P can complete a work in 15 days and Q in 24 days. They began the work together, but Q left the work 2 days before its completion. In how many days, was the work completed?

Options

A. 8
B. 10
C. 9
D. 11

Answer

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

### Step 1: Determine the work done by P and Q in one day
- P can complete the work in 15 days, so the work done by P in one day is:
\[ \text{Work done by P in one day} = \frac{1}{15} \text{ of the work} \]
- Q can complete the work in 24 days, so the work done by Q in one day is:
\[ \text{Work done by Q in one day} = \frac{1}{24} \text{ of the work} \]

### Step 2: Find the combined work done by P and Q in one day
- To find the combined work done by P and Q in one day, we can add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{15} + \frac{1}{24} \]
- To add these fractions, we need a common denominator. The LCM of 15 and 24 is 120. Thus:
\[ \frac{1}{15} = \frac{8}{120} \quad \text{and} \quad \frac{1}{24} = \frac{5}{120} \]
- Therefore,
\[ \text{Combined work rate} = \frac{8}{120} + \frac{5}{120} = \frac{13}{120} \]

### Step 3: Calculate the total work done before Q leaves
- Let \( x \) be the number of days they worked together before Q leaves. Since Q leaves 2 days before the work is completed, P works alone for 2 days after Q leaves.
- In \( x \) days, the total work done by both P and Q is:
\[ \text{Work done together} = x \times \frac{13}{120} \]
- In the 2 days that P works alone, the work done is:
\[ \text{Work done by P alone} = 2 \times \frac{1}{15} = \frac{2}{15} \]

### Step 4: Set up the equation for total work
- The total work is equal to 1 (the whole work), so we can set up the equation:
\[ x \times \frac{13}{120} + \frac{2}{15} = 1 \]

### Step 5: Solve for \( x \)
- First, convert \(\frac{2}{15}\) to a fraction with a denominator of 120:
\[ \frac{2}{15} = \frac{16}{120} \]
- Now, substitute back into the equation:
\[ x \times \frac{13}{120} + \frac{16}{120} = 1 \]
- Multiply through by 120 to eliminate the denominator:
\[ 13x + 16 = 120 \]
- Rearranging gives:
\[ 13x = 120 - 16 = 104 \]
- Solving for \( x \):
\[ x = \frac{104}{13} = 8 \]

### Step 6: Calculate the total time taken to complete the work
- Since they worked together for 8 days and P worked alone for 2 days:
\[ \text{Total time} = 8 + 2 = 10 \text{ days} \]

### Final Answer:
The work was completed in **10 days**.

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