Rounding
Rounding is a way of simplifying awkward numbers into nice ones. For example, the number 98 is needlessly complicated and we could simplify it to the lovely number 100.
Similar numbers
The point of rounding is to take a number that is exact but too complicated, and turn it into a number that has a similar value and is simpler. Usually we don’t need to know that a car is going at 61.49 km/h; it is quite enough to know that it is doing about sixty. This process of turning 61.49 into 60 is called rounding.
The principle of rounding
First we’ll show how to round numbers to the nearest ten. This means that the resulting number must be exactly divisible by ten. The number 57 is not exactly divisible by ten, but 80 and 150 are. All numbers divisible by ten have a zero in the units place (the last digit).
Now the question is which number we should round 57 to. We go by closeness – which are the nearest numbers divisible by ten? There are basically two candidates, one number smaller than 57 and one larger than 57. The first number smaller than 57 that is divisible by ten is 50. On the other side we get 60.
But 57 is closer to 60 than to 50, because the distance of 60 from 57 is 3 (60 − 57) and the distance of 57 from 50 is 7 (57 − 50). So 57 rounded to the nearest ten is 60. The number 60 is the nearest number that is divisible by ten.
But what happens when we have to round 55 to the nearest ten? That number is just as far from 50 as from 60, because it is 5 away from both. In that case we round up, that is, to the higher number. So we would round 55 to 60. (Sometimes, mainly for technical reasons, other methods are used.)
The rules of rounding
Again we will assume that we are rounding to the nearest ten.
There are basically two ways to round. The first is rounding down, which means we make the original number smaller. We round down when the last digit is between 0 and 4. For example, we round 23 to 20, because the last digit is a three. We round 54 to 50, because the last digit is a four. We round 80 to 80. Careful: we definitely don’t round down to 70 when there is a zero – the number does not change.
The second way is rounding up, which makes the original number bigger. We round up when the last digit is between 5 and 9. So we round 19 to 20, 56 to 60 and 88 to 90. We also round up when the number ends in the digit 5, so we round 15 to 20, 75 to 80 and so on.
If the number ends in 0, 1, 2, 3 or 4, we round down. If it ends in 5, 6, 7, 8 or 9, we round up.
Rounding to different places
So far we have always rounded only to the nearest ten. But we can choose any other place to round to. By place we mean the tens, hundreds, thousands and so on.
If we round a number to the nearest hundred, it means that we want the number to be exactly divisible by a hundred. The number 1,853 is not divisible by a hundred, but the number 7,200 is. The same goes for the other places. Such numbers are always recognisable by a certain number of zeros at the end. A number divisible by ten has at least one zero at the end (it may have more – the number 7,200 is of course also divisible by ten), a number divisible by a hundred has two zeros, a number divisible by a thousand has three zeros, and so on.
If we want to round a number to the nearest hundred, we look for the nearest number that is divisible by a hundred. So we would round the number 389 to 400, because this number is closer to 389 than 300 is.
The rules for rounding to different places
The method is practically the same as for rounding to the nearest ten, except that we don’t look at the last digit of the number, but at the first digit that will become zero.
For tens this is the first digit from the right, for hundreds the second digit from the right, for thousands the third digit from the right, and so on.
Let’s try the method on the number 6,482, which we will round to the nearest hundred. The resulting number must end in at least two zeros, so we look at the second digit from the right. That is the digit 8. The digit 8 tells us to round up, just as before. So we increase the value of the third digit from the right, that is, the digit 4, by one, and set the last two digits to zero – we get 6,500. It is the nearest higher number divisible by a hundred.
Let’s try to round the same number, 6,482, to the nearest thousand. We should get a number whose last three digits are zero. We look at the third digit from the right, that is, 4. It tells us to round down. This time the value of the fourth digit from the right does not change; we just put a 0 in place of the last three digits, so we get 6,000.
Decimals
We can also round decimals, again depending on the place we are rounding to. For example, we can round the number 96.6 to the nearest whole number, 97. It works the same way for smaller numbers. So take the number 0.327. If we want to round this number to two decimal places, we look at the third digit after the decimal point and find a seven there, which rounds up, so we round to 0.330. With decimals we don’t need to write the zeros at the end. The number 3.200 is the same as 3.2. So the previous result after rounding can be rewritten as 0.33. If we wanted to round the original number to one decimal place, the result would be 0.3.
Rounding in stages
When we have to round a number to some place, we always look only one place lower. We don’t go step by step, rounding to every place in turn. Take the number 746, for example. When we round it to the nearest hundred, we get 700, because we look at the second digit from the right, which is 4, and that rounds down.
But we could, in theory, first round the number to the nearest ten, which would give 750, because the digit 6 tells us to round up. Only now would we round to the nearest hundred – we would get 800, not 700, because we are now rounding according to the digit 5, not the digit 4.
This process is called rounding in stages and we don’t usually use it, because it is longer and less accurate.
