Question

P and Q can do a piece of work in 8 and 12 days respectively. If they work for a day alternately, beginning with P, then the work will be completed in how many days?

• A. 8
• B. \(9\frac{1}{2}\)
• C. 10
• D. \(9\frac{2}{3}\)

Hint:

For alternate day problems, calculate the work done in a 2-day cycle. Use this to find how many full cycles fit into the total work, then handle the remaining work day by day.

Answer 1

The Correct Option is D

Step 1: Find the total work and individual rates. Let the total work be LCM(8, 12) = 24 units. Rate of P = \( \frac{24}{8} = 3 \) units/day. Rate of Q = \( \frac{24}{12} = 2 \) units/day.

Step 2: Calculate the work done in one cycle (2 days). The work is done on alternate days, starting with P. Day 1 (P works): 3 units done. Day 2 (Q works): 2 units done. In one cycle of 2 days, a total of \( 3 + 2 = 5 \) units of work is completed.

Step 3: Find how many full cycles are needed. Total work is 24 units. Number of cycles = \( \lfloor \frac{24}{5} \rfloor = 4 \) full cycles. Work completed in 4 cycles = \( 4 \times 5 = 20 \) units. Time taken for 4 cycles = \( 4 \times 2 = 8 \) days.

Step 4: Calculate the remaining work and the time to complete it. Remaining work = \( 24 - 20 = 4 \) units. After 8 days, it is P's turn to work (on the 9th day). P can do 3 units in one day. So, on the 9th day, P completes 3 units. Work remaining after 9 days = \( 4 - 3 = 1 \) unit. Time taken so far = 9 days. Now it is Q's turn to work (on the 10th day). Q's rate is 2 units/day. Time for Q to do the remaining 1 unit of work = \( \frac{\text{Remaining Work}}{\text{Q's Rate}} = \frac{1}{2} \) of a day. Wait, let me recheck. P:3, Q:2. Cycle=5 units in 2 days. Total=24. 4 cycles \textrightarrow 20 units in 8 days. Remaining = 4. Day 9 (P's turn): P does 3 units. Remaining = 1. Time = 9 days. Day 10 (Q's turn): Q needs to do 1 unit. Time = 1/2 day. Total time = 9.5 days. This is option (B). Let's re-read the question. P starts. P(8), Q(12). Day 1: P (3) Day 2: Q (2) \textrightarrow Total 5 ... Day 8: Q \textrightarrow Total 20. Day 9: P (3) \textrightarrow Total 23. Remaining work = 1 unit. Day 10: Q's turn. Q's rate is 2 units/day. Time to do 1 unit = 1/2 day. Total time = 9.5 days. Let me check the other option \(9\frac{2}{3}\). Remaining 1 unit. Q does it. Time = 1/2 day. Total 9.5 Where could \(9\frac{2}{3}\) come from? Maybe if P had to do the last bit. Let's say after 9 days (23 units done), Q works. Remaining work is 1. Q's rate is 2. Time is 1/2. Maybe I miscalculated the rates. P=1/8, Q=1/12. Work = 1. 2 days work = 1/8 + 1/12 = (3+2)/24 = 5/24. In 8 days (4 cycles), work done = 4 (5/24) = 20/24. Remaining = 4/24 = 1/6. Day 9 (P's turn): P's rate is 1/8. \(1/8>1/6\) is false. P can complete his work. Remaining after P works: \(1/6 - 1/8 = (4-3)/24 = 1/24\). This means P works for the full 9th day. Work left is 1/24. Day 10 (Q's turn): Q's rate is 1/12. Time to do 1/24 work = \( (1/24) / (1/12) = 1/2 \). Total time = 9.5 days. The answer seems to be consistently 9.5 days. Let's re-examine \(9\frac{2}{3}\). This means after 9 days, 2/3 of a day is needed. Whose turn? Q's turn. Work done by Q in 2/3 day = \( \frac{2}{3} \times \frac{1}{12} = \frac{1}{18} \). The math does not support \(9\frac{2}{3}\). There must be an error in the provided options/answer. My calculation points to 9.5 days.