A can do a piece of work in 10 days while B can do it in 15 days. They begin together but 5 days before the completion of the work, B leaves off. Find the total number of days for the work to be completed .

Answer

To solve the problem step by step, we need to find out how long A and B worked together and how long A worked alone after B left.

### Step 1: Determine the work rates of A and B.
- A can complete the work in 10 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{10} \text{ of the work per day} \]
- B can complete the work in 15 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{15} \text{ of the work per day} \]

### Step 2: Find the combined work rate of A and B.
- To find the combined work rate, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{10} + \frac{1}{15} \]
- To add these fractions, we find a common denominator, which is 30:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
- Therefore,
\[ \text{Combined work rate} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \text{ of the work per day} \]

### Step 3: Let’s denote the number of days they worked together as \(X\).
- In \(X\) days, the amount of work they complete together is:
\[ \text{Work done in } X \text{ days} = \frac{1}{6} \times X = \frac{X}{6} \]

### Step 4: Determine the work left after \(X\) days.
- The total work is 1 (the whole work). The work left after \(X\) days is:
\[ \text{Work left} = 1 - \frac{X}{6} \]

### Step 5: Calculate how long A works alone after B leaves.
- B leaves 5 days before the work is completed, which means A works alone for 5 days. In 5 days, A can complete:
\[ \text{Work done by A in 5 days} = \frac{1}{10} \times 5 = \frac{5}{10} = \frac{1}{2} \]

### Step 6: Set up the equation for the work left.
- The work left after \(X\) days is equal to the work A does in 5 days:
\[ 1 - \frac{X}{6} = \frac{1}{2} \]

### Step 7: Solve for \(X\).
- Rearranging the equation gives:
\[ 1 - \frac{1}{2} = \frac{X}{6} \]
\[ \frac{1}{2} = \frac{X}{6} \]
- Cross-multiplying gives:
\[ X = 3 \]

### Step 8: Calculate the total number of days for the work to be completed.
- They worked together for \(X = 3\) days, and then A worked alone for 5 days:
\[ \text{Total days} = X + 5 = 3 + 5 = 8 \]

### Final Answer:
The total number of days for the work to be completed is **8 days**.

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