P can complete a work in 12 days working 8 hours a day. Q can complete the same work in 8 days working 10 hours a day. If both P and Q work together, working 8 hours a day, in how man days can they complete the work ?

Options

A. `5 5/(11)`
B. `5 6/(11)`
C. `6 5/(11)`
D. `6 6/(11)`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the total work done by P and Q in hours.
- P can complete the work in 12 days, working 8 hours a day.
- Total hours worked by P = 12 days * 8 hours/day = 96 hours.

- Q can complete the work in 8 days, working 10 hours a day.
- Total hours worked by Q = 8 days * 10 hours/day = 80 hours.

### Step 2: Determine the work done by P and Q in one hour.
- Work done by P in one hour = 1/96 (since he completes the work in 96 hours).
- Work done by Q in one hour = 1/80 (since he completes the work in 80 hours).

### Step 3: Calculate the combined work done by P and Q in one hour.
- Combined work done by P and Q in one hour = (1/96 + 1/80).
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 96 and 80 is 480.

Calculating the fractions:
- Work done by P in one hour = 1/96 = 5/480.
- Work done by Q in one hour = 1/80 = 6/480.

Now, adding these:
- Combined work = (5/480 + 6/480) = 11/480.

### Step 4: Calculate the total time taken to complete the work when they work together for 8 hours a day.
- In one hour, P and Q together complete 11/480 of the work.
- In 8 hours, they complete:
\[ 8 \times \frac{11}{480} = \frac{88}{480} = \frac{11}{60} \text{ of the work.} \]

### Step 5: Calculate the total days required to complete the work.
- To find out how many days it will take to complete the entire work (1 unit of work), we set up the equation:
\[ \text{Total days} = \frac{1 \text{ (total work)}}{\frac{11}{60} \text{ (work done in one day)}} \]
\[ \text{Total days} = \frac{60}{11} \text{ days.} \]

### Step 6: Convert the total days into mixed fraction.
- Dividing 60 by 11 gives us 5 with a remainder of 5.
- Therefore, the answer in mixed fraction form is:
\[ 5 \frac{5}{11} \text{ days.} \]

### Final Answer:
P and Q can complete the work together in \( \frac{60}{11} \) days or \( 5 \frac{5}{11} \) days.

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