P, Q, and R enter into a partnership by investing certain amounts for 12 months, *T* months, and (12 – *T*) months respectively. Find the value of *T* given the following statements:

Statement I: Q invested 50\% more than the amount invested by P, and R invested twice the amount invested by Q.

Statement II: The total profit earned at the end of the partnership is Rs. 700, and Q's share of the profit is Rs. 100.

Statement III: The ratio of profit shares of P to R is 1 : 2.

Which of the following combinations of statements is sufficient to find *T*?

Question

P, Q, and R enter into a partnership by investing certain amounts for 12 months, *T* months, and (12 – *T*) months respectively. Find the value of *T* given the following statements:

Statement I: Q invested 50\% more than the amount invested by P, and R invested twice the amount invested by Q.

Statement II: The total profit earned at the end of the partnership is Rs. 700, and Q's share of the profit is Rs. 100.

Statement III: The ratio of profit shares of P to R is 1 : 2.

Which of the following combinations of statements is sufficient to find *T*?

Accepted Answer

**Solution:**

1. Let the amounts invested by P, Q, and R be *P*, *Q*, and *R* respectively.

2. From Statement I:
   - Q invested 50\% more than P, so $Q = 1.5P$
   - R invested twice the amount invested by Q, so $R = 2Q = 3P$

3. The time periods of investment are:
   - P: 12 months
   - Q: *T* months
   - R: (12 – *T*) months

4. The profit share is proportional to the product of amount invested and time invested.

5. Using Statement II:
   - Total profit = Rs. 700
   - Q's profit share = Rs. 100

Q's share ratio = $(Q 	imes T)$
   Total investment-time product = $P 	imes 12 + Q 	imes T + R 	imes (12 - T)$

Substitute $Q = 1.5P$ and $R = 3P$:
   $$	ext{Total} = 12P + 1.5P 	imes T + 3P 	imes (12 - T) = 12P + 1.5P T + 36P - 3P T = 48P - 1.5P T$$

Q's share ratio:
   $$1.5P 	imes T$$

Profit share ratio for Q:
   $$\frac{1.5P T}{48P - 1.5P T} = \frac{100}{700} = \frac{1}{7}$$

Simplify:
   $$\frac{1.5 T}{48 - 1.5 T} = \frac{1}{7}$$

Cross-multiplied:
   $$7 	imes 1.5 T = 48 - 1.5 T$$
   $$10.5 T = 48 - 1.5 T$$
   $$10.5 T + 1.5 T = 48$$
   $$12 T = 48$$
   $$T = 4$$

6. Using Statement III:
   - Ratio of profit shares of P to R is 1 : 2

Profit shares:
   - P: $P 	imes 12$
   - R: $R 	imes (12 - T)$

Substitute $R = 3P$:
   $$\frac{12P}{3P (12 - T)} = \frac{1}{2}$$

Simplify:
   $$\frac{12}{3(12 - T)} = \frac{1}{2}$$
   $$\frac{12}{36 - 3T} = \frac{1}{2}$$

Cross-multiplied:
   $$24 = 36 - 3T$$
   $$3T = 36 - 24 = 12$$
   $$T = 4$$

**Conclusion:**
- Using Statements I and II together, we can find *T*.
- Using Statements I and III together, we can also find *T*.
- Statement II and III without I are insufficient because we don't know the investment amounts.

**Answer:** Either I and II or I and III

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