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!