## What does it mean to round a number?

Rounding means to make a number shorter or simpler, but keeping it as close in value as possible to the original number. Let’s take a closer look:

## Rounding to the nearest integer

The most common type of rounding is to round 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). If the digit in the tenths place is less than 5, then **round down**, which means the units digit remains the same; if the digit in the tenths place is 5 or greater, then round up, which means you should increase the unit digit by one.

**Here are a couple other things to know:**

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:

**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.

## What do you have to know about how to round to the nearest integer 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.

## Other cases of rounding

Very occasionally, a GRE math 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 GRE math, 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.

*Editor’s Note: This post was originally published in November 2012 and has been updated for freshness, accuracy, and comprehensiveness.*

Hi Mike,

This isn’t just a good article but a GREAT ONE! Thanks a lot! And thanks for wonderful Magoosh Quantitative Lessons

Hi, Mike, I saw a question related to the standard deviation in the book published by ETS, it related to penguin, I remembered the possibility from the chart is 16%, and it asks to round to the nearest 0.05, and the answer is 15% from the ETS guid, how to do that?

Hi Yiming,

I’m afraid I don’t really understand your question! The problems in the Official Guide are very specific, so I can’t answer this question without more context. It would help if you write the problem

exactlyas written in the Official Guide (NOT just paraphrased) so that I can see what you mean 🙂The figure above shows a normal distribution with mean m and standard deviation d, including approximate percents of the distribution corresponding to the six regions shown.

Suppose the heights of a population of 3,000 adult penguins are approximately normally distributed with a mean of 65 centimeters and a standard deviation of 5 centimeters.

(b) If an adult penguin is chosen at random from the population, approximately what is the probability that the penguin’s height will be less than 60 centimeters? Give your answer to the nearest 0.05.

http://math.stackexchange.com/questions/865770/finding-the-range-from-standard-deviation-and-gaussian-curve

Hi, sorry I cannot directly insert the picture to this blog, but I put the website below, the thing is how to round 16% to the nearest 0.05.

Thank you

If I’m understanding correctly, this problem is treating percentage as a decimal expression. In that case, 16% = 0.16. Rounded to the nearest 0.5, this would be 15%. (Similarly, 2.36 is rounded down to 2.35 in the example in this post.)

If the percentage is being treated as a whole number, on the other hand, then 16% = 16.0%. In that case, it would be unchanged when rounded to the nearest 0.5, as it is already divisible by 0.5. On the GRE, a question along these lines would very likely be multiple choice, and the choices would clearly show whether 16% is a decimal or whole number. Similarly, a GRE numeric entry question would clearly state whether you needed to treat the percentage as a decimal or a whole number.

hlo mike, what is 3.194 rounded to 0.5? is it 3.0 becoz 1 gets rounded up because of 9 and since 2 is closer to 0 rather than 5. Also what is value of 105.6 rounded to nearest 7? is it 105 or 112 m bit confused I think its 105, am I correct?

Hi Kunal,

Honestly, these are not going to be the type of questions you get on the GRE. They are going to ask you to round to the nearest integer or a certain decimal place (e.g., tenth, hundredth, thousandth, etc.). They will definitely not ask you to round to the nearest 7.

What would be the number 3.494 rounded to the digit immediately right to the decimal point? 3.4 or 3.5?

Hi Proshno,

The digit immediately to the right of the decimal is the tenths digit. If we were asked to round to the nearest tenth, our rounding convention would dictate that 3.494 rounds to 3.5! I hope that helps. 🙂

Hi

My answer is 1.1587 and what will the answer if it was mentioned as “Give your answer to the nearest 0.1 percent.”

Could You plz. Explain too??

Hi Satya,

If are are asked to round to the nearest percent, it means that you have to convert your answer first to a percent. 1.1587 as a percent would be 115.87%. If you have to round the to the nearest .1, it means that you should only have one number after the decimal point. In order to know how to round the 8, we look at the 7. Since the 7 is higher than 5, we round the 8 up to 9. So, this answer rounded to the nearest .1 percent would be 115.9 %.

But the correct answer is 15.8% not 115.9%.

Hi Mashael,

Thanks for your message 🙂 If you could provide the actual question, I think we’ll be able to work this out more easily 🙂 Based on the answer you provided, 1.1587, if we wanted to make this a percent, we would multiply the value by 100%, giving us 115.87%. From there, as mentioned in my earlier response, rounding to the nearest 0.1 percent, we would end up with 115.9%. Again, this is based off the answer you provided. That said, if the original question is from Magoosh or is an official ETS question, I’d be happy to take a look at how to get to the provided answer of 15.8% 🙂

Hi,

The original question is from an official ETS question. It’s a numeric entry question.

