whole number and to one decimal place; read, write, order and compare numbers with up to three decimal places, Solve problems involving number up to three decimal places, Recognise the per cent symbol (%) and understand that per cent relates to number of parts per hundred; write percentages as a fraction with denominator 100 and as a decimal, Solve problems which require knowing percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 and fractions with a denominator of a multiple of 10 or 25.

Exercise P Do you remember?

Non-calculator

FWhat is the value of the 4 in 2,450?

Done

FWork out 4.6 × 10.

Done

FWrite 3 tenths as an equivalent fraction with a denominator of 100.

Done

RWhich is greater, 2/5 or 3/10? Explain your answer.

Done

FWrite 25 hundredths as a fraction in its simplest form.

Done

7.1

Tenths, hundredths and thousandths

Year 5
Read and write decimal numbers as fractions, Recognise and use thousandths and relate them to tenths, hundredths and decimal equivalents

Video lesson: Tenths, hundredths and thousandthsComing soon

A decimal is a fraction written with a point. The first digit after the point is in the tenths column, the second is in the hundredths column and the third is in the thousandths column. Each column is ten times smaller than the one on its left.

Ten equal parts, with 6 shaded. Six tenths is 0.6.

A hundredth is one of 100 equal parts. The square below has 36 parts shaded, so it shows 0.36, which is 36/100. In its simplest form, 36/100 is 9/25.

Each small square is one hundredth. Count the shaded squares: 36.

A thousandth is one of 1,000 equal parts. The number 0.047 has 0 tenths, 4 hundredths and 7 thousandths. So 0.047 is 47/1,000. The place-value chart shows where each digit goes.

The 7 is in the thousandths column, so the last digit tells us the bottom number is 1,000.

Key ideas

A decimal is a fraction. Count the decimal places to find the bottom number: 1 place = tenths (10), 2 places = hundredths (100), 3 places = thousandths (1,000).
Write the digits after the point over the place value, then simplify if you can.
Adding a zero at the end does not change the value: 0.5 = 0.50 = 0.500 = 5/10 = 50/100 = 500/1,000.
A whole part stays in front: 3.25 is 3 and 25/100, which is 3 1/4.

Worked example 7.1.1

Write 0.36 as a fraction in its simplest form.

Key words. A decimal uses a point (the decimal point) to show parts of a whole. The places after the point are tenths (one of ten equal parts), hundredths (one of a hundred equal parts) and thousandths (one of a thousand equal parts).

Method. Write the digits as the top number and the place value as the bottom number. Then simplify.

Read the digits after the point. The last digit is in the hundredths column, so the bottom number is 100.
0.36: the digits after the point are 36
Write the digits over the place value. The top number is the digits. The bottom number is the place value of the last digit.
³⁶⁄₁₀₀
Simplify the fraction. The highest common factor (HCF) of 36 and 100 is 4. Divide the top and the bottom by 4.
³⁶⁄₁₀₀ = ⁹⁄₂₅
Write the answer. This is the answer.
⁹⁄₂₅

Check. 36 ÷ 100 = 0.36, so 0.36 is right. ✓

Watch out. Count the decimal places: 2 places means a bottom number of 100. Do not forget the zeros.

Worked example 7.1.2

Write 0.047 as a fraction.

Method. Write the digits as the top number and the place value as the bottom number. Then simplify.

Read the digits after the point. The last digit is in the thousandths column, so the bottom number is 1,000.
0.047: the digits after the point are 047
Write the digits over the place value. The top number is the digits. The bottom number is the place value of the last digit.
⁴⁷⁄₁₀₀₀
Write the answer. This is the answer.
⁴⁷⁄₁₀₀₀

Check. 47 ÷ 1,000 = 0.047, so 0.047 is right. ✓

Watch out. Count the decimal places: 3 places means a bottom number of 1,000. Do not forget the zeros.

Worked example 7.1.3

Write 3.25 as a mixed number.

Method. Write the digits as the top number and the place value as the bottom number. Then simplify.

Read the digits after the point. The last digit is in the hundredths column, so the bottom number is 100.
3.25: the digits after the point are 25
Write the digits over the place value. The top number is the digits. The bottom number is the place value of the last digit.
3 and ²⁵⁄₁₀₀
Simplify the fraction. The highest common factor (HCF) of 25 and 100 is 25. Divide the top and the bottom by 25.
²⁵⁄₁₀₀ = ¹⁄₄
Write the answer. Read the whole number and the fraction together.
3 ¹⁄₄

Check. 25 ÷ 100 = 0.25, so 3.25 is right. ✓

Watch out. Count the decimal places: 2 places means a bottom number of 100. Do not forget the zeros.

Watch out

Do not write 0.07 as 7/10. The 7 is in the hundredths column, so it is 7/100. The number of decimal places tells you the bottom number, not the number of digits.

7.2

Comparing and ordering decimals

Year 5
Read, write, order and compare numbers with up to three decimal places

Video lesson: Comparing and ordering decimalsComing soon

To compare two decimals, first write them with the same number of decimal places. Add zeros at the end: 0.6 = 0.60. Then compare the digits from the left. The first place where the digits are different decides which number is bigger.

