### What are Decimals?

Decimals tell us the exact value of a number.

Decimals look like:

1.5

7.89

5.501

They all have digits **after** their **decimal point**. These digits sit in the tenths column, the hundredths column, the thousandths column… and so on. Look back at the **place value** lesson if you need to remind yourself where these decimal places sit.

Here is a number line showing the tenths between 4 and 5:

4 itself can be written as a decimal, it is 4.0, it has 4 units and 0 in the tenths column. It could also be written 4.00000000, it’s all the same but we just write ‘4’ as the other values are all zero so it would be silly to write them endlessly!

**4.1** is **4 units** and **1 tenth**. It is has a slightly bigger value than 4, one tenth more to be exact. It is 0.1 bigger than 4.

**4.5** is **halfway between 4 and 5**, it has **4 units** and **5 tenths**. Each unit is split in to 10 tenths so 5 tenths is halfway.

**4.9** has **4 units** and **9 tenths** so we can see it is slightly less than 5.

We can see even more precise values using **hundredths**, look at this number line showing the hundredths between 4.5 and 4.6:

**4.55** is halfway between **4.5** and **4.6**, it is 5 hundredths bigger than 4.5.

**4.51** is **1 hundredth bigger** than **4.5**.

**4.58** is **2 hundredths smaller** than **4.6**.

### Adding and Subtracting Decimals

To **add and subtract decimals** we use the same methods as we do with whole numbers the only difference is the **decimal point** which must be lined up and stay in it’s place.

Don’t forget about the **decimal point** in your answer. It is a good idea to write it in to the answer box at the beginning of the sum.

Have a look at the examples below.

Example – Adding

**Question**: What is 4.5 + 2.8?

First write out the sum with the **units above the units** and the **tenths above the tenths**. Also put the **decimal point** in to the **answer box** in line with the other decimal points:

Now we can **add up** the numbers, starting with the column **furthest to the right**. In this case it is 5 + 8 = 13.

The **3** goes in the answer box and the **1** is **carried over** to the **units** column:

Now we can add up the numbers in the units column and remember to add on the **1** that has been **carried over**:

Example – Subtracting

**Question**: What is 3.1 – 2.2?

We write out the sum with the bigger number on top, and with the **units above the units **and the **tenths above the tenths**. We can also put the decimal place in the answer box so that it doesnt get forgotten about:

Now we can start subtracting the **tenths** in the right hand column. As the digit on the top row is smaller than the digit on the bottom row, we need to **borrow 1** from the **3** in the units column:

Next we **subtract** the **units column**, remember the 3 has become a 2 as we **borrowed** from it. This column is then 2 – 2 = 0, which we can write in the answer box:

### Ordering Decimals

Ordering decimals is like ordering whole numbers, but you must remember the **decimal point**.

Here is a **method** to follow when ordering decimals:

- Write the decimals down in a list one on top of the other with the
**decimal points in a line**. - Next, arrange the list by the
**size**of the**digits before the decimal point**as you would arrange a group of numbers – ignoring the decimal places for now. - For numbers who share the same digits before the decimal point they can be written in a
**group**to be arranged in the next step. - Next, for groups of numbers who shared the same digits before the decimal point move right one place to the tenths column and arrange the grouped numbers according to this next digit, again grouping numbers who share this digit.
- Repeat the process moving one place right and arranging the grouped numbers by size until there are no groups left and the numbers will then be in size order.

Example

Question: Arrange the following decimals by order of size.

1.66

1.7

8.9

4.32

Start by writing the decimals in a list with the decimal points in a line:

1.66

1.7

8.9

4.32

Now start at the left and arrange the decimals in order according to the number before the decimal point:

8.9

4.32

1.66

1.7

There are two numbers with the same first digit, so for those two we need to move to the next place to the right and order the decimals by the size of this digit:

8.9

4.32

1.7

1.66

As there were no digits the that were the same that time we now know the numbers are in order.

### Rounding Decimals

Rounding decimals is similar to rounding whole numbers, in fact the only difference is the decimal point and the words used to ask you to do it.

You might be asked to round to **one decimal place**, this means the first digit after the decimal point needs to be rounded, the digit in the tenths column.

If you are asked to round to **two decimals places** you are being asked to round off the digit in the second place after the decimal point, the hundredths column.

**3 decimals places** is the 3rd place after the decimal point, the hundredths column.

To round the number you use the same method as when rounding whole numbers. You look at the number one place to the right of the number you are being asked to round. If this number is 5 or more you round up. If it is less than 5 you round down.

For example:

**Question**: Round 2.56883 to two decimal places.

Two decimal places means we are looking at the 6 here 2.5__6__883

The digit to its right is **8** so we round up.

Our answer is: **2.57**, we have rounded the number to 2 decimal places so all the digits after the hundredths column are now gone.

### Money

Money is often displayed as a decimal.

For example, 1 pound and 25 pence can be written:

£1.25

This is a decimal, we have one unit (one pound), 2 tenths of a pound (20 pence) and 5 hundredths of a pound (5p).

To add money you can just use the same method as with adding any decimal.

Forget about the pounds and pence but remember the decimal point and remember to put the pound sign back in at the end if you need to.

### Common Questions

This question plays on your knowledge of decimals and multiplication. There is a quick way to answer it.

You have been told that 27 x 38 = 1026.

So 2.7 x 3.7 will give us the same answer except that there are two decimal places now being used: 2.7 x 3.7.

This means that the answer will have 2 decimal places:

10.26

With this type of question you simply count the difference in decimal places between the first and second sum and then give the same number of decimal places to the answer.