If r, s are positive integers, is r/s a terminating decimal?

Decimal is terminating when it can be expressed as \(\frac{K}{2^n5^m}\), where n, m are non-negative integers and k is an integer (e. g. 0.1 can be expressed as \(\frac{1}{2^15^1}\); 0.015 can be expressed as \(\frac{15}{2^35^3}\) or \(\frac{3}{2^35^2}\) etc)

1) \(\frac{1}{r}\) is a terminating decimal, thus, can be expressed as \(\frac{1}{2^n5^m}\), \(r=2^n5^m\).
If s=5, then \(\frac{r}{s}=\frac{2^n5^m}{5}\), terminating.
If s=3, then \(\frac{r}{s}=\frac{2^n5^m}{3}\), not terminating.
Insufficient.

2) \(s=2^n5^m\)
\(\frac{r}{s}=\frac{r}{2^n5^m}\). Since r, s are positive integers, then the expression is a terminating decimal for any r,s.

Sufficient.

B

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If r, s are positive integers, is r/s a terminating decimal?

1) 1/r is a terminating decimal
2) 1/s is a terminating decimal


When you modify the original condition and the question, a terminating decimal is a fraction that only 2 and 5 are the prime factors of a denominator. So, you only need to figure out s from r/s.
Therefore, the answer is B.


 Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

Target question: Is r/s a terminating decimal?
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.
So, r/s will be a terminating decimal if the prime factorization of s (the denominator) contains only 2's and/or 5's.

Statement 1: 1/r is a terminating decimal
If 1/r is a terminating decimal, then we know that the prime factorization of r contains only 2's and/or 5's.
This, however, tells us nothing about about the value of the denominator in the fraction r/s
Consider these two pairs of conflicting values of r and s:
Case a: r = 2 and s = 4, in which case r/s = 2/4 = 0.5, which is a terminating decimal
Case b: r = 2 and s = 3, in which case r/s = 2/3 = 0.666.., which is not a terminating decimal
Since we can't answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 1/s is a terminating decimal
If 1/s is a terminating decimal, then we know that the prime factorization of s contains only 2's and/or 5's.
Since the prime factorization of s (the denominator) contains only 2's and/or 5's, we know that the decimal version of the fraction will 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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