## If r and s are positive integers, is r/s a terminating decim

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### If r and s are positive integers, is r/s a terminating decim

by Max@Math Revolution » Tue Oct 04, 2016 6:11 am
If r and s are positive integers, is r/s a terminating decimal?
1) r is a factor of 100
2) s is a factor of 500

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by Brent@GMATPrepNow » Tue Oct 04, 2016 3:57 pm
Max@Math Revolution wrote:If r and s are positive integers, is r/s a terminating decimal?

1) r is a factor of 100
2) s is a factor of 500
Target question: Is r/s a terminating decimal?

Statement 1: r is a factor of 100
There are several pairs of values that meet this condition. Here are two:
Case a: r = 1 and s = 4, in which case r/s = 1/4 = 0.25, which is a terminating decimal
Case b: r = 1 and s = 3, in which case r/s = 1/3 = 0.333.., which is not a terminating decimal
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: s is a factor of 500
There's a nice rule that says something like,
If the prime factorization of the denominator contains only 2's and/or 5's, then the decimal version of the fraction will be a terminating decimal.
Since 500 = (2)(2)(5)(5)(5), any factor of 500 will contain only 2's and/or 5'2 (or the denominator can be 1, in which case the decimal will definitely terminate).
Since the denominator of r/s must contain only 2's and/or 5's, r/s must be a terminating decimal
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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by TayoUmar » Sun Feb 19, 2017 5:24 am
Brent@GMATPrepNow wrote:
Max@Math Revolution wrote:If r and s are positive integers, is r/s a terminating decimal?

1) r is a factor of 100
2) s is a factor of 500
Target question: Is r/s a terminating decimal?

Statement 1: r is a factor of 100
There are several pairs of values that meet this condition. Here are two:
Case a: r = 1 and s = 4, in which case r/s = 1/4 = 0.25, which is a terminating decimal
Case b: r = 1 and s = 3, in which case r/s = 1/3 = 0.333.., which is not a terminating decimal
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: s is a factor of 500
There's a nice rule that says something like,
If the prime factorization of the denominator contains only 2's and/or 5's, then the decimal version of the fraction will be a terminating decimal.
Since 500 = (2)(2)(5)(5)(5), any factor of 500 will contain only 2's and/or 5'2 (or the denominator can be 1, in which case the decimal will definitely terminate).
Since the denominator of r/s must contain only 2's and/or 5's, r/s must be a terminating decimal
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
Hi Brent,

I thought since we do not know the value of r, the statement is automatically insufficient.

Thanks.

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by Brent@GMATPrepNow » Sun Feb 19, 2017 7:51 am
TayoUmar wrote:
Brent@GMATPrepNow wrote:
Max@Math Revolution wrote:If r and s are positive integers, is r/s a terminating decimal?

1) r is a factor of 100
2) s is a factor of 500
Target question: Is r/s a terminating decimal?

Statement 1: r is a factor of 100
There are several pairs of values that meet this condition. Here are two:
Case a: r = 1 and s = 4, in which case r/s = 1/4 = 0.25, which is a terminating decimal
Case b: r = 1 and s = 3, in which case r/s = 1/3 = 0.333.., which is not a terminating decimal
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: s is a factor of 500
There's a nice rule that says something like,
If the prime factorization of the denominator contains only 2's and/or 5's, then the decimal version of the fraction will be a terminating decimal.
Since 500 = (2)(2)(5)(5)(5), any factor of 500 will contain only 2's and/or 5'2 (or the denominator can be 1, in which case the decimal will definitely terminate).
Since the denominator of r/s must contain only 2's and/or 5's, r/s must be a terminating decimal
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
Hi Brent,

I thought since we do not know the value of r, the statement is automatically insufficient.

Thanks.
Good question.
Consider the fraction r/2.

If r is a positive integer, will this fraction be a terminating decimal?
Let's see what happens with various values of r.
- If r = 5, then r/2 = 2.5 (a terminating decimal)
- If r = 9, then r/2 = 4.5 (a terminating decimal)
- If r = 8, then r/2 = 4 (a terminating decimal)
- If r = 1, then r/2 = 0.5 (a terminating decimal)
.
.
.
etc.
In fact, for ANY value of r (where r is an integer), r/2 will ALWAYS be a terminating decimal.
In other words, even though we don't know the value of r, we can be certain that r/2 will be a terminating decimal.

Does that help?

Cheers,
Brent
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by DavidG@VeritasPrep » Sun Feb 19, 2017 7:52 am
TayoUmar wrote:
Brent@GMATPrepNow wrote:
Max@Math Revolution wrote:If r and s are positive integers, is r/s a terminating decimal?

1) r is a factor of 100
2) s is a factor of 500
Target question: Is r/s a terminating decimal?

Statement 1: r is a factor of 100
There are several pairs of values that meet this condition. Here are two:
Case a: r = 1 and s = 4, in which case r/s = 1/4 = 0.25, which is a terminating decimal
Case b: r = 1 and s = 3, in which case r/s = 1/3 = 0.333.., which is not a terminating decimal
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: s is a factor of 500
There's a nice rule that says something like,
If the prime factorization of the denominator contains only 2's and/or 5's, then the decimal version of the fraction will be a terminating decimal.
Since 500 = (2)(2)(5)(5)(5), any factor of 500 will contain only 2's and/or 5'2 (or the denominator can be 1, in which case the decimal will definitely terminate).
Since the denominator of r/s must contain only 2's and/or 5's, r/s must be a terminating decimal
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
Hi Brent,

I thought since we do not know the value of r, the statement is automatically insufficient.

Thanks.
I'll leave it to Brent to explain the mechanics of his solution, but I suspect you're struggling with the distinction between YES/NO questions and VALUE questions. In a YES/NO question (as we have here), we don't necessarily need a unique value to have sufficiency, just a definitive "YES" or a definitive "NO."

To take a simple example, imagine you're asked, "Is x > 0?" If a statement told you that "x > 10," that statement would be sufficient to answer YES to the question, as we'd know definitively that x was positive, even though there'd be an infinite number of possibilities for the value of x.
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by Scott@TargetTestPrep » Mon Feb 27, 2017 10:41 am
Max@Math Revolution wrote:If r and s are positive integers, is r/s a terminating decimal?
1) r is a factor of 100
2) s is a factor of 500
When solving this problem, we should remember that there is a special property about fractions that allows their decimal equivalents to terminate. This property states:

In its most-reduced form, any fraction with a denominator whose prime factorization contains only 2s, only 5s, or both 2s and 5s, produces decimals that terminate. A denominator with any other prime factors produces decimals that do not terminate.

We must determine whether r/s is a terminating decimal, or in other words, whether s has only 2s, 5s, or both as prime factors.

Statement One Alone:

r is a factor of 100.

Since we do not have any information about s, statement one alone is not sufficient to answer the question.

Statement Two Alone:

s is a factor of 500.

Since 500 = 2^2 x 5^3 and s is a factor of 500, s will contain only 2s, 5s, or both as prime factors. If r/s is already in its most-reduced form, then r/s is a terminating decimal. If r/s is not in its most-reduced form, then the most-reduced form of r/s, say r'/s', will also be a terminating decimal since s' will then be a factor of s and it will contains only 2s, 5s or both as prime factors. Statement two alone is sufficient to answer the question.