# Rounding – Revision for KS3 and GCSE

Welcome back (or hello to newcomers!) to another revision-based learning guide that will assist you through your study. Today, we look at rounding.

But, before we start our tuitional style blurb, we should probably ask the question: what is rounding?

Rounding is simplifying a number so that it is shorter and easier to use, but is still close to the original number.

Of course, there isn’t just one method for rounding. There are several situations in which we will need to round so there are also several techniques for reaching the correct answer.

In this text, we’ll talk you through a range of rounding techniques for the following:

In true revision style, we’ll cover the basics before detailing step-by-step procedures for each subtopic. The information will suit you if you’re working through KS3 Maths or studying/revising the foundations for your GCSE Maths course.

So, let’s get into it!

There are two key rounding rules which you should know:

• If the deciding digit is less than 5 (0, 1, 2, 3 or 4), we round down.
• If the deciding digit is 5 or more (5, 6, 7, 8 or 9), we round up.

### Rounding to Significant Figures

In maths, significant means ‘to have value’. For example, in the number 2795, the 2 is the most significant digit, because it tells us that the number is 2 thousand and something. However, in the number 0.052, the 5 is the most significant digit.

A ‘significant figure’ is any of the following:

• A number other than 0 is always significant.
• 0 is significant if it is between two significant numbers (e.g. 1012).
• 0 is significant if it is the rightmost number after a decimal point (e.g. 2.50).

## Rounding to the Nearest 10, 100 and 1000

Let’s look at this in practice.

#### Rounding to the Nearest 10

Round 376 to the nearest 10.

There are a couple of ways in which we can approach this question.

Method 1

We could use a number line to help us. For example, we can identify that the digit in the tens position in 376 is 7, which is worth 70.

As we are rounding to the nearest 10, we are thinking about the multiples of 10. The 10 value before 376 is 370, and the 10 value after 376 is 380. This makes 375 the midpoint, halfway between 370 and 380. Using a number line, we can see that 376 is after 375; in other words, it is closer to 380 than to 370.

Therefore, we can say that 376 rounded to the nearest 10 is 380.

However, we don’t always have to use a diagram…

Method 2

If we take the same example, 376, we begin by identifying the column where we are being asked to round. In this case, we are rounding to the nearest 10 so we find the digit in the 10s position, which is 7.

Now, look at the digit in the column to the right of the 7. This digit will decide what happens to the 7 and is known as the ‘decider’. It will determine whether the number will round up or round down, and therefore whether the digit in the rounding position will increase or stay the same.

3  7 | 6

In this case, the column immediately to the right of the tens column is the units column and contains the ‘decider’ digit, 6. If the decider digit is 5 or more, it tells us that we will round up. If it is 4 or less, it tells us that we will round down. Here, we are rounding up, so the 7 needs to be increased to 8; in other words, 376 is closer to 380 than to 370 (just like the diagram).

It is vital that the place value of the original digits remains the same. For whole numbers, you will need to replace any rounded digits with the correct number of zeros.

Therefore, 376 rounded to the nearest 10 is 380.

#### Rounding to the Nearest 100

Round 89 234 to the nearest 100.

Method 1

Begin by identifying the digit in the hundreds position: 2 (which is worth 200).

This time, we are rounding to the nearest 100, so we are thinking about the multiples of 100. The 100 value before 89 234 is 89 200, and the 100 value after 89 234 is 89 300. This makes 89 250 the midpoint.

Using a diagram, we can see that 89 234 is less than 250. In other words, 89 234 is closer to 89 200 than to 89 300, as we have not yet passed the midway point.

Therefore, 89 234 rounded to the nearest 100 is 89 200.

Method 2

Again, start by identifying the column where we are being asked to round. In this case, we are rounding to the nearest 100 so we find the digit in the 100s position, which is 2.

Now, look at the digit and column to the right of the 2. This is the decider digit and determines whether we will be rounding up or down.

8  9   2 | 3  4

We can see that the decider digit here is 3. If the deciding digit is less than 5, then we are rounding down. This means the digit in the hundreds column (2) remains the same, and the following two numbers are replaced by 0s. If you’re rounding down to the nearest hundred, you do not change the number in the hundreds column. Don’t forget to include the correct number of zeros after the hundreds column, so the place value of the original number remains the same.

89 234 rounded to the nearest 100 is 89 200.

#### Rounding to the Nearest 1000

Round 39 875 to the nearest 1000.

Rounding to the nearest 1000 follows similar steps to rounding to the nearest 10 or 100. Whichever method we choose, we start by identifying the digit in the thousands column: 9.

Method 1

39 875 appears after the midway point, therefore, it’s closer to 40 000.

39 875 rounded to the nearest 1000 is 40 000.

Method 2

Alternatively, find the decider digit in the number 39 875 and use this to determine the answer.

3  9 | 8  7  5

By applying the rounding rules, the decider digit (8) tells us that we are rounding up. This means the digit in the rounding position (9) needs to be increased. We need to be careful here. If we increase 9 by one we get 10, but we can’t just squeeze 10 into the gap. Instead, we replace the 9 with a 0, and add the 1 to the column to the left (3). As usual, the 3 digits to the right of the thousands column are replaced by 0s, giving us 40 000.

39 875 rounded to the nearest 1000 is 40 000.

You can round off your skills nicely by using the skills learnt with this sampler material and take them into the full resource pack below!

Rounding to the Nearest 10, 100 and 1,000

## Rounding to a Given Number of Decimal Places

#### Rounding to One Decimal Place

Write 135.72 correct to the nearest tenth.

