When A alone does a piece of work , he takes 25 days more than the time taken (A+B) to completes that particular work, while B alone takes 49 days more than the time taken by (A + B) to finish the same work. A andBtogether will take what time to finish this work?

Options

A. 35 days
B. 25 days
C. 15 days
D. 45 days

Answer

To solve the problem step by step, we need to first define the variables and understand the relationships given in the question.

### Step 1: Define the Variables
Let:
- \( x \) = the number of days taken by A and B together to complete the work.
- \( A \) = the number of days taken by A alone to complete the work.
- \( B \) = the number of days taken by B alone to complete the work.

From the problem, we know:
- \( A = x + 25 \) (A takes 25 days more than A and B together)
- \( B = x + 49 \) (B takes 49 days more than A and B together)

### Step 2: Use the Work Formula
The work done by A and B together can be expressed in terms of their individual work rates. The work done in one day by A is \( \frac{1}{A} \) and by B is \( \frac{1}{B} \). Therefore, the work done by A and B together in one day is:
\[ \frac{1}{A} + \frac{1}{B} = \frac{1}{x} \]

### Step 3: Substitute A and B
Substituting the expressions for A and B from Step 1 into the equation:
\[ \frac{1}{x + 25} + \frac{1}{x + 49} = \frac{1}{x} \]

### Step 4: Find a Common Denominator
To solve the equation, we will find a common denominator for the left-hand side:
\[ \frac{(x + 49) + (x + 25)}{(x + 25)(x + 49)} = \frac{1}{x} \]
This simplifies to:
\[ \frac{2x + 74}{(x + 25)(x + 49)} = \frac{1}{x} \]

### Step 5: Cross Multiply
Cross multiplying gives us:
\[ (2x + 74)x = (x + 25)(x + 49) \]

### Step 6: Expand Both Sides
Expanding both sides:
\[ 2x^2 + 74x = x^2 + 74x + 1225 \]

### Step 7: Simplify the Equation
Subtract \( x^2 + 74x + 1225 \) from both sides:
\[ 2x^2 + 74x - x^2 - 74x - 1225 = 0 \]
This simplifies to:
\[ x^2 - 1225 = 0 \]

### Step 8: Solve for x
Factoring gives:
\[ (x - 35)(x + 35) = 0 \]
Thus, \( x = 35 \) (we discard the negative solution as time cannot be negative).

### Conclusion
A and B together will take **35 days** to finish the work.

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