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Video: Rounding Decimals

Tim Burnham

We are here explaining the process of rounding to a given number of decimal places and looking at increasingly difficult cases when rounding up.

11:28

Video Transcript

We’re gonna look at how to go about rounding decimal places in numbers, for example to two decimal places or five decimal places. The basic method is quite simple but there are a few tricky examples and special cases that lots of people often get wrong so it’s well worth taking a few moments to just practice this really important maths skill.

Sometimes numbers have a lot of digits. Sometimes they even have an infinite number of digits, like the irrational number 𝜋. Obviously in these cases, it would be impossibly time-consuming and use way too much ink to write out the number in full. We have to decide how accurately we’d like to record the number, for example to how many decimal places of accuracy.

Let’s say when measuring the width of a flower pot, if we’re using a ruler or a tape measure, we can’t measure more accurately than about a tenth or a sixteenth of an inch, so it wouldn’t make sense to try to write down thirty decimal places. We just round our measurement to one decimal place. We can call this expressing the answer correct to one decimal place or rounding to one decimal place. Okay let’s look at a method for rounding to a given number of decimal places.

In our first example we’re gonna round the number one point two four nine three to one decimal place. First, we’re gonna write out the whole number as it was written in the question. Then, we’re gonna use a piece of paper or a ruler or even our finger to hide everything after that first decimal place. So we want the answer to one decimal place, so we cover up everything after that one decimal place. Now that’s not the answer; that’s kind of our first guess of the answer. What we need to do is check the contents of the next digit along to see whether or not that two can stay as a two or whether it can be rounded up to a three.

So we slide back the cover just that one extra decimal place, and we’ll still ignore all the rest of the digits, but the next digit was a four. And the rule is if that next number is five or above, then this number here, our last decimal place that we’re looking at, would round up to the next decimal place. If it’s less than five, then we’re gonna leave it as it is. And in this case, four is less than five so we can leave the two as it is.

Now we’ve finished looking at the second decimal place we can slide back the cover and cover it up again. And our answer is one point two correct to one decimal place. In our second example, we’re gonna round the same number one point two four nine three to two decimal places.

So first write out the number, then cover everything up after that second decimal place. And one point two four is our first guess at what the answer might be. Now we need to slide back the cover and take a sneaky peek at that next digit to see whether the four can stay as a four or whether we have to round it up to a five.

Well in this case that next digit is a nine, so that is five or bigger so we’re gonna have to round the four up to a five. So we add one to the four to make it a five. And this basically means that one point two four nine three is closer to one point two five than it was to one point two four.

So hopefully that explains what rounding is all about. What we’re saying is which number that’s got two decimal places is the closest to the number that we started off with. And in this case, it- was it one point two four or was it one point two five? It was in fact one point two five.

So next, round fifty-eight point zero zero nine four to two decimal places. So we’ve just written the number out, and then we cover up everything after that second decimal place. Now we’re gonna need to slide the cover back so we can have a look at that third digit to see whether the second digit in the decimals has to be rounded up to one or whether it can stay as zero.

And the third decimal place is nine, which is five or bigger, so that means that the zero is gonna have to have one added to it which is gonna make it one. So we cover up the third decimal place again, and then we look at the answer: fifty-eight point zero one to two decimal places. So again the number that’s got two decimal places in which is closest to the number we started off with is fifty-eight point zero one.

So what would happen if the question had been round fifty-eight point zero zero nine four to one decimal place? We’d write out the number, cover up everything after that first decimal place, slide back the cover to reveal that second decimal place. And it’s zero, which is less than five, so we’re not going to have to add anything to our first decimal place. So this zero here can remain as a zero.

And our answer is fifty-eight point zero to one decimal place. So hopefully this is fairly straightforward as we said, but there are just a couple of issues that we need to think about before we move on. The question asked for one decimal place and we’ve included one decimal place, so that’s right. But a lot of people get tempted to say, “Oh well if it’s only a zero, I won’t bother writing. I’ll just write it as fifty-eight to one decimal place.” Now that would be wrong. If it asks for one decimal place, it wants one decimal place so you have to put it in there even if it’s not- even if it’s only a zero.