1. Identify the digit in the rounding position. In this case, we are asked to write the number correct to the nearest tenth. Thinking about place values, the tenths column is the first column after the decimal point, so rounding to the nearest tenth is the same as rounding to one decimal place. In this question, the first digit after the decimal point is 7, therefore this is the digit in the rounding position.
2. To determine whether the number needs to be rounded up or down, we must look at the decider. This is the digit immediately to the right of the rounding position, in this case, a 2.
3. If the decider is 4 or less, we round the number down. If it is 5 or more, we round up. As the decider is 2 we are rounding down, the 7 in the rounding position remains the same.
4. Finally, any digits after the 7 are removed. Although the 7 hasn’t changed, removing these digits means we have rounded down, as our new number is smaller than our original number.

137.72 correct to the nearest tenth is 137.7. We do not need any zeros after the 7 as that would change the number of decimal places.

#### Rounding to Two Decimal Places

Write 1024.596 correct to 2 decimal places.

In this case we are told how many decimal places to round to, and we don’t have to think about place values. To round to 2 decimal places, we follow similar steps to rounding to 1 decimal place.

1. Identify the digit in the rounding position. We are asked to write the number correct to 2 decimal places, so we find the second digit after the decimal point: 9
2. Find the decider immediately to the right of the rounding position: 6
3. As the decider is 5 or more, we need to round up. This means the number in the rounding position increases. However, we can’t turn 9 into 10, it won’t fit. Instead, we replace the 9 with 0 and add the 1 to the number to the left. In this example, that turns the 5 into a 6, leaving us with 1024.60… (in some cases, that number to the left will also be a 9. In that case, you replace that 9 with a 0, and add 1 to the next digit to the left).
4. As before, we remove all numbers after the rounding position, giving us 1024.60.

1024.596 correct to 2 decimal places is 1024.60

#### Rounding to Three Decimal Places

Write 0.0584 correct to 3 decimal places.

Once we know how to round to 1 or 2 decimal places, we can easily round to 3 decimal places.

1. Identify the digit in the rounding position: 8
2. Locate the decider digit: 4
3. Apply the rounding rules: 4 tells us that the digit in the rounding position remains the same.
4. Remove the numbers after the rounding position, as this means the number is rounded down.

0.0584 correct to 3 decimal places is 0.058

Why not take this learning and further improve your level of accuracy using the teaching materials below?

Rounding to a Given Number of Decimal Places

## Rounding to Significant Figures

#### Rounding to One Significant Figure

Example 1: Round 37 to 1 significant figure.

We must firstly identify the column where we are being asked to round. In this case, we are being asked to round the number to one significant figure. When we round to significant figures, we start counting as soon as we reach a number that is not zero. In this example, it is the digit 3.

Now, look at the digit in the column to the right of the 3. This digit will decide what happens to the 3 and is known as the ‘decider’.

It will determine whether we will be rounding up or down, and therefore whether the digit in the rounding position will increase or stay the same. If the decider digit is 5 or more, it tells us that we will round up. If it is 4 or less, it tells us that we will round down.

By applying this rule, we see that we are rounding up, so the 3 needs to be increased to 4. This is because 37 is closer to 40 than it is to 30.

It is vital that the place value of the original digits remains the same. For whole numbers, you will need to replace any rounded digits with the correct number of zeros.

Therefore, 37 rounded to 1 significant figure is 40. Notice how the number now only has one ‘significant’ figure (4) whilst retaining its original place value (the 0 has no value, so is not counted as a significant figure).

#### Rounding to Two Significant Figures

Example 2: Round 91 378 to two significant figures.

Again, start by identifying the column where we are being asked to round. In this case, we are rounding to two significant figures. Remember we start counting as soon as we reach a number that is not zero. The first significant figure is 9. The second significant figure is 1.

Now, look at the digit and column to the right of the 1. This is the decider digit and determines whether we will be rounding up or down

We can see that the decider digit here is 3. If the deciding digit is less than 5, then we are rounding down. This means the digit in the rounding position, 1, remains the same and the following three digits are replaced by 0s. Although we haven’t changed the value of the 1, we have rounded down, as our number is smaller than our original number, but keeps its original place value.

91 378 rounded to 2 significant figures is 91 000.

#### Rounding to Three Significant Figures

Example 3: Round 0.05697 to three significant figures.

Rounding to three significant figures follows similar steps to rounding to one or two significant figures.

Begin by identifying the digit in the rounding position. This time we have a decimal number which contains 0s at the beginning. Remember, we start counting as soon as we reach a number that is not zero.

By applying the rounding rules, the decider digit (7) tells us that we are rounding up. This means the digit in the rounding position (9) needs to be increased. We need to be careful here. If we increase 9 by one, we get 10 but we can’t just squeeze 10 into the gap. Instead, we replace the 9 with a 0, and add the 1 to the column to the left (6).

0.05697 rounded to three significant figures is 0.0570. You might think that we don’t need the final zero; however, a trailing zero after a decimal place is significant, because it gives us information about how accurately we know the number (0.60 is more accurate than 0.6, for example).

Example 4: Round 14 032.25 to three significant figures.

As before, we identify our rounding digit by counting to the third significant figure, 0. In this case, 0 is significant, because it is between other significant figures and if we removed it we would change their value (the 4 is currently worth 4000, if we removed the 0 the 4 would be worth 400). We look at the next digit, 3, which tells us to round down. We leave the 0, replace the 3 and 2 with 0s, and remove the 0s after the decimal point. This gives us 14 000.

You may notice that 14 000 actually only has two significant figures. In this situation, you can think of rounding to three significant figures as rounding up to three significant figures. You will never have to use more significant figures, but sometimes you may have to use less.