Which of the Following Fractions has a Decimal Equivalent that is a Terminating Decimal? GMAT Problem Solving

Question: Which of the following fractions has a decimal equivalent that is a terminating decimal?

10/189
15/196
16/225
25/144
39/128

“Which of the following fractions has a decimal equivalent that is a terminating decimal?''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT quant section measures the candidates’ skill and logical thinking abilities to solve quantitative problems. The students must answer the question by calculating it thoroughly with accurate mathematical understanding. The students must know fundamental knowledge of mathematical calculations to solve GMAT Problem Solving questions. The mathematical problems of the GMAT Quant topic in the problem-solving part is a very important part and can be solved by suitable quantitative skills.

Solution and Explanation:
Approach Solution 1:

The problem statement asks to find out the fractions which have a decimal equivalent which is basically a terminating decimal.

To solve the problem, it is important to understand the theoretical concept behind terminating decimals. For this, the concept of reduced fractions is to be understood.

A reduced fraction a/b, implying a fraction which is reduced to its lowest terms, is expressed as a terminating decimal if only the denominator, b, is of the form \(2^n5^m\), where m and n are non-negative integers.

For instance: 7/250 is a terminating decimal 0.028 which is because 250 is the denominator which basically equals to \(2 * 5^2\). Fraction 3/30 is also a terminating decimal, as it is 3/30 = 1/10 and denominator 10 is equal to 2*5.

It is essential to note that if the denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example \(x/2^n5^m\) (where x, n and m are integers) will always be the terminating decimal.

We need a reduction in the case when we have the prime in a denominator other than 2 or 5 to see whether it could be reduced. For example, 6/15 has 3 as the prime denominator and it is important to know whether it can be reduced.

Considering the question that is required to identify which fraction has a terminating decimal, it can be said that when the fraction is reduced to its lowest form and has the denominator as \(2^n5^m\).

Here this suggests 39/128 = 39/2^7

Correct Answer: E

Approach Solution 2:

The problem statement asks to find out the fractions which have a decimal equivalent which is basically a terminating decimal.

For a fraction to produce a terminating decimal, its denominator needs to be composed ONLY of powers of 2 and/or of powers of 5 when the fraction exists in its MOST REDUCED FORM.

In any multiple of 3, the sum of the digits is a multiple of 3.
Therefore, the denominators of A (189), C (225), and D (144) are all multiples of 3.
Any of these answer choices cannot be further reduced.
Since all of these options are in their most reduced form, and the denominator of each of these answer choices is something other than a power of 2 or a power of 5, none of these will yield a terminating decimal.
Therefore, we can eliminate A, C and D.

Answer choice B: 15/196 = (3*5)/(2*2*7*7)
Since in its most reduced form this fraction has in its denominator something other than a power of 2 or a power of 5, the resulting decimal will not be terminating.
Therefore, we can eliminate B.

Answer choice E: 39/128 = (3*13)/(2^7).
Since in its most reduced form this fraction has in its denominator only a power of 2, the resulting decimal will be terminating.

Correct Answer: E

Approach Solution 3:
The problem statement asks to find out the fractions which have a decimal equivalent which is basically a terminating decimal.

Reduced fraction a/b (it means that fraction is already reduced to its lowest term) can be defined as terminating decimal if and only the denominator b is of the form 2^n5^m, where n and m are non-negative integers.

For example: 7/250 is a terminating decimal 0.028, as 250 (denominator) equals 2∗522∗52. Fraction 3/30 is also a terminating decimal, as 3/30=1/10 and denominator 10=2∗5.

A. 10/189=10/(3^3∗7) --> denominator holds primes other than 2 and 5 in its prime factorization, Hence it is a repeating decimal;

B. 15/196=15/(2^2∗7^2) --> denominator holds primes other than 2 and 5 in its prime factorization, hence it is a repeating decimal;

C. 16/225=16/(3^2∗5^2) --> denominator holds primes other than 2 and 5 in its prime factorization, hence it is repeating decimal;

D. 25/144=25/(2^4∗3^2) --> denominator holds primes other than 2 and 5 in its prime factorization, hence it is repeating decimal.

E. 39/128=39/2^7, the denominator has only prime factor 2 in its prime factorization, hence this fraction will be terminating.

Correct Answer: E

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