The number line shows the decimals from 0 to 1 in tenths. Between 0.3 and 0.4 there are numbers such as 0.35, 0.36 and 0.392. The more places a decimal has, the more exact its position.

The point B is at 0.35, which is halfway between 0.3 and 0.4.

Key ideas

Write the numbers with the same number of decimal places before you compare them.
Compare the whole numbers first, then the tenths, the hundredths and the thousandths.
Use > for “is greater than” and < for “is less than”. The symbol always points to the smaller number.
Zeros at the end of a decimal do not change its value: 0.5 = 0.50 = 0.500.

Worked example 7.2.1

Which is greater, 0.6 or 0.58? Use the symbol > or < to show it.

Key words. Place value tells you what each digit is worth. We compare numbers by starting at the left and finding the first place where the digits are different. > means “is greater than”, and < means “is less than”.

Method. Line up the place values, then compare from the left.

Write both numbers with the same number of decimal places. Zeros at the end do not change a number. They make the columns line up.
0.6 = 0.60
0.58 = 0.58
Compare the columns from the left. The first column where the digits are different decides the answer.
ones: 0 and 0, the same.
tenths: 6 and 5, so 0.6 is bigger.
Write the answer. The symbol points to the smaller number.
0.6 > 0.58

Check. Read it the other way round: 0.58 < 0.6. ✓

Watch out. Do not compare the digits after the point as if they were whole numbers. 0.6 is 0.60, which is bigger than 0.58.

Worked example 7.2.2

Put these in order, smallest first: 0.305, 0.35, 0.053.

Method. Write the numbers with the same number of decimal places, then compare them column by column from the left.

Write every number with the same number of decimal places. Adding zeros at the end does not change a decimal. It lines up the place values.
0.305 = 0.305
0.35 = 0.350
0.053 = 0.053
Put them in order, smallest to largest. Start with the whole numbers. If they are the same, compare the tenths, then the hundredths, then the thousandths.
0.053 < 0.305 < 0.350
Write the answer with the original numbers.
0.053, 0.305, 0.35

Check. Read the list back. Each number is < the next one. ✓

Watch out. Do not sort by the number of digits. 0.45 and 0.405 look like 45 and 405, but 0.45 = 0.450 is the bigger number.

Worked example 7.2.3

Which is greater, 4.072 or 4.7?

Method. Line up the place values, then compare from the left.

Write both numbers with the same number of decimal places. Zeros at the end do not change a number. They make the columns line up.
4.072 = 4.072
4.7 = 4.700
Compare the columns from the left. The first column where the digits are different decides the answer.
ones: 4 and 4, the same.
tenths: 0 and 7, so 4.7 is bigger.
Write the answer. The symbol points to the smaller number.
4.072 < 4.7

Check. Read it the other way round: 4.072 < 4.7. ✓

Watch out. Do not compare the digits after the point as if they were whole numbers. 0.6 is 0.60, which is bigger than 0.58.

Watch out

Do not say 0.45 is bigger than 0.5 because 45 is bigger than 5. Write 0.5 as 0.50: it is 50 hundredths, which is more than 45 hundredths. Also do not say 0.405 is bigger than 0.45 because it has more digits. 0.45 = 0.450, which is bigger.

Exercise 7B Comparing and ordering decimals

Non-calculator

FUse <, > or = to complete each statement.

(a)0.4 ? 0.6

(b)0.73 ? 0.37

(d)0.25 ? 0.205

Done

FWhich is greater, 0.7 or 0.65?

Done

FPut these in order, smallest first: 0.45 0.4 0.405 0.054

Done

FPut these in order, largest first: 2.5 2.25 2.205 2.052

Done

PSThe digit in the box is in the hundredths place. 0.6? is greater than 0.64 and less than 0.67. Which digits could go in the box?

Method. Try each digit in the hundredths place. Keep the numbers that are greater than 0.64 and less than 0.67.

Try the digits near 0.64 and 0.67. Only the digits between 4 and 7 can work, so we check those.
Number Between 0.64 and 0.67?
0.64 no
0.65 yes
0.66 yes
0.67 no
Write the digits that work. Two digits give a number that is between the two.
The digits are 5 and 6.

Number	Between 0.64 and 0.67?
0.64	no
0.65	yes
0.66	yes
0.67	no

Check. 0.65 and 0.66 are both greater than 0.64 and less than 0.67. ✓

Watch out. Check the end numbers. 0.64 is not greater than 0.64, and 0.67 is not less than 0.67.

Done

FWhich is greater, 0.509 or 0.51?

Done

PSWhich decimal is exactly halfway between 0.3 and 0.4?

Done

FThe points A, B, C and D are on the number line. Which letter shows each decimal?

(a)0.35

(b)0.9

Done

PSAli ran in 12.45 seconds, Bea in 12.5 seconds and Chen in 12.405 seconds. Who was the fastest?

