A man and a boy working together can complete a work in 24 days. If for the last 6 days, the man alone does the work, then it is completed in 26 days. How long will the boy take to complete the work alone ?

Options

A. 72 days
B. 73 days
C. 49 days
D. 62 days

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Determine the work done by the man and the boy together
The man and the boy together can complete the work in 24 days. This means that their combined work rate is:
\[ \text{Work rate of (Man + Boy)} = \frac{1}{24} \text{ of the work per day} \]

### Step 2: Determine the work done by the man alone in the last 6 days
According to the problem, the man works alone for the last 6 days. Therefore, the work done by the man in these 6 days is:
\[ \text{Work done by Man in 6 days} = 6 \times \text{Work rate of Man} = 6M \]

### Step 3: Determine the total work done
The total work can also be expressed in terms of the work done by both the man and the boy together for the first 20 days and the man alone for the last 6 days:
\[ \text{Total Work} = \text{Work done in 20 days by (Man + Boy)} + \text{Work done in 6 days by Man} \]
\[ \text{Total Work} = 20 \times \left(\frac{1}{24}\right) + 6M \]
\[ \text{Total Work} = \frac{20}{24} + 6M = \frac{5}{6} + 6M \]

### Step 4: Equate the total work expressions
Since the total work can also be expressed in terms of the man and boy working together for 24 days:
\[ \text{Total Work} = 1 \text{ (whole work)} \]
Setting the two expressions for total work equal gives:
\[ \frac{5}{6} + 6M = 1 \]

### Step 5: Solve for M
Rearranging the equation:
\[ 6M = 1 - \frac{5}{6} \]
\[ 6M = \frac{1}{6} \]
\[ M = \frac{1}{36} \]
This means the man can complete \(\frac{1}{36}\) of the work in one day.

### Step 6: Find the work rate of the boy
Using the combined work rate:
\[ \frac{1}{24} = M + B \]
Substituting \(M = \frac{1}{36}\):
\[ \frac{1}{24} = \frac{1}{36} + B \]
To find \(B\), we can rearrange:
\[ B = \frac{1}{24} - \frac{1}{36} \]
Finding a common denominator (72):
\[ B = \frac{3}{72} - \frac{2}{72} = \frac{1}{72} \]
This means the boy can complete \(\frac{1}{72}\) of the work in one day.

### Step 7: Calculate the time taken by the boy to complete the work alone
To find out how long it will take the boy to complete the work alone:
\[ \text{Time taken by Boy} = \frac{1}{B} = \frac{1}{\frac{1}{72}} = 72 \text{ days} \]

### Final Answer
The boy will take **72 days** to complete the work alone.

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