# KS3 Bitesize

Maths

Decimals

Decimals play an important part in everyday life - particularly through the use of money - so it's crucial that you know how to use them.

## Introduction

This Revision Bite covers:

## Place value and ordering decimals

### Decimal place values

We use a decimal point to separate units from parts of a whole (like tenths, hundredths, thousandths, etc).

• 0.1 is a tenth, 1/10 of a unit
• 0.01 is a hundredth, 1/100 of a unit
• 0.001 is a thousandth. 1/1000 of a unit

In 52.13, the value of the figure 1 is 1/10 , and the value of the figure 3 is 3/100.

### Ordering decimals

When ordering numbers, always compare the left digits first.

Eg Which is greater, 2.301 or 2.32?

Units   Tenths Hundredths Thousandths
2 . 3 0 1
2 . 3 2

Both numbers have two units and three tenths, but 2.301 has no hundredths, whereas 2.32 has two hundredths. Therefore, 2.32 is greater than 2.301.

Another way to look at it is to add a zero to the end of 2.32 (this doesn't change its value as it's after the decimal point).

The two numbers are now 2.320 and 2.301 and it is quite easy to see that 2.320 is bigger (just as 2 320 is bigger than 2 301).

Question

Q1. In the number 3.546, what is the value of the figure 4?

Q2. Place the following numbers in order, smallest first: 3.2, 3.197, 3.02, 3.19

A1. The value of the figure 4 is 4/100.

A2. Did you get 3.02, 3.19, 3.197, 3.2?

All the numbers have three units, so start by comparing the tenths. 3.02 has no tenths, 3.197 and 3.19 have one tenth, and 3.2 has two tenths. Therefore, 3.02 is the smallest and 3.2 is the largest.

You could also write the numbers as 3.020, 3.190, 3.197 and 3.200 and compare them.

When adding and subtracting decimals, remember is to keep the decimal points in line in the question and the answer.

Question

David is doing some DIY. He buys a 2m length of wood. He needs to cut two pieces of wood - one of length 0.6m and one of length 1.02m.

What is the total length of wood that David needs to cut?

So the total length of wood that David needs to cut is 1.62m.

You can add zeros to the end of a decimal without affecting its value, so 0.6 is the same as 0.60.

### Subtracting decimals

Question

David originally had 2m of wood. What is the length of the piece of wood that is left?

David cut off 1.62m, so we need to calculate 2 - 1.62

So there is 0.38m of wood left.

## Multiplying decimals by 10, 100 and 1000

### Multiplying by 10

When a decimal is multiplied by 10, every figure moves one place to the left.

#### Multiplying a decimal by 10

What is 4.25 x 10?

#### Multiplying a decimal by 10

Each figure moves 1 place to the left, so 4.25 x 10 = 42.5

### Multiplying by 100

When multiplying by 100, every figure moves two places to the left.

#### Multiplying a decimal by 100

What is 0.103 x 100?

#### Multiplying a decimal by 100

Each figure moves 2 places to the left. The zero at the front is no longer needed so we ignore it. So 0.103 x 100 = 10.3

### Multiplying by 1000

When multiplying by 1000, every figure moves three places to the left.

#### Multiplying a decimal by 1000

What is 0.04 x 1000?

#### Multiplying a decimal by 1000

Each figure moves 3 places to the left. The two zero's in the thousands and hundred are no longer needed so we ignore them. We don't have any units, so we place a '0' in the units column. So we get, 0.04 x 1000 = 40

Question

Which is bigger: 0.005 × 10 or 0.0004 × 1000?

The answer is 0.0004 × 1000
If you didn't get it right, remember that multiplying by 10 moves each digit one place to the left.

Therefore, 0.005 × 10 = 0.05.
Multiplying by 1000 moves each digit three places to the left.
Therefore, 0.0004 × 1000 = 0.4.
0.4 is bigger than 0.05.

