Malini can do a piece of work in 20 days, while Shalini can do it in 25 days. Shalini started the work and after 5 days Malini joined her.They together completed the remaining work. How ,many days did they take to complete the whole work?

Options

A. `8(1)/(8)`
B. 8
C. `8(8)/(9)`
D. `13(8)/(9)`

Explanation

To solve the problem step by step, we will first determine the work done by Malini and Shalini individually, then calculate the work done by Shalini in the first 5 days, and finally find out how long it takes for both of them to complete the remaining work together.

### Step 1: Determine the work done by Malini and Shalini in one day.
- Malini can complete the work in 20 days. Therefore, her one day work is:
\[ \text{Malini's one day work} = \frac{1}{20} \text{ of the work} \]
- Shalini can complete the work in 25 days. Therefore, her one day work is:
\[ \text{Shalini's one day work} = \frac{1}{25} \text{ of the work} \]

### Step 2: Calculate the total work done by each in one day.
To make calculations easier, let's consider the total work as 100 units.
- Malini's one day work:
\[ \text{Malini's one day work} = \frac{100}{20} = 5 \text{ units} \]
- Shalini's one day work:
\[ \text{Shalini's one day work} = \frac{100}{25} = 4 \text{ units} \]

### Step 3: Calculate the work done by Shalini in the first 5 days.
Since Shalini works alone for the first 5 days:
\[ \text{Work done by Shalini in 5 days} = 5 \times 4 = 20 \text{ units} \]

### Step 4: Calculate the remaining work after 5 days.
The total work is 100 units, and Shalini has completed 20 units. Therefore, the remaining work is:
\[ \text{Remaining work} = 100 - 20 = 80 \text{ units} \]

### Step 5: Calculate the combined work done by Malini and Shalini in one day.
When Malini joins Shalini, they work together:
\[ \text{Combined one day work} = \text{Malini's one day work} + \text{Shalini's one day work} = 5 + 4 = 9 \text{ units} \]

### Step 6: Calculate the number of days required to complete the remaining work.
To find out how many days they need to finish the remaining 80 units:
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Combined one day work}} = \frac{80}{9} \approx 8.89 \text{ days} \text{ or } 8 \frac{8}{9} \text{ days} \]

### Step 7: Calculate the total time taken to complete the work.
The total time taken is the time Shalini worked alone plus the time they worked together:
\[ \text{Total time} = 5 \text{ days} + \frac{80}{9} \text{ days} = 5 + 8 \frac{8}{9} = 13 \frac{8}{9} \text{ days} \]

### Final Answer:
Thus, the total number of days taken to complete the whole work is:
\[ \text{Total days} = 13 \frac{8}{9} \text{ days} \]

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