A can do a piece of work in 40 days. He works on it for 5 days and then B completes it in 21 days. How long will A and B together take to complete the work?

Options

A. 10 days
B. 15 days
C. 20 days
D. 25 days

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine A's Work Rate
A can complete the work in 40 days. Therefore, A's work rate (efficiency) is:
\[ \text{Work rate of A} = \frac{1 \text{ work}}{40 \text{ days}} = \frac{1}{40} \text{ work/day} \]

### Step 2: Calculate Work Done by A in 5 Days
In 5 days, A will complete:
\[ \text{Work done by A in 5 days} = 5 \times \frac{1}{40} = \frac{5}{40} = \frac{1}{8} \text{ of the work} \]

### Step 3: Calculate Remaining Work
The total work is considered as 1 unit. After A works for 5 days, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{8} = \frac{7}{8} \]

### Step 4: Determine B's Work Rate
Let B's work rate be \( b \). Since A and B together complete the remaining work in 21 days, we can express this as:
\[ \text{Work done by B in 21 days} = 21b \]
Since B completes the remaining work of \( \frac{7}{8} \):
\[ 21b = \frac{7}{8} \]

### Step 5: Solve for B's Work Rate
To find B's work rate, rearrange the equation:
\[ b = \frac{7}{8 \times 21} = \frac{7}{168} = \frac{1}{24} \text{ work/day} \]

### Step 6: Calculate A and B's Combined Work Rate
Now we can find the combined work rate of A and B:
\[ \text{Combined work rate} = \frac{1}{40} + \frac{1}{24} \]
To add these fractions, find a common denominator (which is 120):
\[ \frac{1}{40} = \frac{3}{120}, \quad \frac{1}{24} = \frac{5}{120} \]
Thus,
\[ \text{Combined work rate} = \frac{3}{120} + \frac{5}{120} = \frac{8}{120} = \frac{1}{15} \text{ work/day} \]

### Step 7: Calculate Time Taken by A and B Together
To find out how long A and B together will take to complete the entire work:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{1}{\frac{1}{15}} = 15 \text{ days} \]

### Final Answer
A and B together will take **15 days** to complete the work.

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