Done

PSFour bags of rice weigh 1.25 kg, 1.052 kg, 1.5 kg and 1.205 kg. Write the weights in order, heaviest first.

Done

RAlways, sometimes or never? A decimal with more digits after the point is bigger. Explain with an example.

Done

RUse each of the digits 2, 5 and 7 once to make a decimal with three decimal places. Make it as close as possible to 0.5. Show your working.

Method. Write every decimal you can make, then find the one with the smallest difference from 0.5.

List the possible decimals. Use each digit once in the tenths, hundredths and thousandths places.
Decimal Difference from 0.5 Closest?
0.527 0.027 yes
0.572 0.072 no
0.275 0.225 no
0.725 0.225 no
0.257 0.243 no
0.752 0.252 no
Find the smallest difference. The closest decimal to 0.5 is the one with the smallest difference.
The smallest difference is 0.027.
Write the answer.
0.527

Decimal	Difference from 0.5	Closest?
0.527	0.027	yes
0.572	0.072	no
0.275	0.225	no
0.725	0.225	no
0.257	0.243	no
0.752	0.252	no

Check. 0.527 is 0.027 away from 0.5. ✓

Watch out. Put the 5 in the tenths place, so the number starts 0.5. Then the other digits decide how close it is.

Done

7.3

Rounding decimals

Year 5
Round decimals with two decimal places to the nearest whole number and to one decimal place

Video lesson: Rounding decimalsComing soon

To round a number, find the two numbers it lies between, then decide which one it is closer to. Look at one digit: the digit just after the place you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.

The number line shows 3.67. It is between 3.6 and 3.7. The halfway point is 3.65, and 3.67 is above it, so 3.67 rounds up to 3.7 to one decimal place.

The point x is 3.67. The halfway number between 3.6 and 3.7 is 3.65.

Key ideas

To the nearest whole number, look at the tenths digit: 5 or more rounds up (4.5 rounds up to 5).
To one decimal place, look at the hundredths digit: 5 or more rounds the tenths up.
Rounding 2 decimal places to 1 decimal place: 6.25 rounds to 6.3 and 6.24 rounds to 6.2.
Money is rounded in the same way: £7.65 is about £8 to the nearest pound.

Worked example 7.3.1

Round 4.36 to the nearest whole number.

Key words. To round a number is to change it to the nearest value in a given place, such as the nearest whole number or one decimal place. The halfway number is exactly between two numbers. A digit of 5 or more rounds up; a digit of 4 or less rounds down.

Method. Find the two numbers it lies between, find the halfway number, then see which one it is nearer to.

Find the two numbers it is between. We round to the nearest whole number. These are the two numbers it could round to.
4.36 is between 4 and 5
Find the halfway number. The halfway number is exactly between the two. A number at or above halfway rounds up.
Halfway = 4.5
Look at the digit after the place you are rounding to. The digit is 3. 4 or less: round down.
4.36 is below halfway, so round down.
Write the answer. Keep the place you rounded to, and drop the digits after it.
4

Check. The difference is 0.36, which is at most half a step (0.5). ✓

Watch out. Look at ONE digit only. Do not round twice: 3.449 is 3.4 to one decimal place, not 3.5.

Worked example 7.3.2

Round 7.85 to one decimal place.

Method. Find the two numbers it lies between, find the halfway number, then see which one it is nearer to.

Find the two numbers it is between. We round to one decimal place. These are the two numbers it could round to.
7.85 is between 7.8 and 7.9
Find the halfway number. The halfway number is exactly between the two. A number at or above halfway rounds up.
Halfway = 7.85
Look at the digit after the place you are rounding to. The digit is 5. 5 or more: round up.
7.85 is exactly on halfway, so round up.
Write the answer. Keep the place you rounded to, and drop the digits after it.
7.9

Check. The difference is 0.05, which is at most half a step (0.05). ✓

Watch out. Look at ONE digit only. Do not round twice: 3.449 is 3.4 to one decimal place, not 3.5.

Worked example 7.3.3

A pen costs £12.48. Round the price to the nearest pound.

Method. Find the two numbers it lies between, find the halfway number, then see which one it is nearer to.

Find the two numbers it is between. We round to the nearest whole number. These are the two numbers it could round to.
12.48 is between 12 and 13
Find the halfway number. The halfway number is exactly between the two. A number at or above halfway rounds up.
Halfway = 12.5
Look at the digit after the place you are rounding to. The digit is 4. 4 or less: round down.
12.48 is below halfway, so round down.
Write the answer. Keep the place you rounded to, and drop the digits after it.
£12

Check. The difference is 0.48, which is at most half a step (0.5). ✓

Watch out. Look at ONE digit only. Do not round twice: 3.449 is 3.4 to one decimal place, not 3.5.

Watch out

Round only once. The number 3.449 rounds to 3.4 to one decimal place: the hundredths digit is 4, so round down. Do not round to 3.45 first and then to 3.5.

