**What does it mean to round a number? What do you have to know about rounding for the GRE? **

Not many GRE questions will say, “Here’s a number: round it to the nearest such-and-such.” By contrast, many questions, in the course of asking something else, could ask you to round your answer to the nearest such-and-such. In this way, rounding is one math skill you need to know for the GRE. There are a few tricky issues, which I will address here.

## Rounding to the nearest integer

The most common type of rounding is round decimals to the nearest integer. The rule for rounding is simple: look at the digits in the tenth’s place, the first digit to the right of the decimal point, *that digit and that digit only*. If the digit in the tenths place is {0, 1, 2, 3, 4}, then round down, which means the units digit remains the same; if the digit in the tenths place is {5, 6, 7, 8, 9}, then round up, which means the unit digit increases by one. Here are a couple things to notice:

Observation #1: under most circumstances, rounding changes the decimal to whatever integer is closer. For example, 4.3 is rounded to 4, and 4.9 is rounded to 5. The exception is when the decimal is smack dab between two integers: 4.5 is exactly equidistant to both 4 and 5, but because of the “tie-breaker” rule of rounding, anything with a 5 in the tenths digit is rounded up. This is the only case in which the “go to the closer integer” interpretation will fail.

Observation #2: **Do NOT double-round**. Some people look at a number like, say, 7.49, and they erroneously think — well, that 9 would round the 4 up to 5, and then a 5 gets round up, to this number would round to 8. WRONG! Never round a number “in stages.” Rounding is a one-shot deal, a one-step process. When the number we need to round is 7.49, we only need notice that the tenth’s digit is a 4, which means the number is rounded down to 7. One step, case closed. In fact, all of the following numbers get rounded to 7:

7.499

7.499999

7.49999999999999999999999999999999999999999999999999999999

Here’s the truly mind-boggling part: how many numbers would there be larger than this last number, but still lower than 7.5? INFINITY! No matter how many additional 9’s we slap on to the end of that number, there’s still a continuous infinity of decimals larger than that number and below 7.5 No matter how finely we chop up the real number line, each tiny fragment of the line, no matter how small, still contains a continuous infinite of numbers.

Observation #3: the “tie-breaker” rule can be tricky with negative values. For example, +2.5 gets rounded up to 3, but –2.5 gets rounded *down* … to –3. As with positive numbers, the negative number ending in .5 is rounded to the higher absolute value integer, but with negatives, that’s rounding *down*. (This is not the only way to formulate this rule, but this is the convention that ETS follows.)

## Rounding to any other decimal place

Rounding to the nearest integer is really rounding to the nearest units place. Sometimes, you will be asked to round to the nearest hundreds, or to the nearest hundredths — to some decimal place other than the units place. The rule is just a more generalized version of the previous rounding rule.

Suppose we are asked to round to some specific decimal place — call this the “target place.” You always look at only one digit, the digit immediately to the right of the target place. If this digit immediately to the right is {0, 1, 2, 3, 4}, then you “round down”, and the digit in the target place remains unchanged. If this digit immediately to the right is {5, 6, 7, 8, 9}, then you “round up”, and the digit in the target place increases by 1.

## Other cases of rounding

Very occasionally, a GRE question may ask you not to round to a particular decimal place, but rather to the nearest multiple of something. For example, suppose you are asked to round, say, to the nearest 0.05 — how do you do that?

Well, let’s think about the results first. The result of rounding to the nearest 0.05 would be something divisible by 0.05 — that is to say, a decimal with either a 0 or a 5 in the hundredth place, no digits to the right of that, and any digits to the left of that. The following are examples of numbers which could be the result of rounding to the nearest 0.05:

- 0.35
- 1.40
- 3.15
- 5.2
- 8

Notice: the second, (b) is the square root of 2 (sqrt{2} = 1.414213562….) rounded to the nearest 0.05, and the third, (c), is pi rounded to the nearest 0.05.

