terminating decimal

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maria: Senior | Next Rank: 100 Posts; Posts: 39; Joined: Tue May 06, 2008 11:49 am; Thanked: 1 times

by maria » Thu Jun 05, 2008 6:47 pm

any decimal that has only a finite number of nonzero digits is a terminating decimal. for example, 24, 0.82, and 5.096 are three terminating numbers. If r and s are positive integers and the ratio is r/s is expressed as a decimal, is r/s a terminating decimal?
1. 90<r< 100
2. s = 4

Thanks

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Source: Beat The GMAT — Data Sufficiency | Thu Jun 05, 2008 6:47 pm

chidcguy: Master | Next Rank: 500 Posts; Posts: 438; Joined: Mon Feb 12, 2007 9:44 am; Thanked: 26 times

by chidcguy » Thu Jun 05, 2008 7:52 pm

Answer is B

If we can express the denominator in the form of 2^x X 5^y the decimal will terminate.

4 is 2 ^ 2 X 5 ^ 0

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phumbert: Junior | Next Rank: 30 Posts; Posts: 24; Joined: Thu Jul 03, 2008 7:33 am; GMAT Score:710!

by phumbert » Thu Aug 14, 2008 4:32 pm

chidcguy wrote:Answer is B

If we can express the denominator in the form of 2^x X 5^y the decimal will terminate.

4 is 2 ^ 2 X 5 ^ 0

Would anyone be willing to elaborate on this a little for me? I think I get the explanation, but on a broader scope, Im not sure that Im grasping the concept, and what I would do if the answer set were different. Thanks for anyone that can help!

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rhymes_with_luck: Senior | Next Rank: 100 Posts; Posts: 37; Joined: Fri Aug 01, 2008 6:17 am; Thanked: 1 times

by rhymes_with_luck » Thu Aug 14, 2008 6:15 pm

man think of 1/4, 2/4, 3/4. 4/4 ....etc, have you ever seen decimals recurring for this fractions ?

Hopefully, got the point.

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pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Fri Aug 15, 2008 1:21 am

With any number divided by 4 the remainder will be 0, 1, 2, or 3. I think we agree on that. So their quotients are 0.0, 0.25, 0.5, 0.75
Any number divided by 4 always has terminating decimals.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Fri Aug 15, 2008 3:58 am

phumbert wrote:
chidcguy wrote:Answer is B

If we can express the denominator in the form of 2^x X 5^y the decimal will terminate.

4 is 2 ^ 2 X 5 ^ 0
Would anyone be willing to elaborate on this a little for me? I think I get the explanation, but on a broader scope, Im not sure that Im grasping the concept, and what I would do if the answer set were different. Thanks for anyone that can help!

I'll try. Think first of how you would write a terminating decimal as a fraction. If you saw, for example:

0.203

you'd write that as 203/1000. That is, any terminating decimal can be written with a power of 10 in the denominator. Here, we've used 10^3; prime factorize 10^3 and you have (2^3)*(5^3). That is, any terminating decimal can be written with only 2s and 5s in the prime factorization of the denominator. Some of the 2s and 5s might cancel; using a different example:

0.204 = 204/1000 = 51/250 = 51/[(2^1)*(5^3)]

but the point is, if you only see 2s and/or 5s in the denominator of a completely reduced fraction, the fraction definitely represents a terminating decimal, because you could multiply the numerator and denominator by 2s or 5s to get a power of 10 in the denominator. For example, if you had 11/125, which is 11/5^3, you could multiply by 2^3 to get 1000 in the denominator:

11/125 = 11/(5^3) = [11*(2^3)]/[(2^3)(5^3)] = 88/1000 = 0.088

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sumithshah: Master | Next Rank: 500 Posts; Posts: 165; Joined: Wed May 28, 2008 11:25 pm; Thanked: 9 times; GMAT Score:730

by sumithshah » Sat Aug 16, 2008 11:08 am

Im not sure if this rule applies here ( im too sleepy to solve this ) but anything that is divided by a power of 5 or 2 in the denominator will ALWAYS terminate.