Exercise 7C Rounding decimals

Non-calculator

FRound each number to the nearest whole number.

(a)3.27 →

(b)5.86 →

(d)9.49 →

Done

FRound each number to one decimal place.

(a)4.82 →

(b)0.37 →

(d)2.96 →

Done

FRound each price to the nearest pound.

(a)£7.65 →

(b)£12.49 →

(c)£0.95 →

(d)£3.50 →

Done

FQuick-fire! Round each number to the nearest whole number.

(a)8.09 →

(b)15.5 →

(d)20.44 →

Done

RTick the two numbers that round to 5 when rounded to the nearest whole number. 4.6 5.49 5.5 4.49

Method. Round each number to the nearest whole number, then check which ones give 5.

Round each number. Look at the tenths digit. 5 or more rounds up.
Number Rounded Gives 5?
4.6 5 yes
5.49 5 yes
5.5 6 no
4.49 4 no
Write the numbers that round to 5. Only these two give 5.
4.6 and 5.49

Number	Rounded	Gives 5?
4.6	5	yes
5.49	5	yes
5.5	6	no
4.49	4	no

Check. 4.6 rounds to 5, and 5.49 rounds to 5. 5.5 rounds up to 6, so it is not included. ✓

Watch out. 5.5 rounds UP to 6, but 4.49 rounds DOWN to 4.

Done

FThe point x is at 3.67 on the number line. Round 3.67 to one decimal place.

Done

PSJug A holds 2.75 litres. Round the amount to the nearest whole litre, and then to one decimal place.

(a)Nearest whole litre

(b)One decimal place

Done

PSJake is 1.36 m tall. How tall is he to one decimal place?

Done

PSA walk is 6.84 km long. Round the length to the nearest kilometre.

Done

PSA pencil is 14.36 cm long. Round its length to the nearest centimetre.

Done

RElla rounds 3.449 to one decimal place and gets 3.5. What has she done wrong? Explain.

Done

RExplain why 7.49 and 7.5 round to different whole numbers.

Done

PSRound 2.75 to one decimal place. Explain what the digit 5 tells you.

Done

RFind the number with two decimal places that rounds to 7.3 (one decimal place) and has a 9 in the hundredths place. Is there only one?

Method. Try each number with a 9 in the hundredths place. Round each one to one decimal place and see which gives 7.3.

List the numbers with a 9 in the hundredths. The number is 7 and something, then the hundredths digit is 9.
7.09, 7.19, 7.29, 7.39, 7.49, 7.59, 7.69, 7.79, 7.89, 7.99
Round each one to one decimal place. Look at the hundredths digit. A 9 there always makes the tenths go up by one, so the tenths digit must be 2 for the answer to be 7.3.
Number To 1 dp Gives 7.3?
7.09 7.1 no
7.19 7.2 no
7.29 7.3 yes
7.39 7.4 no
7.49 7.5 no
7.59 7.6 no
7.69 7.7 no
7.79 7.8 no
7.89 7.9 no
7.99 8.0 no
Write the answer. Only one number is left.
7.29

Number	To 1 dp	Gives 7.3?
7.09	7.1	no
7.19	7.2	no
7.29	7.3	yes
7.39	7.4	no
7.49	7.5	no
7.59	7.6	no
7.69	7.7	no
7.79	7.8	no
7.89	7.9	no
7.99	8.0	no

Check. 7.29 rounds to 7.3. The numbers 7.19 and 7.39 do not. ✓

Watch out. A 9 in the hundredths place always rounds the tenths UP. Work out which tenths it starts from.

Done

7.4

Adding, subtracting and solving problems

Year 5
Solve problems involving numbers up to three decimal places, Add and subtract decimals with up to three decimal places

Video lesson: Adding, subtracting and solving problemsComing soon

Adding and subtracting decimals uses the same column method as whole numbers, with one rule: line up the decimal points. Write zeros at the end of the shorter number so that every column has a digit. Then work from the right.

Two bags of flour. We need the total weight, so we add the two parts.

Key ideas

Line up the decimal points, one under the other.
Fill empty places with zeros so that each number has the same number of decimal places.
Work column by column from the right, and exchange (carry) when a column is more than 9 or less than 0.
Put the decimal point in the answer straight down, under the others. Check with the opposite operation.

Worked example 7.4.1

Work out 3.45 + 1.728.

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.
```
  3.45 
+ 1.728
-------
```
Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.
```
  3.450
+ 1.728
-------
```
Add column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.
```
  3.450
+ 1.728
-------
  5.178
```
Write the decimal point in the answer. Bring the point straight down, under the others.
5.178

Check. Check with the opposite operation: 5.178 − 1.728 = 3.45. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Worked example 7.4.2

Work out 4.6 − 2.375.

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.
```
  4.6  
− 2.375
-------
```
Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.
```
  4.600
− 2.375
-------
```
Subtract column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.
```
  4.600
− 2.375
-------
  2.225
```
Write the decimal point in the answer. Bring the point straight down, under the others.
2.225

Check. Check with the opposite operation: 2.225 + 2.375 = 4.6. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Worked example 7.4.3

A jug holds 1.5 litres. Ben pours out 0.275 litres. How much is left in the jug?

