# 2 Rounding

The world population in April 2021 was estimated to be 7.86 billion. That is not an exact value but it does give you an indication of the world’s population at the time. This is an example of a rounded or estimated number.

In newspapers and elsewhere, numbers are often rounded so that people can get a rough idea of the size, without getting lost in details. You may use rounding yourself, such as prices in a supermarket (£2.99 is about £3) or distances (18.2 miles is about 20 miles).

Using approximate values is also useful if you want to get a rough idea of an answer before you get out scrap paper or a calculator. This estimation acts as a check on your calculation and may help you to catch any errors you may have.

The general rules for rounding are:

- Locate the place value to be rounded.
- Look at the next digit to the right.
- If that digit is a 5 or greater, ‘round up’ the previous place value digit – which means increase it by one.
- If that digit is a 4 or less, leave the previous place value digit unchanged.
- Replace all digits after the place value digit with 0, but
*only*if these are to the left of the decimal point.

This last rule may be a little harder to understand, so here’s an example.

If you are rounding 5423 to the nearest one hundred this would be 5400 – the tens and units places being replaced by zeros.

With decimals we often refer to the number of decimal places to round to rather than a digit’s place value. So, instead of asking you to round to the nearest tenth, you might see instructions to ‘round to 1 decimal place’ (1 d.p.). The decimal places are numbered consecutively from the right of the decimal point. So, 1.54 has 2 decimal places, and 34.8942 has 4 decimal places.

Whether you are rounding using place value or decimal places, the general rules remain the same.

Before you have a look at some examples, you might like to view this video on rounding.

#### Transcript

**Round 24 259 to the nearest hundred.**You're going to find that doing these problems are pretty straightforward, but what I want to do is just think about what it means to round to the nearest hundred. So what I'm going to do is I'm going to draw a number line. Let me draw a number line here, and I'm just going to mark off the hundreds on the number line. So maybe we have 24 100, and then we go to 24 200, then we go to 24 300, and then we go to 24 400. I think you see what I mean when I'm only marking off the hundreds. I'm going up by increments of 100. Now, on this number line, where is 24 259? So if we look at the number line, it's more than 24 200 and it's less than 24,300. And it's 259, so if this distance right here is 10, 059 is right about there, so that is where our number is. That is 24 259. So when someone asks you to round to the nearest hundred, they're literally saying round to one of these increments of 100 or round to whichever increment of 100 that it is closest to. And if you look at it right like this, if you just eyeball it, you'll actually see that it is closer to 24 300 than it is to 24 200. So when you round it, you round to 24 300. So if you round to the nearest hundred, the answer literally is 24 300. Now that's kind of the conceptual understanding of why it's even called the nearest hundred.

Here are a few more examples to have a look at before you have a go yourself.

Imagine you want to round the world population figure of 7.86 billion to the nearest billion. The place value that you want to round in this case is the units as these represent the billions, which is a 7. The next number to the right of this is a 8, which is 5 or more, so you need to round up and the 7 becomes an 8.

So 7.86 billion rounded to the nearest billion is 8 billion.

Now suppose you want to round 1.72 to 1 decimal place. The 7 is in the first decimal place and to the right of this is a 2. This is 4 or less, so the 7 stays as it is.

So 1.72 rounded to 1 decimal place is 1.7. Now it’s your turn.

## Activity 2 Rounding

Round each of the following numbers as stated.

(a) Round 126.43 to the nearest tenth.

### Answer

- Locate the place value to be rounded – in this case, the number 4.
- Look at the next digit to the right – in this case, it’s 3.
- 3 is less than 5, so you round down, and the answer is 126.4.

(b) Round 0.015474 to the nearest thousandth.

### Answer

- Locate the place value to be rounded – in this case, the number 5.
- Look at the next digit to the right – in this case, it’s 4.
- Because 4 is less than 5, you round down, and the answer is 0.015.

(c) Round 1.5673 to 2 decimal places (2 d.p.).

### Answer

- Locate the decimal place to be rounded – in this case the number is 6.
- Look at the next digit to the right – in this case, it’s 7.
- Because 7 is greater than 5, you round up, and the answer is 1.57.

Now complete Activity 3.

## Activity 3 How much is it?

Suppose you want to buy a new pick-up truck. When you visit the dealership’s website, it says the cost of the vehicle you are looking at is £19 748. When you call the dealership on Thursday, the sales representative, Joe, tells you the cost is £19 750. However, the next day another sales representative, Tina, informs you that the vehicle costs £19 700. Why did the quote you were given on the cost of the pick-up truck change?

Hint: Round the website cost to the nearest £10. Then round the website cost to the nearest £100.

### Answer

Here’s how the variation in the quotes happened:

Rounding to the nearest £10 (this is the same as rounding to the tens place): the digit 4 is in the tens place. Looking to the place value to the right of the 4, the digit is an 8. Since 8 is larger than 5, you round up the 4 to 5 in the tens place. Finally, you replace the units digit with 0 (because this place is to the left of the decimal point). The rounded number is £19 750. This explains Joe’s quote.

Rounding to the nearest £100 (this is the same as rounding to the hundreds): the digit 7 is in the hundreds place. Looking to the tens place (to the right of the 7), the digit is a 4. Since 4 is less than 5, you leave the digit 7 unchanged and replace the digits after the 7 with zeros. The rounded number is £19 700. This explains Tina’s quote.

You’re now going to take some time to get to know your calculator. Calculators can save you a lot of time, but it’s important to know how to get the most from them.