A takes 5 hours more time than that taken by B to complete a work. If working together they can complete a work in 6 hours, then the number of hours that takes to complete the work individually is

Options

A. 15
B. 12
C. 10
D. 9

Answer

To solve the problem step by step, we will denote the time taken by B to complete the work as \( x \) hours. Since A takes 5 hours more than B, the time taken by A will be \( x + 5 \) hours.

### Step-by-Step Solution:

1. **Define the Work Rates**:
- The work rate of B is \( \frac{1}{x} \) (work done in one hour).
- The work rate of A is \( \frac{1}{x + 5} \).

2. **Combined Work Rate**:
- When A and B work together, their combined work rate is:
\[ \frac{1}{x} + \frac{1}{x + 5} \]

3. **Set Up the Equation**:
- According to the problem, working together they can complete the work in 6 hours, which means their combined work rate is \( \frac{1}{6} \). Therefore, we can set up the equation:
\[ \frac{1}{x} + \frac{1}{x + 5} = \frac{1}{6} \]

4. **Find a Common Denominator**:
- The common denominator for the left side is \( x(x + 5) \). Thus, we rewrite the equation:
\[ \frac{(x + 5) + x}{x(x + 5)} = \frac{1}{6} \]
- This simplifies to:
\[ \frac{2x + 5}{x^2 + 5x} = \frac{1}{6} \]

5. **Cross Multiply**:
- Cross multiplying gives:
\[ 6(2x + 5) = x^2 + 5x \]
- Expanding this results in:
\[ 12x + 30 = x^2 + 5x \]

6. **Rearranging the Equation**:
- Rearranging all terms to one side yields:
\[ x^2 + 5x - 12x - 30 = 0 \]
- This simplifies to:
\[ x^2 - 7x - 30 = 0 \]

7. **Factoring the Quadratic**:
- We need to factor the quadratic equation. We look for two numbers that multiply to -30 and add to -7. The numbers are -10 and 3:
\[ (x - 10)(x + 3) = 0 \]

8. **Finding the Roots**:
- Setting each factor to zero gives:
\[ x - 10 = 0 \quad \text{or} \quad x + 3 = 0 \]
- Thus, \( x = 10 \) or \( x = -3 \).

9. **Selecting the Valid Solution**:
- Since time cannot be negative, we discard \( x = -3 \) and take \( x = 10 \).

10. **Calculating A's Time**:
- Since \( x = 10 \), the time taken by A is:
\[ x + 5 = 10 + 5 = 15 \text{ hours} \]

### Final Answer:
- B takes **10 hours** to complete the work individually.
- A takes **15 hours** to complete the work individually.