Shahu can do a work in 18 days, Yash can do the same work
in 24 days and Dixit can do the whole work in 36 days. If
Shahu & Yash work for first A days together after that Dixit
also joined them, remaining work is completed in `(A+4 4/5)`
days. Find for how many days all three worked together?

Options

A. `6 4/5 days`
B. `5 4/5 days`
C. `4 4/5 days`
D. `8 4/5 days`

Explanation

To solve the problem step by step, we will first determine the work efficiency of each person, then calculate the total work done, and finally find out how many days all three worked together.

### Step 1: Calculate the work efficiency of each person
- Shahu can complete the work in 18 days. Therefore, his work efficiency is:
\[ \text{Efficiency of Shahu} = \frac{1}{18} \text{ (work per day)} \]

- Yash can complete the work in 24 days. Therefore, his work efficiency is:
\[ \text{Efficiency of Yash} = \frac{1}{24} \text{ (work per day)} \]

- Dixit can complete the work in 36 days. Therefore, his work efficiency is:
\[ \text{Efficiency of Dixit} = \frac{1}{36} \text{ (work per day)} \]

### Step 2: Find the combined efficiency of Shahu and Yash
When Shahu and Yash work together, their combined efficiency is:
\[ \text{Combined efficiency of Shahu and Yash} = \frac{1}{18} + \frac{1}{24} \]
To add these fractions, we need a common denominator. The LCM of 18 and 24 is 72.
\[ \frac{1}{18} = \frac{4}{72}, \quad \frac{1}{24} = \frac{3}{72} \]
Thus,
\[ \text{Combined efficiency} = \frac{4}{72} + \frac{3}{72} = \frac{7}{72} \text{ (work per day)} \]

### Step 3: Calculate the work done by Shahu and Yash in A days
If Shahu and Yash work together for A days, the total work done by them is:
\[ \text{Work done by Shahu and Yash in A days} = \text{Combined efficiency} \times A = \frac{7}{72} \times A \]

### Step 4: Calculate the remaining work after A days
The total work is considered as 1 unit. Therefore, the remaining work after A days is:
\[ \text{Remaining work} = 1 - \frac{7}{72} \times A \]

### Step 5: Calculate the combined efficiency of all three working together
When all three work together, their combined efficiency is:
\[ \text{Combined efficiency of Shahu, Yash, and Dixit} = \frac{1}{18} + \frac{1}{24} + \frac{1}{36} \]
Finding a common denominator (LCM of 18, 24, and 36 is 72):
\[ \frac{1}{18} = \frac{4}{72}, \quad \frac{1}{24} = \frac{3}{72}, \quad \frac{1}{36} = \frac{2}{72} \]
Thus,
\[ \text{Combined efficiency} = \frac{4}{72} + \frac{3}{72} + \frac{2}{72} = \frac{9}{72} = \frac{1}{8} \text{ (work per day)} \]

### Step 6: Calculate the time taken to complete the remaining work
The remaining work is completed in \( A + 4 \frac{4}{5} \) days, which can be converted to an improper fraction:
\[ 4 \frac{4}{5} = \frac{24}{5} \implies A + 4 \frac{4}{5} = A + \frac{24}{5} = \frac{5A + 24}{5} \]
The work done in this time is:
\[ \text{Work done by all three} = \text{Combined efficiency} \times \text{Time} = \frac{1}{8} \times \frac{5A + 24}{5} \]

### Step 7: Set up the equation
The remaining work can be expressed as:
\[ 1 - \frac{7}{72}A = \frac{1}{8} \times \frac{5A + 24}{5} \]

### Step 8: Solve the equation
Cross-multiplying gives:
\[ (1 - \frac{7}{72}A) \times 40 = 5A + 24 \]
Expanding and simplifying:
\[ 40 - \frac{280}{72}A = 5A + 24 \]
Multiplying through by 72 to eliminate the fraction:
\[ 2880 - 280A = 360A + 1728 \]
Combining like terms:
\[ 2880 - 1728 = 360A + 280A \]
\[ 1152 = 640A \implies A = \frac{1152}{640} = \frac{36}{20} = \frac{9}{5} = 1.8 \]

### Step 9: Calculate the total days all three worked together
Now substituting A back into the time they worked together:
\[ A + 4 \frac{4}{5} = \frac{9}{5} + \frac{24}{5} = \frac{33}{5} = 6.6 \text{ days} \]

### Final Answer
Thus, the total number of days all three worked together is:
\[ \text{Total days} = 6.6 \text{ days} \]