X can complete one-third of a certain work in 6 days, Y can complete one-third of the same work in 8 days and Z can complete three-fourth of the same work in 12 days. All of them work together for n days and then X and Z quit and Y alone finishes the remaining work in 8 2/3 days. What is n equal to ?

(a) 3
(b) 4 ✓ UPSC's answer
(c) 5
(d) 6

Why the answer is (b)

• Calculate individual daily work rates: X completes 1/3 work in 6 days, so X's rate is (1/3)/6 = 1/18 of the work per day.

• Y completes 1/3 work in 8 days, so Y's rate is (1/3)/8 = 1/24 of the work per day.

• Z completes 3/4 work in 12 days, so Z's rate is (3/4)/12 = 1/16 of the work per day.

• Determine the work done by Y alone in the final 8 2/3 days: Y's rate (1/24) multiplied by 8 2/3 (which is 26/3) equals (1/24) * (26/3) = 26/72 = 13/36 of the total work.

• Calculate the work completed by X, Y, and Z together in n days: The total work is 1, so the work done together is 1 - 13/36 = 23/36.

• Find the combined daily rate of X, Y, and Z: 1/18 + 1/24 + 1/16. The LCM of 18, 24, and 16 is 144. Converting fractions: 8/144 + 6/144 + 9/144 = 23/144.

• Solve for n: Work done = Rate * Time, so 23/36 = (23/144) * n. Dividing both sides by 23/144 gives n = (23/36) * (144/23) = 4.

Why the other options are wrong

(a) 3: Option (a) is incorrect because substituting n=3 into the equation yields a work completion of 23/48, which does not match the required 23/36.
(c) 5: Option (c) is incorrect because substituting n=5 into the equation yields a work completion of 115/144, which exceeds the required 23/36.
(d) 6: Option (d) is incorrect because substituting n=6 into the equation yields a work completion of 23/24, which exceeds the required 23/36.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.