Reshmi can do a piece of work in 16 days. Ravina can do the same work in `12 (4)/(5)`days , while gitika can do it in 32 days. All of them started to work together but reshmi leaves after 4 days. Ravina leaves the job 3 days before the completion of the work . how long would the work last ?

Options

A. 9 days
B. 6 days
C. 16 days
D. 5 days

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the work done by each person in one day.
- Reshmi can complete the work in 16 days. Therefore, her work rate is:
\[ \text{Reshmi's work rate} = \frac{1 \text{ work}}{16 \text{ days}} = \frac{1}{16} \text{ work/day} \]

- Ravina can complete the work in \(12 \frac{4}{5}\) days. First, convert \(12 \frac{4}{5}\) into an improper fraction:
\[ 12 \frac{4}{5} = \frac{12 \times 5 + 4}{5} = \frac{60 + 4}{5} = \frac{64}{5} \text{ days} \]
Therefore, her work rate is:
\[ \text{Ravina's work rate} = \frac{1 \text{ work}}{\frac{64}{5} \text{ days}} = \frac{5}{64} \text{ work/day} \]

- Gitika can complete the work in 32 days. Therefore, her work rate is:
\[ \text{Gitika's work rate} = \frac{1 \text{ work}}{32 \text{ days}} = \frac{1}{32} \text{ work/day} \]

### Step 2: Calculate the total work done by all three in one day.
Now we will add their work rates together to find the total work done by all three in one day:
\[ \text{Total work rate} = \text{Reshmi's work rate} + \text{Ravina's work rate} + \text{Gitika's work rate} \]
\[ = \frac{1}{16} + \frac{5}{64} + \frac{1}{32} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 16, 64, and 32 is 64. We convert each fraction:
\[ \frac{1}{16} = \frac{4}{64}, \quad \frac{5}{64} = \frac{5}{64}, \quad \frac{1}{32} = \frac{2}{64} \]
Now, adding them together:
\[ \text{Total work rate} = \frac{4}{64} + \frac{5}{64} + \frac{2}{64} = \frac{11}{64} \text{ work/day} \]

### Step 3: Calculate the work done in the first 4 days.
In the first 4 days, the total work done by all three is:
\[ \text{Work done in 4 days} = 4 \times \frac{11}{64} = \frac{44}{64} = \frac{11}{16} \text{ work} \]

### Step 4: Calculate the remaining work.
The total work is 1 (the whole work), so the remaining work after 4 days is:
\[ \text{Remaining work} = 1 - \frac{11}{16} = \frac{5}{16} \text{ work} \]

### Step 5: Determine how many days Ravina and Gitika work.
Ravina leaves 3 days before the completion of the work. Let \(X\) be the total number of days the work lasts. Therefore, Ravina works for \(X - 3\) days.

Since Reshmi leaves after 4 days, only Ravina and Gitika work for the remaining days. The work done by Ravina and Gitika in one day is:
\[ \text{Ravina's work rate} + \text{Gitika's work rate} = \frac{5}{64} + \frac{1}{32} = \frac{5}{64} + \frac{2}{64} = \frac{7}{64} \text{ work/day} \]

### Step 6: Set up the equation for the remaining work.
The remaining work is done by Ravina and Gitika:
\[ \text{Remaining work} = \text{Work rate} \times \text{Days worked} \]
\[ \frac{5}{16} = \frac{7}{64} \times (X - 4 - 3) \]
\[ \frac{5}{16} = \frac{7}{64} \times (X - 7) \]

### Step 7: Solve for \(X\).
Cross-multiply to solve for \(X\):
\[ 5 \times 64 = 7 \times 16 \times (X - 7) \]
\[ 320 = 112(X - 7) \]
\[ 320 = 112X - 784 \]
\[ 112X = 320 + 784 \]
\[ 112X = 1104 \]
\[ X = \frac{1104}{112} = 9.857 \text{ days} \approx 10 \text{ days} \]

### Final Answer:
The work lasts approximately **10 days**.

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