A can do a work in 12 days and B can do it in 16 days. A and B started the work jointly and A left 2 days before the work is finished. Find the number of days they took to finish the work.

Options

A. 6 days
B. 7 days
C. 9 days
D. 8 days

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 12 days, so A's work rate is \( \frac{1}{12} \) of the work per day.
- B can complete the work in 16 days, so B's work rate is \( \frac{1}{16} \) of the work per day.

### Step 2: Find the combined work rate of A and B.
- To find the combined work rate, we need to find a common denominator. The LCM of 12 and 16 is 48.
- A's work in 48 hours: \( \frac{48}{12} = 4 \) units of work.
- B's work in 48 hours: \( \frac{48}{16} = 3 \) units of work.
- Therefore, together they can do \( 4 + 3 = 7 \) units of work in one day.

### Step 3: Determine the total work to be done.
- The total work is equivalent to 48 units (as calculated from the LCM).

### Step 4: Calculate the time taken if A leaves 2 days before the work is finished.
- Let the total number of days taken to finish the work be \( x \).
- Since A leaves 2 days before the work is finished, A works for \( x - 2 \) days, while B works for the entire \( x \) days.

### Step 5: Set up the equation for total work done.
- Work done by A in \( x - 2 \) days: \( 4(x - 2) \)
- Work done by B in \( x \) days: \( 3x \)
- Total work equation:
\[ 4(x - 2) + 3x = 48 \]

### Step 6: Solve the equation.
1. Expand the equation:
\[ 4x - 8 + 3x = 48 \]
2. Combine like terms:
\[ 7x - 8 = 48 \]
3. Add 8 to both sides:
\[ 7x = 56 \]
4. Divide by 7:
\[ x = 8 \]

### Conclusion:
- The total number of days taken to finish the work is **8 days**.

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