A can do a certain work in the same time in which B and C together can do it. If A and B together could do it in 10 days and C alone in 50 days, then B alone could do the work in

Options

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

Answer

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

### Step 1: Understand the given information
- A and B together can complete the work in 10 days.
- C alone can complete the work in 50 days.
- We need to find out how many days B alone would take to complete the work.

### Step 2: Calculate the work done by A and B together
If A and B can complete the work in 10 days, then the amount of work they can do in one day is:
\[ \text{Work done by A and B in one day} = \frac{1}{10} \text{ (of the total work)} \]

### Step 3: Calculate the work done by C alone
C can complete the work in 50 days, so the amount of work C can do in one day is:
\[ \text{Work done by C in one day} = \frac{1}{50} \text{ (of the total work)} \]

### Step 4: Set up the equation for A's work
Let the amount of work done by A in one day be \( a \). Therefore, the work done by A and B together can be expressed as:
\[ a + b = \frac{1}{10} \]
where \( b \) is the work done by B in one day.

### Step 5: Set up the equation for B and C's work
Since A can do the work in the same time as B and C together, we can express this as:
\[ a = b + \frac{1}{50} \]

### Step 6: Substitute the value of A's work
Now we have two equations:
1. \( a + b = \frac{1}{10} \)
2. \( a = b + \frac{1}{50} \)

Substituting the second equation into the first:
\[ (b + \frac{1}{50}) + b = \frac{1}{10} \]
\[ 2b + \frac{1}{50} = \frac{1}{10} \]

### Step 7: Solve for B's work
To eliminate the fractions, we can find a common denominator. The least common multiple of 10 and 50 is 50. Rewriting the equation:
\[ 2b + \frac{1}{50} = \frac{5}{50} \]
Subtract \( \frac{1}{50} \) from both sides:
\[ 2b = \frac{5}{50} - \frac{1}{50} \]
\[ 2b = \frac{4}{50} \]
\[ b = \frac{4}{100} = \frac{1}{25} \]

### Step 8: Calculate the time taken by B alone
Since B can do \( \frac{1}{25} \) of the work in one day, the time taken by B to complete the entire work is:
\[ \text{Time taken by B} = \frac{1}{b} = \frac{1}{\frac{1}{25}} = 25 \text{ days} \]

### Final Answer
B alone could do the work in **25 days**.

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