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Round 0.4999999 to integer?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by krusta80 » Wed Feb 29, 2012 8:25 pm
pinktoadette wrote:Dumb question

When you round 0.499999, is it 0 or 1? I thought 1 at first but,...
..if it's repeating 9s? It never reaches 0.5, so 0..?


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by Jim@StratusPrep » Fri Mar 02, 2012 7:39 am
0... .5 is the cutoff. Don't round multiple times.
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by Brent@GMATPrepNow » Sat Mar 03, 2012 12:31 pm
pinktoadette wrote: When you round 0.499999, is it 0 or 1? I thought 1 at first but,...
..if it's repeating 9s? It never reaches 0.5, so 0..?
Your question about the repeating 9's is interesting.

It can be argued that 0.4999...(9's repeating forever) is equal to 0.5

Proof #1
If 0.4999...(forever) does not equal 0.5, then there must exist some number BETWEEN 0.4999... and 0.5
Since no such number exists, we must conclude that 0.4999.... is equal to 0.5, in which case rounding it to the nearest integer would be 1.

Proof #2
Another way to look at it is to say that 0.4999..... = 0.4 + 0.0999...

Well, 0.0999... is equal to 0.1
Here's why.
We know that 1/30 = 0.0333...
So, 3/30 = 0.0999...
But we already know that 3/30 = 0.1
So, 0.0999... must equal 0.1, which means that 0.4999..... = 0.4 + 0.0999... = 0.4 + 0.1 = 0.5


Having said all of that, this would never be tested on the GMAT.


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by vikram4689 » Sat Mar 03, 2012 7:14 pm
Brent,
I think answer would be 0 because in explanation above we round off 2 times which we should not do.

I tried this in EXCEL as well :)
0.4999 = 0.500 = 0.50 = 0.5 = 0 [no. of digits = no. of significant digits allowed, and then i reduced the no. of significant digits one by one]
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by Brent@GMATPrepNow » Sat Mar 03, 2012 10:59 pm
vikram4689 wrote:Brent,
I think answer would be 0 because in explanation above we round off 2 times which we should not do.

I tried this in EXCEL as well :)
0.4999 = 0.500 = 0.50 = 0.5 = 0 [no. of digits = no. of significant digits allowed, and then i reduced the no. of significant digits one by one]
Well, we're already getting into some esoteric/out-of-scope territory, but I will suggest that we're not rounding two times. First, we're recognizing that 0.4999... = 0.5 (no rounding yet), and then we round 0.5 to 1.

Also, I'm having a hard time believing that you entered an infinite number of 9's in Excel. That would have taken quite a while :-)

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by krusta80 » Sat Mar 03, 2012 11:07 pm
Brent@GMATPrepNow wrote:
vikram4689 wrote:Brent,
I think answer would be 0 because in explanation above we round off 2 times which we should not do.

I tried this in EXCEL as well :)
0.4999 = 0.500 = 0.50 = 0.5 = 0 [no. of digits = no. of significant digits allowed, and then i reduced the no. of significant digits one by one]
Well, we're already getting into some esoteric/out-of-scope territory, but I will suggest that we're not rounding two times. First, we're recognizing that 0.4999... = 0.5 (no rounding yet), and then we round 0.5 to 1.

Also, I'm having a hard time believing that you entered an infinite number of 9's in Excel. That would have taken quite a while :-)

Cheers,
Brent
The interesting thing about infinite 9's as a decimal is that it has no equivalent fraction. :)

That said, I'm wondering why the original poster is asking the question? I've seen something like this happen in computer programming when dealing with decimals. In those cases, it's important to realize that the computer isn't rounding as much as it's simply misrepresenting the TRUE value, which in this case would be .5
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by vikram4689 » Sat Mar 03, 2012 11:09 pm
Haha, but we are talking about 0.4999999 and not 0.4999999....infinity.
and 0.4999999 is NOT equal to 0.5 so if we have to answer "0.4999999 to integer?" then it would be 0
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by Brent@GMATPrepNow » Sat Mar 03, 2012 11:11 pm
krusta80 wrote: The interesting thing about infinite 9's as a decimal is that it has no equivalent fraction. :)
I'm not sure I agree with that.

We know that 1/9 = 0.1111...(with infinite 1's)
And 2/9 = 0.2222...(with infinite 2's)
And 3/9 = 0.3333...(with infinite 3's)
...
9/9 = 0.9999...(with infinite 9's)

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by krusta80 » Sat Mar 03, 2012 11:12 pm
Brent@GMATPrepNow wrote:
krusta80 wrote: The interesting thing about infinite 9's as a decimal is that it has no equivalent fraction. :)
I'm not sure I agree with that.

We know that 1/9 = 0.1111...(with infinite 1's)
And 2/9 = 0.2222...(with infinite 2's)
And 3/9 = 0.3333...(with infinite 3's)
...
9/9 = 0.9999...(with infinite 9's)

Cheers,
Brent
But we know that 9/9 is 1. :)

Actually...looks like we're both right...unless of course Wiki is wrong. lol

https://en.wikipedia.org/wiki/0.999...
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by Brent@GMATPrepNow » Sat Mar 03, 2012 11:12 pm
vikram4689 wrote:Haha, but we are talking about 0.4999999 and not 0.4999999....infinity.
and 0.4999999 is NOT equal to 0.5 so if we have to answer "0.4999999 to integer?" then it would be 0
Agreed, but I've been responding to the second part of the original question, that asks what happens if the 9's are repeating.

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by vikram4689 » Sat Mar 03, 2012 11:15 pm
this is what Brent was saying i.e. 0.999... = 1
But we know that 9/9 is 1.
I think we should not delve more as we all agree
0.499999 to integer is 0
0.4999... to integer is 1
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by krusta80 » Sat Mar 03, 2012 11:17 pm
vikram4689 wrote:this is what Brent was saying i.e. 0.999... = 1
But we know that 9/9 is 1.
I think we should not delve more as we all agree
0.499999 to integer is 0
0.4999... to integer is 1
Agreed.
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by Brent@GMATPrepNow » Sat Mar 03, 2012 11:23 pm
It looks like it's unanimous!

Now I can go to bed.... :-)

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by rakesh4u119 » Mon Apr 02, 2012 3:18 am
Try below Funda's..

Pure Recurring to Fractions
FUNDA 1: If a number is of the form of 0.ababab....... then divide the repeating digits with as many 9's as we have repeated digits.
eg. 0.363636... =36/99 =11/4

Mixed Recurring to Fractions

FUNDA 2: If N = 0.abcbcbc.... Then
N =(abc − a)/990
As many 9' s as repeated digits followed by as many zero as non repeated digits
Re peated & non repeated digits Non repeated digits
−
−
eg. 0.25757.....=(257 − 2)/990=255/990 =17/66

FUNDA 3:If N = a.bcbc.... Then
Write N = a + 0.bcbc....
Proceed as Funda 1
5.3636... = 5 + 0.3636... =5 + 36/99 = 11/59
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