A can do a certain work in the same time as B and C together can do. If A and B together can do the work in 8 days and C alone can do the same work in 40 days, in how many days will Balone do the same work?

Options

A. 25 days
B. 30 days
C. 20 days
D. 15 days

Answer

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

### Step 1: Understand the given information
- A and B together can complete the work in 8 days.
- C alone can complete the work in 40 days.
- A can do the work in the same time as B and C together.

### Step 2: Calculate the work done by A and B together
If A and B can complete the work in 8 days, then their combined work rate is:
\[ \text{Work Rate of A and B} = \frac{1}{8} \text{ (work per day)} \]

### Step 3: Calculate the work done by C alone
C can complete the work in 40 days, so C's work rate is:
\[ \text{Work Rate of C} = \frac{1}{40} \text{ (work per day)} \]

### Step 4: Determine the work rate of A
Since A can do the work in the same time as B and C together, we can express the work rate of A as:
\[ \text{Work Rate of A} = \text{Work Rate of B} + \text{Work Rate of C} \]

### Step 5: Set up the equation
Let the work rate of B be \( b \). From the previous steps, we have:
\[ \text{Work Rate of A} = b + \frac{1}{40} \]
And we also know:
\[ \text{Work Rate of A} + \text{Work Rate of B} = \frac{1}{8} \]
Substituting the expression for A's work rate:
\[ (b + \frac{1}{40}) + b = \frac{1}{8} \]
This simplifies to:
\[ 2b + \frac{1}{40} = \frac{1}{8} \]

### Step 6: Solve for B's work rate
To solve for \( b \), first, we need a common denominator. The least common multiple of 40 and 8 is 40. Rewrite the equation:
\[ 2b + \frac{1}{40} = \frac{5}{40} \]
Subtract \( \frac{1}{40} \) from both sides:
\[ 2b = \frac{5}{40} - \frac{1}{40} = \frac{4}{40} = \frac{1}{10} \]
Now divide both sides by 2:
\[ b = \frac{1}{20} \]

### Step 7: Calculate the time taken by B alone
The work rate of B is \( \frac{1}{20} \), which means B can complete the work alone in:
\[ \text{Time taken by B} = \frac{1}{b} = \frac{1}{\frac{1}{20}} = 20 \text{ days} \]

### Final Answer
B alone will take **20 days** to complete the work.

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