A man can do a piece of work in 5 days . With the help of his friend , he can do the same work in 3 days . In how many days can his friend complete the work?

Options

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

Answer

To solve the problem step by step, we will first determine the work done by the man and his friend and then find out how long it will take for the friend to complete the work alone.

### 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:
\[ \text{Work done by Man (M) in one day} = \frac{1}{5} \]

### Step 2: Determine the combined work done by the man and his friend in one day.
When the man works with his friend, they can complete the work in 3 days. Thus, the work done by both in one day is:
\[ \text{Combined work (M + F) in one day} = \frac{1}{3} \]

### Step 3: Set up the equation for the work done by the friend.
Let the work done by the friend in one day be \( F \). Therefore, we can write the equation:
\[ \text{Work done by Man (M)} + \text{Work done by Friend (F)} = \text{Combined work} \]
Substituting the values we have:
\[ \frac{1}{5} + F = \frac{1}{3} \]

### Step 4: Solve for \( F \).
To find \( F \), we need to isolate it in the equation:
\[ F = \frac{1}{3} - \frac{1}{5} \]
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 5 is 15. Thus, we convert the fractions:
\[ F = \frac{5}{15} - \frac{3}{15} = \frac{2}{15} \]

### Step 5: Determine how many days the friend takes to complete the work alone.
If the friend can do \( \frac{2}{15} \) of the work in one day, then the number of days \( D \) it takes for the friend to complete the work alone is the reciprocal of the work done in one day:
\[ D = \frac{1}{F} = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \]

Thus, the friend can complete the work alone in **7.5 days**.

### Summary of Steps:
1. Calculate the work done by the man in one day.
2. Calculate the combined work done by the man and his friend in one day.
3. Set up the equation to find the work done by the friend.
4. Solve for the friend's work rate.
5. Calculate the number of days the friend takes to complete the work alone.