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If x is a positive integer, is x/30 a terminating decimal?

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If x is a positive integer, is x/30 a terminating decimal?

by Max@Math Revolution » Thu May 03, 2018 1:00 am

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[GMAT math practice question]

If x is a positive integer, is x/30 a terminating decimal?

1) x is divisible by 3
2) x is divisible by 4
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Source: — Data Sufficiency |
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by GMATGuruNY » Thu May 03, 2018 3:10 am

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A TERMINATING decimal has a FINITE NUMBER OF DIGITS:
.5
.123
.8730253.

A NON-TERMINATING decimal has an INFINITE NUMBER OF DIGITS:
.33333....
.121212....
.871871871...

To determine whether a fraction will yield a terminating decimal:

1. Put the fraction in its MOST REDUCED FORM.
2. PRIME-FACTORIZE the denominator.

If the prime-factorization of the denominator includes ONLY 2'S AND/OR 5'S, the fraction will yield a TERMINATING decimal.
If the prime-factorization of the denominator includes ANY OTHER PRIME NUMBER, the fraction will yield a NON-TERMINATING decimal.

Case 1: 3/120
In its most reduced form, 3/120 = 1/40.
40 = 2² * 5.
Since the the prime-factorization of the denominator includes only 2's and 5's, 3/120 will yield a TERMINATING DECIMAL:
3/120 = .025

Case 2: 15/110
In its most reduced form, 15/110 = 3/22.
22 = 2*11.
Since the prime-factorization of the denominator includes 11 -- a prime number OTHER THAN 2 OR 5 -- 15/110 will yield a NON-TERMINATING DECIMAL:
15/110 = .1363636...
Max@Math Revolution wrote:[GMAT math practice question]

If x is a positive integer, is x/30 a terminating decimal?

1) x is divisible by 3
2) x is divisible by 4
Statement 1:
Let x = 3a, with the result that x/30 = (3a)/30 = a/10 = a/(2*5).
Since the fraction in blue has only 2's and/or 5's in its denominator, the result is a terminating decimal.
Thus, the answer to the question stem is YES.
SUFFICIENT.

Statement 2:
Let x = 4a, with the result that x/30 = (4a)/30 = (2a)/15.
Case 1: a=3
In this case, (2a)/15 = (2*3)/15 = 2/5 = 0.4.
Since resulting decimal is terminating, the answer to the question stem is YES.
Case 2: a=5
In this case, (2a)/15 = (2*5)/15 = 2/3 = 0.666...
Since resulting decimal is non-terminating, the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.

The correct answer is A.
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by Brent@GMATPrepNow » Thu May 03, 2018 6:36 am

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Max@Math Revolution wrote: If x is a positive integer, is x/30 a terminating decimal?

1) x is divisible by 3
2) x is divisible by 4
Target question: Is x/30 a terminating decimal?
This is a good candidate for rephrasing the target question.
-------------ASIDE------------------
Let's say that x = a/b where the fraction a/b is written in simplest terms.
There's a nice rule that says something like,
If a/b results in a terminating decimal, then the denominator, b, MUST be the product of 2's and 5's only!
So, for example, if b = 20, the fraction a/b will result in a terminating decimal. The same holds true for other values of b such as 4, 5, 25, 40, 2, 8, and so on.
-------BACK TO THE QUESTION--------------------

Given the above information, we can REPHRASE the target question....
REPHRASED target question: Can the denominator of the SIMPLIFIED version of x/30 be written as the product of 2's and 5's only?

Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: x is divisible by 3
This means that x = 3k, for some integer k
So, we can say: x/30 = 3k/30 = k/10
The denominator (10) can be written as follows: 10 = (2)(5)
So, the answer to the REPHRASED target question is YES, the denominator of the simplified version of x/30 CAN be written as the product of 2's and 5's only
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT

Statement 2: x is divisible by 4
There are several values of x that satisfy statement 2. Here are two:
Case a: x = 12. In this case, x/30 = 12/30 = 2/5. Here, the denominator (5) CAN be written as the product of 2's and 5's only
Case b: x = 4. In this case, x/30 = 4/30. Here, the denominator (30) CANNOT be written as the product of 2's and 5's only
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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by Max@Math Revolution » Mon May 07, 2018 1:04 am

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=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Note that 30 = 2*3*5. Therefore, for x/30 to be a terminating decimal, 3 must be a factor of x (then the 3 from the denominator will cancel out with a 3 from the numerator). So, the question is asking whether x is a multiple of 3.

As this is precisely condition 1), condition 1) is sufficient.

Note that condition 2) does not tell us whether x is a multiple of 3.

Therefore, A is the answer.
Answer: A

In cases where 3 or more additional equations are required, such as for original conditions with "3 variables", or "4 variables and 1 equation", or "5 variables and 2 equations", conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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