What is the greatest length x such that 3 1/2 m and 8 3/4 m are integral multiples of x?

(a) 1 1/2 m
(b) 1 1/3 m
(c) 1 1/4 m
(d) 1 3/4 m ✓ UPSC's answer

Why the answer is (d)

• Convert the mixed fractions to improper fractions: 3 1/2 m = 7/2 m and 8 3/4 m = 35/4 m.

• The problem asks for the greatest length x that divides both lengths exactly, which is the Highest Common Factor (HCF) of the two fractions.

• The HCF of two fractions is calculated as the HCF of their numerators divided by the LCM of their denominators.

• The HCF of the numerators 7 and 35 is 7.

• The LCM of the denominators 2 and 4 is 4.

• Therefore, the greatest length x is 7/4 m, which converts to the mixed fraction 1 3/4 m, matching option (d).

Why the other options are wrong

(a) 1 1/2 m: 1 1/2 m (3/2 m) does not divide 8 3/4 m (35/4 m) evenly, as (35/4) / (3/2) equals 35/6, which is not an integer.
(b) 1 1/3 m: 1 1/3 m (4/3 m) does not divide 3 1/2 m (7/2 m) evenly, as (7/2) / (4/3) equals 21/8, which is not an integer.
(c) 1 1/4 m: 1 1/4 m (5/4 m) does not divide 3 1/2 m (7/2 m) evenly, as (7/2) / (5/4) equals 14/5, which is not an integer.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.