From 2011 to 2012, Jack’s annual salary increased by 10 percent and Arnie’s annual salary decreased by 5 percent. If their annual salaries were equal in 2012, then Arnie’s annual salary in 2011 was what percent greater than Jack’s annual salary in 2011

Give your answer to the nearest 0.1 percent

Thanks, Pooja!

OK, based on that problem, here’s my explanation:

The trick is to assign the correct numbers and variables. We’ll say that Jack’s original 2011 annual salary is one variable, J. And we’ll call the other variable, Arnie’s salary for 2001, A.

Jack’s salary increased by 10% in 2012. So his 2012 salary is 100% of his 2011 salary, plus an extra 10%. This means his 2012 salary is 110% of J, or is 1.1J.

Arnie’s salary in 2012 dropped by 55. So Arnie’s new salary is 100% of A, minus 5%. In other words, Arnie’s salary in 2012 is 95% of A, or 0.95A.

Now, since the salaries are equal to each other by 2012, you can say that 1.1J = 0.95A. Let’s solve this for A:

0.95A = 1.1J

A = (1.1J)/0.95

A= (1.1/0.95)*J

Now, 1.1/0.95 = 1.1578, so:

A = 1.1578J

Or, to put this in plan speech, Arnie’s original 2011 salary can be rounded to 1.158 times Jack’s original salary.

Recall that a percentage increase is 1 plus a decimal times the original amount, with the decimal being the actual percent increase, and the 1 being the original 100%. So Arnie’s salary is 115.8% of Jack’s salary, or is 15.8% higher than Jack’s salary.

Thanks a lot

How about 113.0973355 and also 19.09814324 to one decimal place

Happy to help 🙂

Rounded to one decimal place, we get

113.0973355 –> 113.1

19.09814324 –> 19.1

That’s because the following digit is 9 (which is greater than or equal to 5).

Hope this helps 🙂

what will 4.447 be rounded to??????????????????????

Hi,

The rule to remember with rounding is to look at the digit only to the immediate right of the digit you’re rounding. For example, if you are rounding 4.447 to the nearest integer, you focus only on 4.4, which results in 4. If you are rounding 4.447 to the nearest tenth, you focus only on 4.44, which results in 4.4. If you are rounding 4.447 to the nearest hundredth, you focus only on 4.447, which results in 4.45. I hope this helps!

Hey Mike ,

so,in a nutshell , heres a shorcut for the .05 rounding problems .

1.Go to the hundredth’s place , let the digit you find be x .

2.if x=1,2 replace x with 0.

3.if x=3,4,5,6,7 , replace x with 5 , keeping everything else constant .

4.if x=8,9 ,replace x with 0,and add 1 to the number preceding x in the original number (which was to be rounded) .

Am i correct ?

Cheers,

Abhi .

You’ve got it! Great summary!

Hi,

Is it possible to give a summary for 0.01?

Happy to help 🙂 Rounding to the nearest hundredths place (0.01) follows the rules as rounding to the nearest integer. That said, I’ll follow Abhi’s set-up to look at rounding to the nearest hundredth (0.01).

1.Let the hundredths place be x and the thousandths place be y: 0.0xy

2.If y = 0, 1, 2, 3, or 4, then keep x the same.

3.If y = 5, 6, 7, 8, or 9, add 1 to the value of x so that the hundredths place is equal to x+1

As you can see, this is the same basic procedure used to round to the nearest one’s place. The only difference is that we’re dealing with decimal digits when looking at 0.01.

Hope this helps 🙂

I found the following mental process is very useful to myself.

Example “rounding 0.16 to the nearest 0.05”:

Imagine a bug starts at 0, and can only take jumps of 0.05. She wants to land as close as possible to 0.16. Three jumps take her to 0.15. Four take her to 0.20. Three jumps get her closer than four, so 0.16 rounded to the nearest 0.05 is 0.15.

Awesome! Thanks for sharing this way to approach rounding 🙂

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 🙂

What is 10.2 rounded to one decimal place is it 0.1

Hi Yae,

I’m not sure I understand. 10.2 is already a number rounded to a single decimal place. Can you explain your question a little more? 🙂

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 🙂