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 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

Answer = B

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

Answer = B

Cheers,
Brent
Hi Brent,

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

Could you please help clarify this?

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

Answer = B

Cheers,
Brent
Hi Brent,

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

Could you please help clarify this?

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

Answer = B

Cheers,
Brent
Hi Brent,

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

Could you please help clarify this?

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.

Answer: B

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