Key words. The column method lines up the decimal points, so that each place value is in the same column. A zero written at the end of a decimal does not change its value.

Method. Line up the decimal points, fill the empty places with zeros, then work column by column from the right.

Choose the operation. Some water was poured out, so we subtract the amount that was poured out.
1.5 litres − 0.275 litres
Line up the decimal points. Write one number under the other so that the digits with the same place value are in the same column.
```
  1.5  
− 0.275
-------
```
Fill the empty places with zeros. Zeros on the right do not change the value. They make both numbers have the same number of decimal places.
```
  1.500
− 0.275
-------
```
Subtract column by column, from the right. Start with the smallest place value. Exchange or carry when you need to.
```
  1.500
− 0.275
-------
  1.225
```
Write the decimal point in the answer. Bring the point straight down, under the others.
1.225 litres

Check. Check with the opposite operation: 1.225 + 0.275 = 1.5. ✓

Watch out. Line up the decimal POINTS, not the last digits. Add zeros so that every number has the same number of decimal places, then each column has one place value.

Watch out

The most common mistake is to line up the last digits instead of the decimal points. When you work out 4.6 − 2.375, write 4.6 as 4.600 first, so that the 6 sits in the tenths column above the 3 of 2.375.

7.5

Per cent

Year 5
Recognise the per cent symbol (%) and understand that per cent relates to parts per hundred, Write percentages as a fraction with denominator 100 and as a decimal

Video lesson: Per centComing soon

The symbol % means per cent, which is “out of one hundred”. The square below has 100 small squares, and 35 of them are shaded. So 35% of the square is shaded.

Each small square is 1%. Count the shaded squares: 35.

A percentage can be written three ways: as a fraction with denominator 100, as a decimal, and with the % sign. For example, 35% = 35/100 = 0.35. In its simplest form, 35/100 is 7/20.

Key ideas

Per cent means out of 100. 35% = 35/100.
A percentage as a decimal: divide by 100, which moves the digits two places to the right. 7% = 0.07.
A decimal as a percentage: multiply by 100, which moves the digits two places to the left. 0.8 = 80%.
Simplify the fraction if you can: 35/100 = 7/20.

Worked example 7.5.1

Write 35% as a fraction in its simplest form.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Per cent means out of one hundred. Write the percentage over 100, then change it to the form you need.

Per cent means out of 100. 35% means 35 out of 100, so write it over 100.
³⁵⁄₁₀₀
Simplify the fraction. The HCF of 35 and 100 is 5. Divide the top and the bottom by 5.
³⁵⁄₁₀₀ = ⁷⁄₂₀
Write the answer. This fraction is in its simplest form.
⁷⁄₂₀

Check. ⁷⁄₂₀ is the same as 35 out of 100. ✓

Watch out. Per cent means per hundred, so 7% is 7 hundredths: 0.07, not 0.7.

Worked example 7.5.2

Write 0.8 as a percentage.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Change the decimal to hundredths. The number of hundredths is the percentage.

Write the decimal as hundredths. Hundredths are the same as per cent: 1 hundredth = 1%.
0.8 = ⁸⁰⁄₁₀₀
Read the number of hundredths as the percentage. Multiplying by 100 moves the digits two places to the left.
0.8 × 100 = 80
Write the answer.
80%

Check. 80% = ⁸⁰⁄₁₀₀ = 0.8. ✓

Watch out. Multiplying by 100 moves the digits two places to the LEFT. Dividing by 100 moves them two places to the right.

Worked example 7.5.3

Write 7% as a decimal.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Per cent means out of one hundred. Write the percentage over 100, then change it to the form you need.

Per cent means out of 100. 7% means 7 out of 100, so write it over 100.
⁷⁄₁₀₀
Divide by 100. Dividing by 100 moves every digit two places to the right, so the answer is a decimal.
7 ÷ 100 = 0.07
Write the answer.
0.07

Check. 0.07 × 100 = 7, so 0.07 is 7%. ✓

Watch out. Per cent means per hundred, so 7% is 7 hundredths: 0.07, not 0.7.

Watch out

7% is 0.07, not 0.7. Per cent means hundredths, so the 7 is in the hundredths column. 0.7 is 70%.

Exercise 7E Per cent as fractions and decimals

Non-calculator

FWrite each percentage as a fraction in its simplest form.

(a)27% =

(b)9% =

(d)100% =

Done

FWrite each percentage as a decimal.

(a)45% =

(b)6% =

(d)3% =

Done

FWrite each decimal as a percentage.

(a)0.72 =

(b)0.9 =

(d)0.4 =

Done

FWhat percentage of each square is shaded?

(a)Shaded:

(b)Shaded:

(c)Shaded:

Done

RWhich is greater, 60% or 0.55? Explain how you know.

Done

FFill in the gaps.

