## What you need to know

**Rounding** is a process of simplifying a number without changing the value of it too much. We do this by taking a number and working out which of the “simpler” numbers it is closest too. For example, if we were to round 6.2 **to the nearest whole number**, then the answer would be 6, since no other whole numbers are closer to 6.2 than 6 is. If instead, we were rounding 6.8 to the nearest whole number, the answer would be 7.

Rounding numbers works as follows.

- Firstly, determine which digit is your cut-off point, e.g. when rounding 6.2 to the nearest whole number, our cut-off point was the 6 – if we went any further then we wouldn’t be dealing with whole numbers.
- Look at the next digit along after the cut-off point, and determine if it is
**below 5**, or if it is**equal to or above 5**.- If it is below 5, then we
**round down**– i.e. the cut-off point digit**stays the same**. - If it is 5 or above, then we
**round up**– i.e. the cut-off point digit**increases by 1**.

- If it is below 5, then we
- Finally, all the numbers after the cut-off point become zeros.

There are 3 ways that you might be asked to round a number.

- To a whole number –
**the nearest whole number, the nearest 10, the nearest 100,** - To a given number of
**decimal places**(often shortened to**dp**). - To a given number of
**significant figures**(often shortened to**sf**).

The way that we outlined rounding above applies to all 3 of these cases, all that differs between them is how you go about choosing the cut-off point. If you aren’t sure about how this will all work don’t worry, it’ll become clearer with some examples.

**Example: **Round 235 to the nearest ten.

So, where is the cut-off point in this case? Well the first digit, 2, is in hundreds, but the second digit, 3, is in tens, so that will be our cut-off point. Now we look to the digit after this point: it is a 5, which means that we **round up**, i.e. the cut-off point digit gets **increased by 1**, so in this case the 3 will become a 4. Finally, making all the digits after the cut-off point zero, we get that

235\,\,\text{ rounded to the nearest 10 is }\,\,240.

**Example: **Round 13.746 to a) 2 decimal places, and b) 1 decimal place.

Finding the cut-off point is very natural in the case of decimal places: if we’re rounding to 2 decimal places then the cut-off point is the 2nd decimal place, i.e. the 2nd number after the decimal point. In this case, that is the 4. The digit after the 4 is a 6, which means that we round up and increase the digit before it by 1 – the 4 becomes a 5. Then, making all the digits after the cut-off zero, we get

13.746\,\,\text{ rounded to 2 decimal places is }\,\,13.75.

Now, we are rounding 13.746 to 1 decimal place, so our cut-off point is the 1st digit after the decimal point: 7. The digit after the 7 is a 4, which means that we **round down**, and the 7 **stays the same**. Then, making all the digits after the cut-off zero, we get

13.746\,\,\text{ round to 1 decimal place is }\,\,13.7

We did make all the digits after the 7 into a zero here, but because they’re at the end of a decimal they don’t need to be included.

**Note: **if we had taken the result of rounding 13.746 to decimal places – 13.75 – and rounded that to 1 decimal place, we would’ve gotten a different answer to the one we just got and would not have been the correct answer for this question. Be __careful__ with your rounding, small errors can have big knock on effects.

**Example: **Round 8,529 to 2 significant figures.

When finding the cut-off point for decimal places, we counted the digits after the decimal place. When finding it for significant figures, we start counting at the **first non-zero term**. In other words, we move along our number until we hit a digit that isn’t zero – this is our **1st significant figure**. Then, the digit after is the 2nd significant figure, the digit after that I the 3rd, and so on. If any of the digits after the 1st significant figures are zero, they **still count as significant figures**, it’s only the first one that must be non-zero.

So, in this case, the first digit of the number is an 8 so we start counting right away: 8 is the first significant figure, so 5 is the second significant figure, therefore the 5 is our cut-off point. The digit after the 5 is a 2. This is less than 5, so we round down and the cut-off digit stays the same. Then, making all the digits after the cut-off zero, we get

8,529\,\,\text{ rounded to 2 significant figures is }\,\,8,500.

**Example: **Round 0.00589 to 2 significant figures.

So, we need to find the first non-zero term. We can see that the first 3 digits are zero, but the 4th digit is a 5, so this is our 1st significant figure. Therefore, the next digit along, the 8, is our 2nd significant figure and thus is our cut-off point. The digit after the 8 is a 9, which is bigger than 5, and so we round up, increasing the cut-off digit by 1, making the 8 into a 9. Then, making all the digits after the cut-off zero, we get

0.00589\,\,\text{ rounded to 2 significant figures is }\,\,0.0059.

This final example explains what to do when you have to round a 9 up (a common occurrence).

**Example: **Round 4.398 to 2 decimal places.

The 2nd digit after the decimal point, the 9, is our cut-off point. The digit after it is an 8, which is above 5, and so we must increase the 9 by 1. The problem is, increasing a 9 by 1 makes 10, so what do we do? Fortunately, it’s not too much trouble. The cut-off digit becomes zero, and the digit **before** it gets increased by 1. So, in this case, the 9 goes to zero and the 3 before it becomes 4. Then, making all the digits after the cut-off point zero, we get

4.398\,\,\text{ rounded to 2 decimal places is }\,\,4.40

Although 4.40 is exactly the same thing as 4.4, it helps to write the answer as 4.40 to remind both yourself and the person marking your work that you have actually rounded your answer to 2 decimal places and not 1.

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### Example Questions

1) Round 560,180 to the nearest thousand.

The cut-off point is whichever digit is in thousands. Here, that is the 3rd digit along, i.e. the first zero. So, the digit after the cut-off point is a 1, which is less than 5, so we round down and the zero stays the same. Making the rest of the digits after the cut-off point zero, we get

560,180\,\,\text{ rounded to the nearest thousand is }\,\,560,000

2) Round 97.96 to 1 decimal place.

The first digit after the decimal point is our cut-off point. The digit after this is a 6, which is above 5, so we round up and cut-off digit, 9, increases by 1. Since we can’t squeeze a 10 into the space of one digit, we must make the cut-off digit into a zero and increase the digit before it by 1, so the 7 becomes an 8. Then, making the digits after the cut-off point zero, we get

97.96\,\,\text{ rounded to 1dp is }\,\,98.0

3) Round 0.02345 to 3 significant figures.

The first non-zero digit is this number is the 2, so this is the first significant figure. Moving two spaces along we see that the 4 is the 3rd significant figure and thus is our cut-off point. The digit after the 4 is a 5, meaning that we must round up and increase the cut-off digit by 1, i.e. the 4 becomes a 5. Then, making the digits after the cut-off point zero, we get

0.02345\,\,\text{ rounded to 3sf is }\,\,0.0235

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