A and B working together can complete 45% of a work in 18 days. A alone can complete the same work in 60 days. A and B work together for 16 days, and then A leaves. B alone will complete the remaining work in

Options

A. 80 days
B. 60 days
C. 72 days
D. 75 days

Answer

To solve the problem step by step, we will first determine the work rates of A and B, and then find out how long B will take to complete the remaining work after A leaves.

### Step 1: Determine the work rate of A and B together
A and B together can complete 45% of the work in 18 days.

To find their combined work rate:
- Work completed in 18 days = 45% of the work
- Therefore, in 1 day, A and B together complete:
\[ \text{Work rate of A and B} = \frac{45\%}{18} = \frac{0.45}{18} = 0.025 \text{ (work per day)} \]

### Step 2: Determine the work rate of A alone
A can complete the entire work in 60 days.

Thus, A's work rate is:
\[ \text{Work rate of A} = \frac{100\%}{60} = \frac{1}{60} \text{ (work per day)} \]

### Step 3: Determine the work rate of B
Let the work rate of B be \( b \).

From the combined work rate of A and B:
\[ \text{Work rate of A} + \text{Work rate of B} = \text{Work rate of A and B} \]
\[ \frac{1}{60} + b = 0.025 \]

To find \( b \):
\[ b = 0.025 - \frac{1}{60} \]
Convert \( \frac{1}{60} \) to a decimal:
\[ \frac{1}{60} \approx 0.01667 \]
So,
\[ b = 0.025 - 0.01667 = 0.00833 \text{ (work per day)} \]

### Step 4: Calculate the total work
Since A and B together can complete 100% of the work in:
\[ \text{Total work} = \frac{1}{\text{Work rate of A and B}} = \frac{1}{0.025} = 40 \text{ days} \]

### Step 5: Calculate the work done by A and B in 16 days
In 16 days, A and B together will complete:
\[ \text{Work done in 16 days} = 16 \times 0.025 = 0.4 \text{ (or 40% of the work)} \]

### Step 6: Calculate the remaining work
The remaining work after 16 days is:
\[ \text{Remaining work} = 100\% - 40\% = 60\% \]

### Step 7: Calculate how long B will take to complete the remaining work
B's work rate is \( 0.00833 \) (work per day). To find the time \( t \) taken by B to complete the remaining 60% of the work:
\[ t = \frac{\text{Remaining work}}{\text{Work rate of B}} = \frac{0.6}{0.00833} \approx 72 \text{ days} \]

### Final Answer
B alone will complete the remaining work in approximately **72 days**.

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