A man can do a piece of work in 5 days. He and his son working together can finish it in 3 days. In how many days can the son do it alone?

Options

A. `6(1)/(2)` days
B. 7 days
C. `7(1)/(2)` days
D. 8 days

Answer

To solve the problem, we need to determine how many days the son can complete the work alone. Let's break it down step by step.

### Step 1: Determine the work done by the man in one day
The man can complete the work in 5 days. Therefore, the work done by the man in one day is:

\[ \text{Work done by the man in one day} = \frac{1}{5} \]

### Step 2: Determine the work done by both the man and his son in one day
When the man and his son work together, they can complete the work in 3 days. Thus, the combined work done by both in one day is:

\[ \text{Work done by both in one day} = \frac{1}{3} \]

### Step 3: Set up the equation for the work done by the son
Let the work done by the son in one day be represented as \( \frac{1}{x} \), where \( x \) is the number of days the son takes to complete the work alone.

Now, we can express the total work done by both the man and the son in one day as:

\[ \text{Work done by the man} + \text{Work done by the son} = \text{Work done by both} \]

Substituting the known values:

\[ \frac{1}{5} + \frac{1}{x} = \frac{1}{3} \]

### Step 4: Solve for \( x \)
To solve for \( x \), we first need to eliminate the fractions. We can do this by finding a common denominator, which in this case is 15. Multiplying the entire equation by 15 gives:

\[ 15 \left(\frac{1}{5}\right) + 15 \left(\frac{1}{x}\right) = 15 \left(\frac{1}{3}\right) \]

This simplifies to:

\[ 3 + \frac{15}{x} = 5 \]

Next, isolate \( \frac{15}{x} \):

\[ \frac{15}{x} = 5 - 3 \]
\[ \frac{15}{x} = 2 \]

Now, cross-multiply to solve for \( x \):

\[ 15 = 2x \]
\[ x = \frac{15}{2} = 7.5 \]

### Conclusion
The son can complete the work alone in **7.5 days**.

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