Question

'A' and 'B' complete a work in 12 days, 'B' and 'C' in 8 days and 'C' and 'A' in 16 days. 'A' left after working for 3 days. In how many days more will 'B' and 'C' finish the remaining work?

• A. 6 \(\frac{1}{6}\) days
• B. 2.7 days
• C. 3.4 days
• D. 4.3 days

Hint:

When given pairwise work rates, add them to find the total for all three, then subtract to get individual rates.

Answer 1

The Correct Option is C

Step 1: Write work rates for each pair.
From the question:
A + B = \(\frac{1}{12}\) work/day
B + C = \(\frac{1}{8}\) work/day
C + A = \(\frac{1}{16}\) work/day
Step 2: Find total of A + B + C.
Adding all three equations:
(A + B) + (B + C) + (C + A) = \(\frac{1}{12} + \frac{1}{8} + \frac{1}{16}\)
2(A + B + C) = \(\frac{4 + 6 + 3}{48} = \frac{13}{48}\)
So A + B + C = \(\frac{13}{96}\) work/day
Step 3: Find each individual rate.
From A + B = \(\frac{1}{12}\), we have: A + B = \(\frac{8}{96}\).
Subtract from A + B + C: C = \(\frac{13}{96} - \frac{8}{96} = \frac{5}{96}\) work/day.
From B + C = \(\frac{1}{8} = \frac{12}{96}\), we get B = \(\frac{12}{96} - \frac{5}{96} = \frac{7}{96}\) work/day.
From C + A = \(\frac{1}{16} = \frac{6}{96}\), we get A = \(\frac{6}{96} - \frac{5}{96} = \frac{1}{96}\) work/day.
Step 4: Work done by A, B, and C in first 3 days.
Daily combined rate = A + B + C = \(\frac{13}{96}\).
Work done in 3 days = \(3 \times \frac{13}{96} = \frac{39}{96}\) of the total work.
Step 5: Remaining work.
Remaining work = \(1 - \frac{39}{96} = \frac{57}{96}\).
Step 6: B and C’s combined rate.
B + C = \(\frac{12}{96}\) work/day.
Step 7: Time required for remaining work.
Time = \(\frac{\frac{57}{96}}{\frac{12}{96}} = \frac{57}{12} = 4.75 \ \text{days}\).
So the exact answer is 4 \(\frac{3}{4}\) days, which is approximately 4.75 days.