P alone can complete a work in 16 days and Q alone can complete the same work in 20 days. P and Q start the work together but Q leaves the work 7 days before the completion of work. In how many days the total work will be completed

Options

A. 9 days
B. 14 days
C. 12 days
D. 18 days

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the work done by P and Q
- P can complete the work in 16 days.
- Q can complete the work in 20 days.

### Step 2: Calculate the amount of work done by P and Q in one day
- Work done by P in one day = Total work / Days taken by P = 1/16 of the work.
- Work done by Q in one day = Total work / Days taken by Q = 1/20 of the work.

### Step 3: Find the LCM of the days taken by P and Q
- The LCM of 16 and 20 is 80.
- We will assume the total work is 80 units.

### Step 4: Calculate the daily work output
- Work done by P in one day = 80 / 16 = 5 units.
- Work done by Q in one day = 80 / 20 = 4 units.

### Step 5: Calculate the combined work done by P and Q in one day
- Combined work done by P and Q in one day = Work done by P + Work done by Q = 5 + 4 = 9 units.

### Step 6: Determine the total work done before Q leaves
- Let the total number of days taken to complete the work be \( x \).
- Q leaves 7 days before the work is completed, which means Q works for \( x - 7 \) days.

### Step 7: Calculate the work done by P and Q together
- Work done by P and Q together for \( x - 7 \) days = 9 * (x - 7).

### Step 8: Calculate the work done by P alone in the last 7 days
- Work done by P alone in the last 7 days = 5 * 7 = 35 units.

### Step 9: Set up the equation for total work
- Total work done = Work done by P and Q together + Work done by P alone
- 80 = 9 * (x - 7) + 35

### Step 10: Solve for \( x \)
1. Rearranging the equation:
\[ 80 = 9x - 63 + 35 \]
\[ 80 = 9x - 28 \]
\[ 9x = 80 + 28 \]
\[ 9x = 108 \]
\[ x = 12 \]

### Conclusion
The total work will be completed in **12 days**.

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