Pooja and Ritu can do a piece of work in `17.(1)/(7)` 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 reasoning provided in the video transcript:

### Step 1: Understand the problem
Pooja and Ritu can complete a piece of work together in \( 17 \frac{1}{7} \) days. We need to find out how many days each of them will take to complete the work alone.

### Step 2: Convert mixed fraction to improper fraction
First, we convert \( 17 \frac{1}{7} \) into an improper fraction:
\[ 17 \frac{1}{7} = \frac{17 \times 7 + 1}{7} = \frac{119 + 1}{7} = \frac{120}{7} \text{ days} \]

### Step 3: Define variables for individual work
Let:
- \( X \) = number of days Pooja takes to complete the work alone
- \( Y \) = number of days Ritu takes to complete the work alone

### Step 4: Write the equations based on work done
Since they can complete the work together in \( \frac{120}{7} \) days, their combined work done in one day is:
\[ \text{Work done in 1 day} = \frac{1}{\frac{120}{7}} = \frac{7}{120} \]

### Step 5: Express individual work rates
The work done by Pooja in one day is \( \frac{1}{X} \) and by Ritu in one day is \( \frac{1}{Y} \). Thus, we can write:
\[ \frac{1}{X} + \frac{1}{Y} = \frac{7}{120} \]

### Step 6: Use the relationship between their work rates
According to the problem, Pooja's one day work is \( \frac{3}{4} \) of Ritu's one day work:
\[ \frac{1}{X} = \frac{3}{4} \cdot \frac{1}{Y} \]
From this, we can express \( \frac{1}{X} \) in terms of \( Y \):
\[ \frac{1}{X} = \frac{3}{4Y} \]

### Step 7: Substitute into the combined work equation
Substituting \( \frac{1}{X} \) into the combined work equation:
\[ \frac{3}{4Y} + \frac{1}{Y} = \frac{7}{120} \]
Combining the left side:
\[ \frac{3 + 4}{4Y} = \frac{7}{4Y} = \frac{7}{120} \]

### Step 8: Cross-multiply to solve for \( Y \)
Cross-multiplying gives:
\[ 7 \cdot 120 = 7 \cdot 4Y \]
Cancelling \( 7 \) from both sides:
\[ 120 = 4Y \]
Dividing both sides by \( 4 \):
\[ Y = 30 \text{ days} \]

### Step 9: Find \( X \)
Now, substitute \( Y \) back to find \( X \):
\[ \frac{1}{X} = \frac{3}{4Y} = \frac{3}{4 \cdot 30} = \frac{3}{120} = \frac{1}{40} \]
Thus,
\[ X = 40 \text{ days} \]

### Conclusion
Pooja takes 40 days to complete the work alone, and Ritu takes 30 days to complete the work alone.