Let’s demonstrate the rounding by means of an example. What numbers, when rounded to the nearest 0.05, would be rounded to 2.35? Well, for starters, 2.35 and other “tenths” around it would be rounded to 2.35

2.32 — rounded down to 2.30

2.33 — rounded up to 2.35

2.34 — rounded up to 2.35

2.35 — stays at value

2.36 — rounded down to 2.35

2.37 — rounded down to 2.35

2.38 — rounded up to 2.40

Now, the tricky regions are those between the values that are rounded in different directions. For example, 2.32 is rounded down and 2.33 is rounded up, so something fishy is happening between those two. Let’s think about the hundredths between 2.32 and 2.33 — exactly between them is 2.325, the midpoint between 2.30 and 2.35, and like all midpoints, according to the “tiebreaker” rule, it gets rounded up. Thus:

2.320 —- rounded down to 2.30

2.321 —- rounded down to 2.30

2.322 —- rounded down to 2.30

2.323 —- rounded down to 2.30

2.324 —- rounded down to 2.30

2.325 —- rounded up to 2.35 (the “tie-breaker” rule)

2.326 —- rounded up to 2.35

2.327 —- rounded up to 2.35

2.328 —- rounded up to 2.35

2.329 —- rounded up to 2.35

2.330 —- rounded up to 2.35

This is all probably far more detail than you will need to know for the GRE, but this does demonstrate the steps you would take to round any decimal to the nearest 0.05. By analogy, you could round any decimal to any specified multiple.

2.32 — rounded down to 2.30

2.33 — rounded up to 2.35

2.34 — rounded up to 2.35

Isn’t 2.33 and 2.34 rounded to 2.30 ?

Hi Mike,

What could be the round off of 2.4 or 2.6 ? can we consider this as 2 or it could be 3 ( if decimal notation is equal to 5 or more than that?

Dear Deepika,

I’m happy to respond. When we round, the crucial part is: to what place are we rounding? The test will

alwaysspecify this. Round to the nearest integer, or round to the nearest tenth, or round to the neatest hundred, to the nearest multiple of 10, to the nearest multiple of 5, to the nearest power of 2, etc. — while the first is the most likely, any are possible. If we are rounding to the nearest integer, then 2.4 rounds to 2 and 2.6 rounds to 3. If we are rounding to the nearest tenth, both 2.4 & 2.6 stay right where they are.Does all this make sense?

Mike

Yep..Got it ..Thank you Mike..:)

Dear Deepika,

You are quite welcome! Best of luck to you, my friend!

Mike

Hello there,

I have found different ways to round off when the digit involved is 5.

In your example,

2.25( to the nearest tenth ) rounds off to 2.3

2.35( to the nearest tenth ) rounds off to 2.4

According to Wolfram-Alpha

2.25( to the nearest tenth ) rounds off to 2.2

Check here – http://www.wolframalpha.com/input/?i=2.25+to+nearest+tenth

2.35( to the nearest tenth ) rounds off to 2.4

Check here – http://www.wolframalpha.com/input/?i=2.35+to+nearest+tenth

The rule they use is always the place that has to rounded off even.

I wanted to know which rule GRE uses?

Thanks in advance.

Dear Bikram,

My friend, there are many different conventions one could follow for rounding. The folks at Wolfram Alpha are very smart, but they follow a convention different from that of the GRE. If you start looking at non-GRE sources for conventions, you are going to get yourself very confused. For positive numbers, a 5 in any decimal position rounds up.

Mike

Hi Mike,

First of all great article, really helpful.

My doubt is that in the last line (absolute last) it says –

‘ 2.330 —- rounded up to 2.35 ‘

Shouldn’t this be 2.330 rounded down to 2.30 ?

Thanks a bunch !

Dear Pranay,

My friend, I don’t think you were reading this clearly. That final passage is about rounding to the nearest 0.05 — remember that the correct answer for rounding depends very much on to what we are rounding. For example

2.330 rounded to the nearest integer is 2

2.330 rounded to the nearest 0.5 is 2.5

2.330 rounded to the nearest 0.1 is 2.3

2.330 rounded to the nearest 0.05 is 2.35

2.330 rounded to the nearest 0.01 is 2.33

Does all this make sense?

