Previously, I discussed how to round a number to the nearest whole number [or tenth, ten, etc.]. I focused there on what to do when you are simply told to do it — what “round to nearest” *means*, and how that determines whether you round a particular number up or down. I also pointed out that in “exact half-way” cases, the choice is arbitrary, and there are several *different rules* that are taught.

Here, I want to look at the different *reasons* for rounding, and how this – the context – determines whether you want to “round to nearest” or something else. It’s important in learning any kind of math not only to learn to *do the math* you are told, but also to *decide what math* you should do for an application.

## Context

Rather than start with a question that is directly about rounding, let’s start with this brief question from 2001:

Remainders What do you do with the remainder in division?

The answer, of course, is to first ask, “Why are you dividing, and what will you do with the answer?” It is the context that determines how to handle the results. So I suggested a few scenarios for which different choices are appropriate. Here is the first:

Sometimes the remainder means just what it says, and should be part of the answer. If I have thirteen apples to divide among five people, and I want to give each one of the people only whole apples so they can take them home, then I divide [13 / 5 = 2 R 3] and say that each person gets two apples, and I have three apples left over - the remainder. The same thing would happen if I wanted to cut a 13-inch board into 5-inch pieces: I would get two pieces, with a 3-inch piece left over that I couldn't use (for this project, at least). If it doesn't matter what's left over (or you aren't asked), you might just ignore it.

So if the remainder is effectively “scrap”, then we can **round down** or **truncate**: The answer here is not 13/5 = 2.6 apples, but just 2 apples, dropping the fractional part.

In my second scenario, we do want the exact answer, 2.6 or 2 3/5, and there is **no rounding** to do at all.

In the third, things are different:

When you want a whole number answer, you may have to look at the remainder just to decide whether to "round up" the answer. For example, if I have 135 students to take on a trip, and each bus can hold 60 students, how many buses will I need? I divide 135 by 60 and find [135 / 60 = 2 R 15]. That means if I used two buses, I would have fifteen students left over who couldn't go! So I add one to the number of buses, and take three buses. If the remainder had been zero, I wouldn't have to do this.

Here, then, we have to **round up**, to make sure there is room for everyone. It was the purpose of the division, and the meaning of the result, that determined which way to round — not some single rule we always follow.

None of these happen to call for rounding to the nearest whole number.

## Always round up?

Here’s another situation that requires rounding:

Rounding Discounts and Sales Tax I wonder why the math book that I teach from states the following: When computing sales tax alwaysround upbefore adding on to the price. Ex: 14.9305 rounds to 14.94. When computing discounts alwaysround downbefore subtracting from the original price. Ex: 3.9975 round to 3.99. This seems confusing to the students. Why can't you follow rounding rules?

These students have been taught to think of rounding as always “to the nearest [penny]”, but here they are told not to follow that “rule”! Why?

Doctor Twe explained that this is a standard practice because of rounding errors. The government taxes based not on each individual item, but on the total amount the store takes in. Rounding to the nearest penny would sometimes result in the total taken in for taxes being less than they owe the government. (It might sometimes be more, too.) By always rounding up, the store is sure to get the benefit. One could argue about the fairness of this, but there is a reason behind it – and the reason is not really mathematical, but practical.

## Rounding in estimation

Another common reason to round is for estimation. Here, it turns out that whatever “rule” you follow is entirely at your own discretion — something that surprises many students.

This student was told to estimate the result of a subtraction by rounding each number to the nearest ten, and is confused by the results:

Are There Times When Following the Rounding Rules Isn't Best? With 32 - 15 the actual answer is 17. However, when rounding to the nearest 10 I am certain that 32 is rounded to 30. What way is the number 15 rounded if it is to be rounded once again to the nearest 10? My teacher is wanting us to round the numbers first before we solve the problem to see if our answer is close to the answer of the estimation to figure out if we got the correct answer. What I don't understand here is the number 15. If I round the 15 up to 20 for my estimation, then my answer to the estimated part of the problem would be 30 - 20 = 10. But if I round the 15 down to 10 as the closest 10 then my estimation is 30 - 10 = 20. My Mom says my teacher is trying to show me that by rounding 15 down, since it is 5 or less in the ones column, then my estimated answer of 20 is only 3 digits away from 17. My Dad is telling me that we should round the 15 up to 20. But that makes my estimated answer of 10 farther away from 17 since it is 7 digits away. Who is correct here? My mom or dad? Is the rule 5 and down rounds down? Or is it 5 and up rounds up? I'm so confused. Help!

