Question

P and Q can complete a job in 24 days working together. P can alone complete it in 32 days. Both of them worked together for 8 days and then P left. The number of days Q will take to complete the remaining job is:

• A. 26 days
• B. 30 days
• C. 64 days
• D. 60 days

Hint:

An alternative approach is using the "Total Work" unit method. Let Total Work be the LCM of 24 and 32, which is 96 units. Efficiency of (P+Q) = 96/24 = 4 units/day. Efficiency of P = 96/32 = 3 units/day. Efficiency of Q = 4 - 3 = 1 unit/day. Work done in 8 days = 4 units/day * 8 days = 32 units. Remaining work = 96 - 32 = 64 units. Time for Q to finish = Remaining work / Efficiency of Q = 64 / 1 = 64 days.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a work and time problem. The standard approach is to work with the rate of work (amount of job done per day).
Rate of work = \( \frac{1}{\text{Time taken to complete the job}} \)
Step 2: Key Formula or Approach:
1. Find the rate of work for P and Q together (\(R_{P+Q}\)).
2. Find the rate of work for P alone (\(R_P\)).
3. Calculate the rate of work for Q alone (\(R_Q = R_{P+Q} - R_P\)).
4. Calculate the amount of work done by P and Q in 8 days.
5. Calculate the remaining work.
6. Calculate the time taken by Q to complete the remaining work.
Step 3: Detailed Explanation:
Rate of P and Q together, \(R_{P+Q} = \frac{1}{24}\) of the job per day.
Rate of P alone, \(R_P = \frac{1}{32}\) of the job per day.
Rate of Q alone, \(R_Q = R_{P+Q} - R_P = \frac{1}{24} - \frac{1}{32}\).
To subtract the fractions, find a common denominator. The LCM of 24 and 32 is 96.
\(R_Q = \frac{4}{96} - \frac{3}{96} = \frac{1}{96}\). So, Q alone can complete the job in 96 days.
Now, P and Q work together for 8 days.
Work done in 8 days = \(R_{P+Q} \times 8 = \frac{1}{24} \times 8 = \frac{8}{24} = \frac{1}{3}\) of the job.
Remaining work = \(1 - \text{Work done} = 1 - \frac{1}{3} = \frac{2}{3}\) of the job.
Now, P leaves and Q has to complete the remaining work (\(\frac{2}{3}\) of the job).
Time taken by Q = \( \frac{\text{Remaining Work}}{R_Q} \)
Time = \( \frac{2/3}{1/96} = \frac{2}{3} \times 96 \)
Time = \( 2 \times \frac{96}{3} = 2 \times 32 = 64 \) days.
Step 4: Final Answer:
Q will take 64 days to complete the remaining job.