(a)14% = ?/100

(b)0.61 = ?%

Done

PSIn a survey of 100 pupils, 62 chose pizza. What percentage of pupils chose pizza?

Done

PSAli scored 84 out of 100 in a spelling test. What is his score as a percentage?

Done

PSA bag has 100 sweets. 9 of them are red. What percentage of the sweets are not red?

Done

PSKofi has spent 0.2 of his money. What percentage has he spent?

Done

RElla says, “7% is the same as 0.7.” Is she right? Explain.

Done

RWhich is the greatest: 0.5, 45% or 4/10?

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Change all three to percentages, then compare them.

Change each one to a percentage. Use the method that fits each number.
Number As a percentage
0.5 50%
45% 45%
4/10 40%
Compare the percentages. The biggest percentage is the greatest number.
50% > 45% > 40%
Write the answer.
0.5 is the greatest.

Number	As a percentage
0.5	50%
45%	45%
4/10	40%

Check. 0.5 = 50%, 45% and 4/10 = 40%. ✓

Watch out. Change all the numbers to the same form before you compare them.

Done

RWrite 1% as a fraction and as a decimal. What do you notice?

Done

7.6

Fractions, decimals and percentages

Year 5
Solve problems which require knowing percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 and fractions with a denominator of a multiple of 10 or 25

Video lesson: Fractions, decimals and percentagesComing soon

Some fractions are easy to write as percentages. The bar below is split into 5 equal parts and 2 are shaded, so it shows 2/5. To change a fraction to a percentage, make the bottom number 100. The top number is then the percentage.

Five equal parts with two shaded: 2/5 of the bar.

Multiply the top and the bottom by the same number to make 100. Since 5 × 20 = 100, we multiply 2/5 by 20/20: 2/5 = 40/100 = 40%. The square shows the same amount: 40 of the 100 squares are shaded.

Forty out of 100 squares are shaded, which is the same as two fifths.

Key ideas

Make the bottom number 100 and the top number is the percentage: 2/5 = 40/100 = 40%.
Useful equivalents: 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 2/5 = 40%, 4/5 = 80%.
If the bottom is 10, 20, 25 or 50, you can multiply it to make 100: 3/10 = 30/100 = 30%, 9/25 = 36/100 = 36%.
Decimals and percentages: 0.25 = 25/100 = 25%.

Worked example 7.6.1

Write 2/5 as a percentage.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Scale the fraction so that the bottom number is 100. The top number is then the percentage.

Make the bottom number 100. Per cent means out of 100. 5 × 20 = 100, so multiply the top and the bottom by 20.
²⁄₅ = ⁴⁰⁄₁₀₀
Read the top number as the percentage. The top number, out of 100, is the percentage.
⁴⁰⁄₁₀₀ = 40%
Write the answer.
40%

Check. 40 ÷ 100 = 0.4, and 2 ÷ 5 = 0.4. Both are the same decimal. ✓

Watch out. Multiply the top AND the bottom by the same number. Do not add to them.

Worked example 7.6.2

Write 3/4 as a percentage.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Scale the fraction so that the bottom number is 100. The top number is then the percentage.

Make the bottom number 100. Per cent means out of 100. 4 × 25 = 100, so multiply the top and the bottom by 25.
³⁄₄ = ⁷⁵⁄₁₀₀
Read the top number as the percentage. The top number, out of 100, is the percentage.
⁷⁵⁄₁₀₀ = 75%
Write the answer.
75%

Check. 75 ÷ 100 = 0.75, and 3 ÷ 4 = 0.75. Both are the same decimal. ✓

Watch out. Multiply the top AND the bottom by the same number. Do not add to them.

Worked example 7.6.3

Write 9/25 as a percentage.

Key words. Per cent means “out of one hundred”. The symbol % shows per cent: 35% means 35 out of 100, which is 35 hundredths.

Method. Scale the fraction so that the bottom number is 100. The top number is then the percentage.

Make the bottom number 100. Per cent means out of 100. 25 × 4 = 100, so multiply the top and the bottom by 4.
⁹⁄₂₅ = ³⁶⁄₁₀₀
Read the top number as the percentage. The top number, out of 100, is the percentage.
³⁶⁄₁₀₀ = 36%
Write the answer.
36%

Check. 36 ÷ 100 = 0.36, and 9 ÷ 25 = 0.36. Both are the same decimal. ✓

Watch out. Multiply the top AND the bottom by the same number. Do not add to them.

Watch out

1/5 is NOT 15%. Make the bottom 100 and you get 20/100 = 20%. Also, 3/4 is not 34%: 3/4 = 75/100.

Exercise 7F Fractions, decimals and percentages

Non-calculator

FWrite each fraction as a percentage.

(a)¹⁄₂ =

(b)¹⁄₄ =

(c)³⁄₄ =

(d)¹⁄₅ =

Done

FWrite each fraction as a percentage.

(a)³⁄₅ =

(b)⁴⁄₅ =

(c)¹⁄₁₀ =

(d)⁹⁄₁₀ =

Done

FWrite each fraction as a decimal.