Following the “rules”, as Dad insists, she gets 30 – 20 = 10, which is farther from the exact answer than what she gets if she “breaks the rule” to get 30 – 10 = 20.

What’s the answer? It’s that in estimating, your **goal** is not to “round to the nearest ten”, but to get the best estimate you can, so you can bend the “rules” to fit your goal. You get to choose the tool you use, and that tool may *not* be “round to nearest”.

For Mary, the main issue was the 15, where we are at that half-way point at which the choice for “nearest” is arbitrary. But the same issue can come up even without that. If the problem had been 32 – 16, I might think this way: When we subtract, changing the numbers in the same direction will cause the least possible change in the difference. (Think of two friends standing in a line both moving forward, so that the difference between them doesn’t change much.) So I’d rather not round 32 **down** to 30 and 16 **up** to 20, even though those are the nearest tens. Instead, I want to round them **both up**, or **both down**; it doesn’t make any difference which, as I get either 40 – 20 = 20, or 30 – 10 = 20. Since the exact answer is 16, this is definitely a better estimate than 10.

The more thinking you do, the better your answer will be. The more you blindly follow rules, the better the chance you will get a less-than-ideal answer.

(By the way, my guess is that the teacher *intended* the students to follow the “round-to-nearest rule” and discover for themselves that this didn’t give the best answer, so they could then be taught something like the ideas I showed. Dad is doing what the teacher said, but Mom is thinking ahead, already seeing the lesson. Both are right, in different ways.)

## Some surprises

Now let’s have some fun:

Subtleties of Rounding There are 480 freshmen at Washington High. If 36% of the freshmen play sports, how many play sports? Is the correct answer 172 or 173? The argument for 172 is that you cannot have .8 of a person so you round down to 172. The argument for 173 is that you should round to the nearest whole number (and that 173 is closer to 36% of 480 than 172 is).

John multiplied 480 by 0.36, giving the answer 172.8. Since we’re dealing with whole people, we clearly have to round to a whole number; but do we round down (ignoring the extra pieces of freshmen that the division seems to give us, like my scrap apple pieces), or round up (as if we were counting buses), or round to nearest (because that’s what you always do)? The answer is to **check the answer**!

Suppose that 172 play sports; what percent is that? 172/480 = 35.83%. Okay, that would have been reported as 36%, so we’re good.

But suppose that 173 play sports? That’s 173/480 = 36.04%, and this, too, is a valid answer.

In fact, what if only 171 play sports? 171/480 = 0.35625%, which again rounds to 36%. So **any of these answers is reasonable**!

Here’s my answer, written after I did that sort of experimentation:

I think rounding to 173 is best if you need a specific answer; but in reality we don't quite know. If someone who knew the actual number calculated the percentage to the nearest percentage point, and told us that it is 36%, then the exact percentage might be anywhere from 35.5% to 36.5%, corresponding to numbers from 170.4 to 175.2. That means that the original number might have been 171, 172, 173, 174, or 175! In each of those cases, the percentage would come to 36% after rounding; so all are equally possible. If we take the given percentage as 36.0%, accurate to the nearest tenth of a percentage point, then there have to be 173 playing sports, since percentages of 35.95% and 36.05% correspond to 172.56 and 173.04. You can see why rounding to the nearest whole number makes sense. If we take the percentage as being _exactly_ 36%, of course, then somebody on the team is missing a few body parts. That's why we HAVE to make some assumption about the precision of the data. I love these little problems that turn out to be a lot more complicated than they seemed on the surface!

So why does it turn out that rounding to the *nearest* person gives the best answer when it makes a difference? Because we’re assuming that the data resulted from rounding the actual percentage to the *nearest* whole percent, and we’re just trying to undo that. So rounding to nearest in both directions makes sense as the most likely guess. But, as we saw, this may be only one of several possible correct answers.

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