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pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Sat Aug 16, 2008 11:19 am

Can you explain further your tip because
1/(3^2)= 0.1111111111

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sumithshah: Master | Next Rank: 500 Posts; Posts: 165; Joined: Wed May 28, 2008 11:25 pm; Thanked: 9 times; GMAT Score:730

by sumithshah » Sat Aug 16, 2008 11:36 am

When I say a power of 5 or 2, I mean 2^X or 5^Y for all x, y E integers && X, Y != 0

so basically A/(2^X) would be terminating and B/(5^X) would be terminating.

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Sun Aug 17, 2008 6:55 am

pepeprepa wrote:Can you explain further your tip because
1/(3^2)= 0.1111111111

1/(3^2) is not equal to 0.1111111111.

1/(3^2) is equal to 0.11111111111111111111............, repeating forever. It is not a terminating decimal; it is what is called a 'recurring' or 'repeating' decimal. You can tell, by looking at the fraction 1/(3^2), that it will be a repeating decimal, because it is a reduced fraction, and there is a prime different from 2 or 5 in the denominator.

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sumithshah: Master | Next Rank: 500 Posts; Posts: 165; Joined: Wed May 28, 2008 11:25 pm; Thanked: 9 times; GMAT Score:730

by sumithshah » Sun Aug 17, 2008 7:23 am

ah! reduced was the key word. So basically if you have a fraction

ABC / XYZ, reduce it and now, if we have p/q and q is solely a power of 5 or 2 or both ( and nothing else - so eg p/15 wont make a terminating decimal) then its terminating.

Gurus - correct me if im wrong

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pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Sun Aug 17, 2008 8:05 am

Yep Ian I am ok that's a non terminating decimal.
It was a counter-example to what I understood wrongly of sumithshah property.
Indeed what sumithsha told was: A/(2^X) would be terminating and B/(5^X) would be terminating.

Finally, any reduced fraction which has a prime at the denominator is a non terminating, am I right?
For example,
1/3
2/3
1/7
8/9

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4meonly: Legendary Member; Posts: 891; Joined: Sat Aug 16, 2008 4:21 amFollowed by: 1 member; Thanked: 27 times; GMAT Score:660(

by 4meonly » Sun Aug 17, 2008 8:48 am

pepeprepa wrote:Yep Ian I am ok that's a non terminating decimal.
It was a counter-example to what I understood wrongly of sumithshah property.
Indeed what sumithsha told was: A/(2^X) would be terminating and B/(5^X) would be terminating.

Finally, any reduced fraction which has a prime at the denominator is a non terminating, am I right?
For example,
1/3
2/3
1/7
8/9

6/3 will terminate, 14/7 will terminate too

but it will not be decimal

if denumerator can be expressed in prime factorisation of 2 and 5 it is always terminal dicimal

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pepeprepa: Legendary Member; Posts: 661; Joined: Tue Jul 08, 2008 12:58 pm; Location: France; Thanked: 48 times

by pepeprepa » Sun Aug 17, 2008 8:53 am

Ok for your principle:
"if denumerator can be expressed in prime factorisation of 2 and 5 it is always terminal dicimal"

But I don't understand what you mean by 6/3 and 14/7 I talk about reduced fractions...

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Ian Stewart: GMAT Instructor; Posts: 2623; Joined: Mon Jun 02, 2008 3:17 amFollowed by: 355 members; Location: Montreal; Thanked: 1090 times; GMAT Score:780

by Ian Stewart » Sun Aug 17, 2008 9:11 am

I'll try to state the rule as unambiguously as possible. If you need to know whether a fraction represents a terminating decimal:

1) Reduce the fraction completely;
2) Prime factorize the denominator;
3) Look at this prime factorization, and ignore the exponents. If there is a prime besides 2 or 5 in the denominator, the fraction represents a recurring (non-terminating) decimal. If the only primes in the denominator are 2, 5, or both, the fraction represents a terminating decimal.

So 9/125, 3/32, 7/80 and 3/30 are all terminating decimals (when you look at 3/30, you must reduce the fraction first, of course), while 7/121, 5/33, 7/60 and 19/99 all represent recurring decimals.

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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