(a)¹⁄₄ =

(b)¹⁄₅ =

(c)⁴⁄₅ =

(d)³⁄₅ =

Done

FWrite each as a percentage. The bottom number is a multiple of 10 or 25.

(a)³⁄₁₀ =

(b)⁷⁄₂₀ =

(c)¹⁄₂₀ =

(d)³⁄₅₀ =

Done

FWrite each fraction as a percentage. Multiply the top and the bottom to make 100.

(a)⁹⁄₂₅ =

(b)¹⁷⁄₂₅ =

(c)¹⁄₂₅ =

(d)³⁄₂₅ =

Done

FWhich is more: 3/4 or 70%?

Done

FPut these in order, smallest first: 0.6 2/5 35%

Key words. Equivalent means the same value written in a different way. A fraction, a decimal and a percentage can all be equivalent to each other.

Method. Change each one to a percentage, then put them in order.

Change each one to a percentage. Use the method that fits each number.
Number As a percentage
0.6 60%
²⁄₅ 40%
35% 35%
Put them in order, smallest first. The smallest percentage is the smallest amount.
35% < 40% < 60%
Write the answer.
35%, 2/5, 0.6

Number	As a percentage
0.6	60%
²⁄₅	40%
35%	35%

Check. 35% = 0.35, 2/5 = 0.4 and 0.6. These are in order. ✓

Watch out. Changing everything to percentages makes the comparison easy.

Done

PSIn a class of 20 pupils, 7 wear glasses. What percentage of the class wear glasses?

Done

PSRavi got 18 out of 25 in a spelling test. What percentage did he get?

Done

PSA jar has 25 sweets. 4 of them are sour. What percentage of the sweets are sour?

Done

PSA quarter of the pupils walk to school. A fifth cycle. What percentage of pupils neither walk nor cycle?

(a)Walk: what percentage of the pupils?

(b)Cycle: what percentage of the pupils?

(c)Neither walk nor cycle

Done

RZara says, “1/5 is the same as 15%.” Is she right? Explain.

Done

RWrite a fraction with denominator 25 that is greater than 30% and less than 35%. Is there only one?

Key words. A fraction has a numerator (the top number) and a denominator (the bottom number). Multiplying the top and the bottom by the same number makes an equivalent fraction.

Method. Write each fraction with denominator 25 as a percentage. Keep the one between 30% and 35%.

Write the fractions with bottom number 25. Each one-25th is 4%, because 25 × 4 = 100.
Each one-25th is 4%
Check the fractions near 30% to 35%. Look at the percentages for the fractions near the range.
Fraction Percentage Between 30% and 35%?
⁶⁄₂₅ 24% no
⁷⁄₂₅ 28% no
⁸⁄₂₅ 32% yes
⁹⁄₂₅ 36% no
¹⁰⁄₂₅ 40% no
Write the answer. Only one fraction is between 30% and 35%.
8/25 = 32%

Fraction	Percentage	Between 30% and 35%?
⁶⁄₂₅	24%	no
⁷⁄₂₅	28%	no
⁸⁄₂₅	32%	yes
⁹⁄₂₅	36%	no
¹⁰⁄₂₅	40%	no

Check. 8/25 = 32%, and 30% < 32% < 35%. ✓

Watch out. Write each fraction as a percentage, not as a decimal. One-25th is always 4%.

Done

Chapter summary

Skill	Method
Decimal as a fraction	Digits over 10, 100 or 1,000. The number of decimal places tells you the bottom number. Simplify: 0.36 = 36/100 = 9/25.
Place value	Each place to the right is ten times smaller: tenths, hundredths, thousandths. Use 0 as a place holder: 0.07 is 7 hundredths.
Fraction as a decimal	Make the bottom 10, 100 or 1,000 (scale by a factor). 3/4 = 75/100 = 0.75; 9/20 = 45/100 = 0.45.
Compare and order	Write the numbers with the same number of decimal places, then compare from the left: 0.6 = 0.60 is greater than 0.58.
Round	Look at one digit: the digit after the place you round to. 5 or more: up. 4 or less: down. 3.449 rounds to 3.4 (one decimal place).
Add and subtract	Line up the decimal points, fill empty places with zeros, then calculate column by column from the right.
Per cent	Per cent means out of 100. 35% = 35/100 = 0.35. Multiply a decimal by 100 to get a percentage: 0.8 = 80%.
Fractions, decimals and percentages	Make the bottom number 100 and the top number is the percentage: 2/5 = 40/100 = 40%. Useful: 1/4 = 25%, 1/5 = 20%, 3/4 = 75%.

Can you…?

I can write a decimal with up to three places as a fraction, and simplify it.
I can say what each digit is worth in tenths, hundredths and thousandths.
I can write a fraction with a bottom number of 10, 100 or 1,000 as a decimal.
I can compare and order decimals with up to three decimal places.
I can round a decimal with two decimal places to the nearest whole number and to one decimal place.
I can add and subtract decimals by lining up the decimal points.
I can change between a percentage, a fraction with denominator 100 and a decimal.
I can write fractions such as 1/4, 1/5, 2/5 and 4/5 as percentages and decimals.

