Pooja and Ritu can do a piece of work in 174 days. If one day work of Pooja be three fourth of one day work of Ritu, find in how many days each will do the work alone.

Answer

To solve the problem step by step, we will follow the information given and derive the required values.

### Step 1: Understand the relationship between Pooja and Ritu's work
We know that Pooja's one day work is three-fourths of Ritu's one day work. Let's denote:
- Ritu's one day work = R
- Pooja's one day work = P

From the information given, we can write:
\[ P = \frac{3}{4}R \]

### Step 2: Calculate the combined work done by Pooja and Ritu
Pooja and Ritu can complete the work together in 174 days. Therefore, their combined one day work is:
\[ P + R = \frac{1}{174} \] (since they complete the work in 174 days)

### Step 3: Substitute P in terms of R
Substituting \( P \) from Step 1 into the equation from Step 2:
\[ \frac{3}{4}R + R = \frac{1}{174} \]

### Step 4: Combine the terms
Combining the terms on the left side:
\[ \frac{3}{4}R + \frac{4}{4}R = \frac{7}{4}R = \frac{1}{174} \]

### Step 5: Solve for R
Now, we can solve for \( R \):
\[ R = \frac{1}{174} \times \frac{4}{7} \]
\[ R = \frac{4}{1218} \]
\[ R = \frac{2}{609} \]

### Step 6: Calculate P
Now substitute \( R \) back to find \( P \):
\[ P = \frac{3}{4}R = \frac{3}{4} \times \frac{2}{609} = \frac{6}{2436} = \frac{1}{406} \]

### Step 7: Find the number of days each will take to complete the work alone
Now we can find the number of days each will take to complete the work alone:
- Days taken by Ritu:
\[ \text{Days by Ritu} = \frac{1}{R} = \frac{1}{\frac{2}{609}} = \frac{609}{2} = 304.5 \text{ days} \]

- Days taken by Pooja:
\[ \text{Days by Pooja} = \frac{1}{P} = \frac{1}{\frac{1}{406}} = 406 \text{ days} \]

### Final Answer
- Pooja will take **406 days** to complete the work alone.
- Ritu will take **304.5 days** to complete the work alone.

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