Pranay, Rohit, and Qureshi bought some marbles. If Rohit gave 2 marbles to Qureshi, the ratio of the number of marbles with Pranay, Rohit, and Qureshi would become 1:3:3. Had Rohit given 2 marbles to Pranay, then the ratio of the number of marbles with them would have become 2:3:2. What will be the ratio of the number of marbles with Pranay, Rohit, and Qureshi when 4 marbles are added to the number of marbles each one of

them already had?

Let's denote the number of marbles with Pranay, Rohit, and Qureshi as \( P \), \( R \), and \( Q \) respectively.

### Step 1: Set Up Equations Based on Given Ratios

**First scenario (Rohit gives 2 marbles to Qureshi):**
After giving 2 marbles to Qureshi:
- Pranay: \( P \)
- Rohit: \( R - 2 \)
- Qureshi: \( Q + 2 \)

The new ratio becomes:
\[
\frac{P}{R - 2} = \frac{1}{3} \quad \text{and} \quad \frac{R - 2}{Q + 2} = \frac{3}{3}
\]

From the first part:
\[
3P = R - 2 \quad \Rightarrow \quad R = 3P + 2 \quad \text{(1)}
\]

From the second part:
\[
R - 2 = Q + 2 \quad \Rightarrow \quad Q = R - 4 \quad \text{(2)}
\]

**Second scenario (Rohit gives 2 marbles to Pranay):**
After giving 2 marbles to Pranay:
- Pranay: \( P + 2 \)
- Rohit: \( R - 2 \)
- Qureshi: \( Q \)

The new ratio becomes:
\[
\frac{P + 2}{R - 2} = \frac{2}{3} \quad \text{and} \quad \frac{R - 2}{Q} = \frac{3}{2}
\]

From the first part:
\[
3(P + 2) = 2(R - 2) \quad \Rightarrow \quad 3P + 6 = 2R - 4 \quad \Rightarrow \quad 2R = 3P + 10 \quad \Rightarrow \quad R = \frac{3P + 10}{2} \quad \text{(3)}
\]

From the second part:
\[
2(R - 2) = 3Q \quad \Rightarrow \quad 2R - 4 = 3Q \quad \Rightarrow \quad Q = \frac{2R - 4}{3} \quad \text{(4)}
\]

### Step 2: Solve the System of Equations

Substituting equation (1) into equations (2) and (3):
- From (2): 
\[
Q = (3P + 2) - 4 = 3P - 2
\]

- Substituting \( R \) from (1) into (3):
\[
3(3P + 2) + 10 = 2R \quad \Rightarrow \quad 9P + 6 + 10 = 2(3P + 2)
\]
\[
9P + 16 = 6P + 4 \quad \Rightarrow \quad 3P = -12 \quad \Rightarrow \quad P = -4 \quad \text{(impossible)}
\]

### Step 3: Verify Correctness and Re-setup Equations

Returning to ratios and verifying, let's isolate \( R \) and \( Q \):
Using equation (1):
Substituting \( R \) in \( Q \):
\[
Q = R - 4 = (3P + 2) - 4 = 3P - 2
\]

Using the second scenario:
Substituting into (4):
\[
Q = \frac{2(3P + 2) - 4}{3} = \frac{6P + 4 - 4}{3} = \frac{6P}{3} = 2P
\]

### Step 4: Find \( P \), \( R \), \( Q \)

Setting the equations for \( Q \):
From both equations:
\[
3P - 2 = 2P \quad \Rightarrow \quad P - 2 = 0 \quad \Rightarrow \quad P = 2
\]

Then:
From (1):
\[
R = 3(2) + 2 = 6 + 2 = 8
\]
From (2):
\[
Q = R - 4 = 8 - 4 = 4
\]

Thus:
\[
P = 2, \quad R = 8, \quad Q = 4
\]

### Step 5: Calculate Ratios After Adding 4 Marbles

Now when each has 4 additional marbles:
- Pranay: \( 2 + 4 = 6 \)
- Rohit: \( 8 + 4 = 12 \)
- Qureshi: \( 4 + 4 = 8 \)

### Final Ratio

Thus, the new ratio is:
\[
\frac{6}{12}{8} = 6:12:8 = 3:6:4
\]

### Conclusion

The final ratio of the number of marbles with Pranay, Rohit, and Qureshi after adding 4 marbles each is:
\[
3:6:4
\]

Would you like more details or have any questions? Here are some related questions:

1. What happens if Rohit gives 3 marbles to Qureshi instead?
2. How would the ratios change if Pranay had started with 5 marbles?
3. Can we find the total number of marbles they originally had?
4. What would be the ratio if all three gave away 2 marbles each?
5. How many marbles does each person have if we only change the addition to 3 marbles? 

**Tip:** Always keep track of how many marbles each person starts with when solving ratio problems; it can help prevent confusion!

Pranay, Rohit, and Qureshi bought some marbles. If Rohit gave 2 marbles to Qureshi, the ratio of the number of marbles with Pranay, Rohit, and Qureshi would become 1:3:3. Had Rohit given 2 marbles to Pranay, then the ratio of the number of marbles with them would have become 2:3:2. What will be the ratio of the number of marbles with Pranay, Rohit, and Qureshi when 4 marbles are added to the number of marbles each one of them already had?

Explore how to solve a marble ratio problem involving Pranay, Rohit, and Qureshi. This problem demonstrates the use of algebraic equations and ratio properties to find the final ratio when each gains 4 marbles. Perfect for students in Grades 9-10.

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