## Dividing decimals by 10, 100, 1000

### Dividing by 10

When you divide by 10, every figure moves one place to the right. Hundreds become tens, tens become units, units become tenths and tenths become hundredths.

#### Dividing a decimal by 10

What is 27 divided by 10?

#### Dividing a decimal by 10

Each figure moves 1 place to the right. So 27 divided by 10 = 2.7

### Dividing by 100

When you divide by 100, every figure moves two places to the right.

#### Dividing a decimal by 100

What is 27 divided by 100?

#### Dividing a decimal by 100

Each figure moves 2 places to the right. We don't have any units and tenths, so we place a '0' in the columns. So 2 divided by 100 = 0.02

### Dividing by 1000

When you divide by 1000, every figure moves three places to the right.

#### Dividing a decimal by 1000

What is 30 divided by 1000?

#### Dividing a decimal by 1000

Each figure moves 3 places to the right. We don't have any units and tenths, so we use '0'. The zero in the thousandths column is no longer needed so we ignore it. So 30 divided by 1000 = 0.03

## Multiplying a decimal by a whole number

Multiplying a decimal by a whole number is the same as multiplying two whole numbers. Remember:

• If there is one digit after the decimal point in the question, there will be one digit after the decimal point in the answer.
• If there are two digits after the decimal point in the question, there will be two digits after the decimal point in the answer.
Question

Calculate:

a) 2.43 × 7
b) 2.4 × 5

a) There were two digits after the decimal point in the question (4 and 3), so you must have two digits after the decimal point in the answer.

b) There was one digit after the decimal point in the question, so you must have one digit after the decimal point in the answer. The answer is therefore 12.0, but this can then be given as 12.

Check that you have a sensible answer by finding an approximate solution.

In the above example you were asked to calculate 2.4 × 5.

2 × 5 = 10, so you are looking for an answer which is slightly bigger than 10. So an answer of 12 seems sensible.

## Dividing a decimal by a whole number

Remember to keep the decimal points aligned in the question and the answer.

### Example

Work out 4.05 divided by 9

Solution:

### Example

Work out 2.4 divided by 5

Solution:

It is sometimes necessary to add a '0' or '0's to the end of a decimal, as in this example (2.40 is the same as 2.4 but the question stays the same)

## Multiplying and dividing by less than 1

### Multiplying by a number between 0 and 1

The multiplication sign can be replaced by 'lots of'.

For example,

2 × 3 means 2 lots of 3
6 × 8 means 6 lots of 8

So, 1/2 × 10 means 1/2 of 10
And 1/3 × 12 means 1/3 of 12

When you multiply by a number greater than 1, you get an answer that is greater than the original number. But when you multiply by a number between 0 and 1, the answer is smaller than the original number.

In general:

m × 1/n = m ÷ n

#### Example

8 × 1/4 = 8 ÷ 4 = 2

20 × 1/5 = 20 ÷ 5 = 4

### Dividing by a number between 0 and 1

Imagine that you had 10 bars of chocolate that you wanted to share amongst some children.

If you gave the children 2 bars each, you would have enough for 5 children.
10 ÷ 2 = 5

If you gave the children 1/2 bar each, you would have enough for 20 children.
10 ÷ 1/2 = 20

#### The pattern

Can you see what's happening?

10 ÷ 2 = 5

10 ÷ 1/2 = 20

When you divide by a whole number the answer is less than the original number. When you divide by 1/2 the answer (20) is greater than the original number (10).

It's the opposite of multiplying. When we divide by a number greater than 1, we get an answer that is less than the original number. But when we divide by a number between 0 and 1 the answer is larger than the original number.

So, 10 ÷ 1/2 = 20

Similarly, 10 ÷ 1/3 = 30 and 10 ÷ 1/4 = 40

In general:

m ÷ 1/n = mn

Question

Q1. What is 10 ÷ 1/7 ?

Q2. Find the value of: 4 ÷ 1/3