If the numerator and denominator of a proper fraction are increased by the same positive quantity which is greater than zero, the resulting fraction is

(a) always less than the original fraction
(b) always greater than the original fraction ✓ UPSC's answer
(c) always equal to the original fraction
(d) such that nothing can be claimed definitely

Why the answer is (b)

• Let the proper fraction be a/b, where 0 < a < b.

• Let the positive quantity added to both numerator and denominator be x, where x > 0.

• The new fraction becomes (a + x) / (b + x).

• To compare, subtract the original fraction: (a + x)/(b + x) - a/b = [b(a + x) - a(b + x)] / [b(b + x)] = (ab + bx - ab - ax) / [b(b + x)] = x(b - a) / [b(b + x)].

• Since a < b, (b - a) is positive; since x > 0, the numerator x(b - a) is positive.

• The denominator b(b + x) is also positive, so the difference is positive, meaning the new fraction is always greater than the original.

Why the other options are wrong

(a) always less than the original fraction: The difference calculation shows the new fraction is larger, not smaller, because the denominator is larger than the numerator in a proper fraction.
(c) always equal to the original fraction: The fractions are not equal because adding the same positive number to a smaller numerator and a larger denominator increases the ratio.
(d) such that nothing can be claimed definitely: A definite mathematical relationship exists; the new fraction is always greater for any proper fraction and any positive x.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.