A and B together can dig a trench in 12 days, which A alone can dig in 28 days , B alone can dig it in.
A और B मिलकर एक खाई को 42 दिन में खोद सकते हैं, जिसे A अकेला 28 दिन में खोद सकता है। B अकेला उसे कितने दिन में खोद्‌ सकता है?

Options

A. A)20 days
B. B)21 days
C. C)22 days
D. D)23 days

Explanation

To find out how many days B alone can dig the trench, we can use the concept of work rates. Here's the step-by-step solution:

### Step 1: Determine the work rates of A and B

1. **Work rate of A**:
- A can dig the trench in 28 days.
- Therefore, A's work rate is \( \frac{1}{28} \) of the trench per day.

2. **Work rate of A and B together**:
- A and B together can dig the trench in 12 days.
- Therefore, their combined work rate is \( \frac{1}{12} \) of the trench per day.

### Step 2: Set up the equation for B's work rate

Let B's work rate be \( \frac{1}{x} \) of the trench per day, where \( x \) is the number of days B takes to dig the trench alone.

From the information given, we can write the equation for their combined work rate:

\[ \text{Work rate of A} + \text{Work rate of B} = \text{Combined work rate} \]

Substituting the values we have:

\[ \frac{1}{28} + \frac{1}{x} = \frac{1}{12} \]

### Step 3: Solve for B's work rate

To solve for \( \frac{1}{x} \), we first isolate it:

\[ \frac{1}{x} = \frac{1}{12} - \frac{1}{28} \]

### Step 4: Find a common denominator

The least common multiple of 12 and 28 is 84. We can convert both fractions to have a common denominator:

\[ \frac{1}{12} = \frac{7}{84} \quad \text{and} \quad \frac{1}{28} = \frac{3}{84} \]

Now substituting these values back into the equation:

\[ \frac{1}{x} = \frac{7}{84} - \frac{3}{84} = \frac{4}{84} \]

### Step 5: Simplify the fraction

\[ \frac{1}{x} = \frac{1}{21} \]

### Step 6: Solve for x

Taking the reciprocal gives us:

\[ x = 21 \]

### Conclusion

B alone can dig the trench in **21 days**.

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