Mike

Got it !

Thanks Mike

Dear Pranay,

You are quite welcome, my friend. Best of luck to you!

Mike

Hi Mike,

Great article…So here’s my question, could you please give an example of a question where you round a number to the nearest multiple of 0.25 …just a question and and answer to it…

Regards,

Manish A.

Dear Manish,

I’m glad you liked the article. Here are a couple of examples,

3.14 rounded to the nearest 0.25 is 3.25.

2.56 rounded to the nearest 0.25 is 2.50

4.87 rounded to the nearest 0.25 is 4.75

4.88 rounded to the nearest 0.25 is 5.00

The GRE is quite unlikely to ask you about this.

Mike

I don’t understand how 4.87 rounded to the nearest 0.25 is 4.75.

Would you please explain in more detail?

Dear Bushra,

Think about it this way. The 0.25’s are the set:

…3.75, 4.00, 4.25, 4.50. 4.75, 5.00, 5.25, …

These are all multiples of 0.25. Anything rounded to the nearest 0.25 will be rounded to something in that set. The number 4.87 is between 4.75 and 5.00 — the question is: to which is it closest?

|4.87 – 5.00| = 0.13

|4.87 – 4.75| = 0.12

Thus, 4.87 is closer to 4.75 than it is to 5.00, so 4.75 is nearest number in the set, the set of all possible multiples of 0.25.

Does this make sense?

Mike

Nice Reply Mike!!:)

Dear Trijot,

Thank you for your kind words. I wish you the best of luck!

Mike

Can some one reply that when 4.447 is rounded to nearest tenths then which will be answer.

I. 4.4

I. 4.5

how about 4.4455 is rounded to nearest tenth

I. 4.4

I. 4.5

Could you please reply.

Dear Shekhar,

I’m happy to respond. For both of those numbers, 4.447 and 4.4455, the number in the hundredths place, the place right after the tenths place, is a 4 — nothing beyond that matters. The hundredths place is a 4, so the tenths place gets “rounded down” to a 4, and both of those numbers, rounded to the nearest tenth, are 4.4.

Think about it visually. Suppose make the scale really big — let’s say there’s a full meter between 4.4 and 4.5. The 4.4 is at the left side, and one meter away is 4.5. Well, on that scale, the 4.447 would be 47 cm from the left side. It would be very close to the middle, but that’s not the question. That point at 47 cm, is closer to the left side — that’s precisely why the left side, 4.4, is the “nearest” tenth.

Does all this make sense?

Mike

Hi Mike,

How to round 10.8 or 10.2 to nearest 0.1 ?

Dear Kavitha,

The numbers 10.8 and 10.2 already ARE rounded to the nearest 0.1. What you are asking is like asking: what is 3 rounded to the nearest integer? 3 already is an integer! If a number is given to the hundredth place or more, 3.76 or 2.7182818284, then we can round to the nearest tenth (3.8 and 2.7 respectively). Does all this make sense?

Mike

Hi Mike,

thanks for ur reply. I got this doubt because in one of the practice test question it was told to round the answer to nearest 0.1. What exactly it means? Sorry if this sounds silly.

Dear Kavitha,

Numbers to the nearest 0.1, that is, to the nearest tenth are numbers such as 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, etc. That’s an infinite list, in both the positive & negative directions: imagine a number line marked off in tenths that is infinite in both directions.

When you are given a number with more than one place after the decimal, such as 1.25992105 (the cube root of 2), you could be asked to “round it to the nearest tenth”, which means, round it to the number on the infinite list above, the number line by tenths, to which it is closest. We would look at the decimal in the second decimal place, the hundredths place, to determine which way to round. Here, the hundredths place is a 5, so we round up to 1.3. That is this number (the cube root of 2) rounded to the nearest tenth. Does all this make sense?

Mike

Thanks Mike !!! Now I understood !!

Kavitha,

You’re quite welcome. Best of luck to you.

