A takes as much time as B and C together to complete a work. While A and B together can do this piece of work in 10 days. C alone can do this piece of work in 50 days. In how many days will this work be completed by B alone?

Answer

To solve the problem step by step, we will use the information given in the question about the work done by A, B, and C.

### Step 1: Understand the relationships between A, B, and C
We know that:
- A takes as much time as B and C together to complete the work.
- A and B together can complete the work in 10 days.
- C alone can complete the work in 50 days.

### Step 2: Determine the work rates
Let's denote the total work as 1 unit of work.

- The work rate of C (C's work rate) is:
\[ \text{Work rate of C} = \frac{1}{50} \text{ (since C completes the work in 50 days)} \]

- If A and B together can complete the work in 10 days, their combined work rate is:
\[ \text{Work rate of A + B} = \frac{1}{10} \]

### Step 3: Express A's work rate in terms of B's work rate
Let B's work rate be \( b \) (work done by B in one day), and A's work rate be \( a \).

From the information given:
- A's work rate can be expressed as:
\[ a = b + \frac{1}{50} \text{ (since A takes as much time as B and C together)} \]

### Step 4: Set up the equation for A and B's work rate
We know:
\[ a + b = \frac{1}{10} \]

### Step 5: Substitute A's work rate in the equation
Substituting \( a \) in the equation:
\[ (b + \frac{1}{50}) + b = \frac{1}{10} \]
This simplifies to:
\[ 2b + \frac{1}{50} = \frac{1}{10} \]

### Step 6: Solve for B's work rate
To solve for \( b \), first, convert \( \frac{1}{10} \) to a fraction with a denominator of 50:
\[ \frac{1}{10} = \frac{5}{50} \]
Now we have:
\[ 2b + \frac{1}{50} = \frac{5}{50} \]
Subtract \( \frac{1}{50} \) from both sides:
\[ 2b = \frac{5}{50} - \frac{1}{50} = \frac{4}{50} = \frac{2}{25} \]
Now divide both sides by 2:
\[ b = \frac{2}{25} \times \frac{1}{2} = \frac{1}{25} \]

### Step 7: Calculate the time taken by B to complete the work alone
Since B's work rate is \( \frac{1}{25} \), it means B can complete the work in:
\[ \text{Time taken by B} = \frac{1}{b} = \frac{1}{\frac{1}{25}} = 25 \text{ days} \]

### Final Answer
B alone will complete the work in **25 days**.

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