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
Answer & explanation
Answer: (b) always greater than the original fraction
For a proper fraction a/b with 0 < a < b, adding the same k > 0 to both parts gives a difference of k(b − a)/[b(b + k)], which is positive. The fraction always increases.
- Let the fraction be a/b with 0 < a < b, and add k > 0 to both.
- (a + k)/(b + k) − a/b = [b(a + k) − a(b + k)]/[b(b + k)] = k(b − a)/[b(b + k)].
- k > 0 and b − a > 0, so the difference is positive: the new fraction is larger.
- Example: 1/2 becomes (1 + 1)/(2 + 1) = 2/3, which is greater than 1/2.
Remember · Adding the same positive number to both parts pulls a fraction towards 1: proper fractions rise, improper ones fall.
Question and answer: UPSC's official GS Paper II (2019, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·