Anand can do a piece of work in 45 days, but Bahuguna can do the same work in 5 days less, than Anand, when working alone. Anand and Bahuguna both started the work together but Bahuguna left after some days and Anand finished the remaining work in 56 days with half of his efficiency but he did the work with Bahuguna with his complete efficiency. For how many days they had worked together?

Options

A. 6
B. 8
C. 9
D. 12

Answer

To solve the problem, we need to determine how many days Anand and Bahuguna worked together before Bahuguna left. Let's break down the solution step by step.

### Step 1: Determine the work rates of Anand and Bahuguna

- Anand can complete the work in 45 days.
- Bahuguna can complete the work in 40 days (5 days less than Anand).

The work done by each person in one day can be calculated as follows:

- Work rate of Anand = \( \frac{1}{45} \) (work per day)
- Work rate of Bahuguna = \( \frac{1}{40} \) (work per day)

### Step 2: Calculate the combined work rate of Anand and Bahuguna

When Anand and Bahuguna work together, their combined work rate is:

\[ \text{Combined work rate} = \text{Work rate of Anand} + \text{Work rate of Bahuguna} = \frac{1}{45} + \frac{1}{40} \]

To add these fractions, we need a common denominator. The least common multiple of 45 and 40 is 360.

\[ \frac{1}{45} = \frac{8}{360}, \quad \frac{1}{40} = \frac{9}{360} \]

Thus,

\[ \text{Combined work rate} = \frac{8}{360} + \frac{9}{360} = \frac{17}{360} \]

This means together they can complete \( \frac{17}{360} \) of the work in one day.

### Step 3: Determine the total work done after Bahuguna leaves

Let \( x \) be the number of days Anand and Bahuguna worked together. The amount of work they completed together in \( x \) days is:

\[ \text{Work done together} = x \times \frac{17}{360} \]

After Bahuguna leaves, Anand finishes the remaining work in 56 days at half his efficiency. His work rate at half efficiency is:

\[ \text{Half efficiency of Anand} = \frac{1}{2} \times \frac{1}{45} = \frac{1}{90} \]

The amount of work Anand completes in 56 days at half efficiency is:

\[ \text{Work done by Anand alone} = 56 \times \frac{1}{90} = \frac{56}{90} = \frac{28}{45} \]

### Step 4: Set up the equation for total work

The total work is equal to the work done together plus the work done by Anand alone:

\[ 1 = x \times \frac{17}{360} + \frac{28}{45} \]

### Step 5: Solve for \( x \)

First, convert \( \frac{28}{45} \) to a fraction with a denominator of 360:

\[ \frac{28}{45} = \frac{28 \times 8}{45 \times 8} = \frac{224}{360} \]

Now substitute this into the equation:

\[ 1 = x \times \frac{17}{360} + \frac{224}{360} \]

Multiply through by 360 to eliminate the denominator:

\[ 360 = 17x + 224 \]

Now, isolate \( x \):

\[ 17x = 360 - 224 \]
\[ 17x = 136 \]
\[ x = \frac{136}{17} = 8 \]

### Conclusion

Anand and Bahuguna worked together for **8 days**.