Now the other point I wanna make is you know why am I doing all this covering up stuff. Some of you might be thinking, “Oh that’s a bit pointless. I can just- I don’t need to cover it up with a ruler or piece of paper or my finger, I can just ignore those digits.”

But one of the most common mistakes in questions like this other example is that people start sort of rounding and carrying too late on in a number. So for example, if we want to round this fifty-eight point zero four nine four to one decimal place, lots of people sort of say, “Oh well I’ve got a nine here so that rounds the four up to a five. I then got a five here which means that that zero needs to be rounded up to a one.” And they would’ve come up with the answer fifty-eight point one because that is completely wrong. If you use this technique of covering up everything after the decimal place that you’re interested in and then just slipping it back doing a sneaky peek at that one next digit, you’d never be tempted to do all this rounding and rounding and rounding and carrying forward and getting the wrong answer.

Okay so next example, round twenty-three point nine five zero two to one decimal place. Write out the number, cover up everything after the first decimal place, and then slide back the cover one decimal place to reveal that next digit.

So this is a five, which is five or above, so that means we’re gonna have to round up the nine; we’re gonna have to add one to it. Well when we add one to nine, it becomes ten. Ah! But ten isn’t a single digit, so what we have to do is we keep the zero, the units number in that column and we carry the one, the tens value, into the next column. So that’s gonna go onto here. Now I have to add that one to the three to make four, so the number we’ve got is two four point zero. So that’s twenty-four point zero. Remember, we must include the zero even though it’s a zero because the question asked for one decimal place.

Okay, next example: round thirty-nine point nine nine seven one eight to two decimal places. First we write out the number, then we cover up everything after the second decimal place, then we do a sneaky peek to reveal that third decimal place. It’s a seven, which is five or above, so we’re going to have to add one to this nine. And when we add one to that nine, it becomes a ten which means we’re gonna have to do the carry thing. So that becomes a zero carry one, and that means we’ve got to add the one to the next nine, which becomes a ten, which means we’ve got to do the carry thing. So that becomes a zero carry one, which means we’ve now got to add one to that nine, which means it becomes a ten, which means we’ve got to do zero and carry one and then we add one to the three to make four. So we’ve ended up with four zero point zero zero.

So sometimes the carrying thing carries and carries and carries and can be quite complicated. Right, here’s one last example of an even more extreme version of that. So nine hundred and ninety-nine point nine nine nine six three, we’re gonna around that to two decimal places. So we write out the number and cover up everything after that second decimal place.

We then slide back the cover one decimal place to reveal the third decimal place to see if we’ve got to do any rounding. And it’s a nine, so we- that is five or above, so we’re going to have to add one to the next- to the second decimal place. And when we add one, that becomes a ten, which is zero carry one; one plus nine is ten, so well that becomes a zero carry one; one plus nine is ten, so that becomes a zero carry one; one plus nine is ten, so that’s a zero carry one; and one plus nine is ten, so that’s a zero carry one. So we’ve now actually carried one into the next column here, so we’ve got one zero zero zero point zero zero.

So this is kind of a monster example. It’s involved lots and lots of lots of carrying over here. In fact, it’s actually moved us up to the next digit, and it’s also got point zero zero at the end which a lot of people are tempted to lose those zeros. So it kind of- it covers lots of different cases.

So let’s just summarise that process: first of all, we write the number out in full; then we cover up everything after the required number of decimal places, so for example if it was two decimal places, we hide everything after that second decimal place; then we slide back the cover to take a sneaky peek at the next digit; and then use the information in that digit to decide whether or not we need to round or possibly even carry some values over to the higher value columns. This is a five, which is five or above, so we do need to round that six by adding one to it up to a seven. Finally, we can put the cover back again and you’ve got your answer, in this case fifty-one point two seven to two decimal places.