Mike

Hi Mike,

Number 1.2599 rounded to nearest tenth is 1.3? Isn’t it? U mentioned the answer as 1.26

Neelam,

Good eye! Yes, that was a mistake, and I corrected it. The number 1.2599 rounded to the nearest hundredth is 1.26, and to the nearest tenth is 1.3.

Mike

Hi Mike

The negative rule I would like to teach to my aspiring L6s from Year 6. Would it be ok to teach that with negative numbers you are moving away from zero therefore you are looking at the nearest whole number moving in that direction. So -3.5 would round to minus 4. Also as a cheat take away the minus sign round it and replace the minus sign. Coming from a non academic background I try to make more challenging maths more accessible, but I am wary that this could confuse later.

Cheers Dave

Dave,

I must say, I am not familiar with what an “L6″ is, and what level these students might be. The important thing to realize is — different authorities will follow different conventions as regards the “rounding with negatives” rule. The standard I explain here is consistent with the standards of the GRE. The standards to which your students will be accountable may or may not follow another, slightly different rule (e.g always round up at the 0.5, whether positive or negative). I would hesitate to say that one way is THE WAY unless you know all the conventional choices your system has made.

Does this make sense?

Mike

Mike Sir,

The last part seems little confusing! Will we encounter such sums on the Gre?

Also, can you give me some other example apart from this?

Anki,

This last example gets into some of the hardest rounding ideas you will possibly see on the GRE. If you find it confusing, I would say — just file away for now that you found this information here, and if you happen to come across a rounding problem like this in your other practice, then come back & re-read this and it may make more sense at that point.

Does this make sense?

Mike

great !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Thanks a lot MIKE!

Dear Chiraag,

You are quite welcome.

Mike

so that means when ever we are asked to round to the nearest multiple , we just have to see what numbers in the options are divisible by that number.

e.g if we have to find to the nearest multiple of .09 . we have 2 options

a) 81

b)85

HERE 81 is divisible by 9 , so it is the required multiple.

is it right?

First of all, NO ONE on the planet is ever going to ask you to round to the nearest 0.09. It’s rare that you are asked to round to anything other than a power of ten, and when it’s not, the number is something very closed related to powers of ten —- 0.05 is halfway between two multiples of 0.01. The whole point of rounding is to get something to a “nicer” number — if it’s close to an integer for example, rounding it to the integer. Rounding things to the nearest 0.09 would round things AWAY FROM integers, which fundamentally contradicts the idea of rounding.

Furthermore, we are talking about rounding to decimals, but then you propose two positive integers, 81 & 85. It doesn’t make sense to talk about rounding to the nearest decimal and then talk about positive integers. The whole goal of rounding decimals is getting to, or closer to, integers.

Does all this make sense?

Mike

hi there Mike!

In the last part, don’t you mean 2.320 –> 2.32 instead of 3.20 and so on?

YES! Fantastic catch! That’s a total goof on my part. I just corrected the mistake in the post. Thank you, thank you, thank you!

Mike

If 2.5 rounded to 3 but -2.5 is rounded to -2 then how – 2.4 and – 2.7 is rounded if we consider a closest nearest integer?

Dear Debadrata,

By ETS convention, -2.5 is rounded down to -3. The number -2.4 is rounded to the closest integer: -2.4 is closer to -2 than to -3, so it is rounded to -2. Similarly, the number -2.7 is closer to -3, so it is rounded to -3. If you draw a number line in the negative region and mark the tenths between -2 and -3, so that you can see on the number line where -2.4 and -2.7 fall, then I think you will see visually why this has to be the case.

Does all this make sense?

Mike

Hi Mike,

Firstly, Great Post !!! Really helped me..

I have one doubt though. In your post you have mentioned that “+2.5 gets rounded up to 3, but –2.5 gets rounded down … to –3 ” . But here you have ,mentioned that “By convention, -2.5 is rounded “up” to -2″ . Now, I’m a little confused as to follow which rule .

Dear Abirami,

Oops! That was a typo. i just corrected it. Everything should be consistent now. Thanks for pointing this out.

Mike