If x is an integer, can the number (5/28)(3.02)(90%)(x) be GMAT Data Sufficiency

bySayantani Barman Experta en el extranjero

Question: If x is an integer, can the number (5/28)(3.02)(90%)(x) be represented by a finite number of non-zero decimal digits?

(1) x is greater than 100
(2) x is divisible by 21

A) Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

"If x is an integer, can the number (5/28)(3.02)(90%)(x) be" – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT data sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

In this question, an expression is given -> (5/28)(3.02)(90%)(x)

It has to find out if the number can be represented by a finite number of non-zero decimal digits.

The theory behind this question is given below.

Reduced fraction a/b (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only bb (denominator) is of the form 2^n*5^m, where m and n are non-negative integers. For example: 7/250 is a terminating decimal, 0.028, as 250 (denominator) equals to 2∗(5^3). Fraction 3/30 is also a terminating decimal, as 3/30=1/10 and denominator 10=2∗5

It should be noted that whether or not the percentage is lowered is irrelevant if the denominator already contains only 2-s and/or 5-s.

For example, x/2^n*5^m, (where x, n and m are integers) will always be a terminating decimal.

(We need reducing in the case when we have the prime in denominator other than 2 or 5 to see whether it could be reduced. For example, fraction 6/15 has 3 as prime in the denominator and we need to know if it can be reduced.)

First of all 5/28∗3.02∗9/10∗x = 5∗302∗9∗x / (28∗100∗10) = 5∗302∗9∗x/(7∗(4∗100∗10))

=5∗302∗9∗x7∗(22∗22∗52∗2∗5)528∗3.02∗910∗x=5∗302∗9∗x28∗100∗10=

5∗302∗9∗x/(7∗(4∗100∗10)) = 5∗302∗9∗x/(7∗(22∗22∗52∗2∗5)). Now, according to the theory above, in order for this number to be a termination decimal, 7 must be reduced by a factor of x (no other number in the numerator has 7 as a factor and all other numbers in the denominator have only 2's and 5's), so it'll be a terminating decimal if x is a multiple of 7.

(1) x is greater than 100. Not sufficient.
(2) x is divisible by 21. Sufficient.

Correct Option: B

Approach Solution 2:

In this question, an expression is given -> (5/28)(3.02)(90%)(x)

It has to find out if the number can be represented by a finite number of non-zero decimal digits.

5/28∗302/100∗90/100∗x

for this to be a terminating decimal the denominator must be in the form of x/2^a∗5^b

Re-writing the question stem numbers to be 5/(7∗2^2)∗ 302/(2^2∗5^2)∗ 90/(2^2∗2^5)∗x

From statement 1) if we have x > 100 its might include a multiple of 7 and it might not. so insufficient.

2) Since x is divisible by 21, it may be any number between 21 and 63. As 21 = 7 * 3, it is a multiple of 21.

This will cancel the 7 in the denominator and lead to the terminating decimal form, which is sufficient.

Correct Option: B

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