Key terms

Decimal point: The point that separates the whole number part from the tenths, hundredths and thousandths.
Place value: The value of a digit, which depends on its column, such as tenths or hundredths.
Tenth, hundredth, thousandth: One part of 10, 100 or 1,000 equal parts. 0.1 is one tenth, 0.01 is one hundredth.
Place holder: A zero that keeps the other digits in the right columns, such as the 0 in 0.07.
Round: To change a number to the nearest value in a given place.
Halfway number: The number exactly between two numbers. 4.5 is halfway between 4 and 5.
Per cent (%): Out of one hundred. 35% means 35 out of 100.
Equivalent: Having the same value, but written in a different way, such as 1/2, 0.5 and 50%.
Highest common factor (HCF): The biggest number that divides exactly into two or more numbers. Used to simplify fractions.
Column method: A written method for adding or subtracting, with the place values in columns.

Total 21 marks. Allow about 21 minutes.

Exercise 7X SATs-style questions

Non-calculator

FWhat is the value of the 7 in 3.478?[1 mark]

Done

FWrite 0.085 as a fraction in its simplest form.[1 mark]

Done

FUse > or < to complete the statement: 0.45 ? 0.405[1 mark]

Done

FRound 6.85 to the nearest whole number. Then round it to one decimal place.[2 marks]

(a)Nearest whole number

(b)One decimal place

Done

FWork out 4.6 + 7.085. Show your working.[2 marks]

Done

FWork out 9 − 3.275. Show your working.[2 marks]

Done

FWrite 35% as a fraction in its simplest form and as a decimal.[2 marks]

(a)As a fraction

(b)As a decimal

Done

PSA bottle holds 1.5 litres. Sam pours out 0.375 litres, then 0.2 litres more. How many litres are left? Show your working.[3 marks]

Done

RTick the numbers that are equal to 0.4: 2/5 4/10 40% 4/100 0.04 Show your working.[3 marks]

Key words. Equivalent means the same value written in a different way. A fraction, a decimal and a percentage can all be equivalent to each other.

Method. Change each one to a decimal, then tick the ones that equal 0.4.

Change each one to a decimal. Make the bottom number 10, 100 or 1,000, or use the percentage.
Number Decimal Equal to 0.4?
²⁄₅ 2/5 = 0.4 yes
⁴⁄₁₀ 4/10 = 0.4 yes
40% 40% = 0.4 yes
⁴⁄₁₀₀ 4/100 = 0.04 no
0.04 0.04 no
Tick the ones that are equal to 0.4. Three of them have the value 0.4.
2/5, 4/10 and 40%
Write the answer. Check that you have not ticked the ones that are not 0.4.
2/5, 4/10 and 40%

Number	Decimal	Equal to 0.4?
²⁄₅	2/5 = 0.4	yes
⁴⁄₁₀	4/10 = 0.4	yes
40%	40% = 0.4	yes
⁴⁄₁₀₀	4/100 = 0.04	no
0.04	0.04	no

Check. 2/5, 4/10 and 40% are all 0.4. ✓

Watch out. 4/100 is 0.04, not 0.4. Check the number of decimal places.

Done

RPut these in order, smallest first. Show your working. 0.4 45% 0.405 3/10[4 marks]

Method. Change every number to a percentage, then put them in order from the smallest.

Change each number to a percentage. Multiply a decimal by 100, or make the fraction out of 100.
Number As a percentage
0.4 40%
45% 45%
0.405 40.5%
³⁄₁₀ 30%
Put them in order. Compare the percentages from the smallest.
30% < 40% < 40.5% < 45%
Write the answer in the original form. The numbers are the same, so write them as they were given.
³⁄₁₀, 0.4, 0.405, 45%

Number	As a percentage
0.4	40%
45%	45%
0.405	40.5%
³⁄₁₀	30%

Check. 0.405 = 40.5%, which is between 40% and 45%. ✓

Watch out. Compare with the same unit. The number 0.405 is 40.5%, not 4.05%.

Done

← Chapter 6: Fractions Chapter 8: Units, perimeter, area, volume and time →

Decimals and percentages

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Tenths, hundredths and thousandths

Key ideas

Watch out

Exercise 7A Decimals as fractions

Comparing and ordering decimals

Key ideas

Watch out

Exercise 7B Comparing and ordering decimals

Rounding decimals

Key ideas

Watch out

Exercise 7C Rounding decimals

Adding, subtracting and solving problems

Key ideas

Watch out

Exercise 7D Adding and subtracting decimals

Per cent

Key ideas

Watch out

Exercise 7E Per cent as fractions and decimals

Fractions, decimals and percentages

Key ideas

Watch out

Exercise 7F Fractions, decimals and percentages

Chapter summary

Can you…?

Key terms

Exercise 7Z Check-up (non-calculator)

Exercise 7X SATs-style questions