Each of the questions below consists of a question and three statements numbered I, II and III given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read all the statements and give answer.
P, Q and R entered into a partnership by investing certain amount for 12 months, T months and (12 – T) months respectively. Find the value of T.
Statement I: Q invested 50% more amount than the amount invested by P and R invested twice of the amount invested by Q.
Statement II: At the end of the partnership, total profit earned by them is Rs. 700 and Q gets Rs. 100 as his share of profit.
Statement III: At the end of the partnership, profit share of P and R is in the ratio of 1 : 2

Options

A. Only the data given in statements I and II together is sufficient to answer the question.
B. Only the data given in statements I and III together is sufficient to answer the question.
C. The data given in any two statements together is sufficient to answer the question.
D. The data given in either statement I and II together or statement I and III together is sufficient to answer the question.

Explanation

To solve the problem, we need to determine the value of T based on the information provided in the statements. Let's break down the solution step by step.

### Step 1: Define the investments
Let the amount invested by P be \( P \).
According to Statement I:
- Q invested 50% more than P, so Q's investment is \( Q = 1.5P \).
- R invested twice the amount invested by Q, so R's investment is \( R = 2Q = 2(1.5P) = 3P \).

### Step 2: Define the time periods
According to the problem:
- P invested for 12 months.
- Q invested for T months.
- R invested for \( (12 - T) \) months.

### Step 3: Calculate the total investment in terms of P
The total investment can be calculated as follows:
- For P: \( 12P \)
- For Q: \( 1.5P \times T \)
- For R: \( 3P \times (12 - T) \)

Thus, the total investment (in terms of P) is:
\[ \text{Total Investment} = 12P + 1.5PT + 3P(12 - T) \]

### Step 4: Simplify the total investment equation
Expanding the equation:
\[ \text{Total Investment} = 12P + 1.5PT + 36P - 3PT \]
Combining like terms:
\[ \text{Total Investment} = (12P + 36P) + (1.5PT - 3PT) = 48P - 1.5PT \]

### Step 5: Use the profit information from Statement II
According to Statement II, the total profit earned is Rs. 700, and Q's share of the profit is Rs. 100.
The share of profit for Q can be expressed as:
\[ \text{Q's Profit} = \frac{\text{Q's Investment} \times \text{Total Profit}}{\text{Total Investment}} = \frac{1.5PT \times 700}{48P - 1.5PT} \]
Setting this equal to Rs. 100:
\[ \frac{1.5PT \times 700}{48P - 1.5PT} = 100 \]

### Step 6: Solve for T
Cross-multiplying gives:
\[ 1.5PT \times 700 = 100(48P - 1.5PT) \]
Expanding this:
\[ 1050PT = 4800P - 150PT \]
Combining terms:
\[ 1050PT + 150PT = 4800P \]
\[ 1200PT = 4800P \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 1200T = 4800 \]
\[ T = \frac{4800}{1200} = 4 \]

### Step 7: Verify with Statement III
According to Statement III, the profit share of P and R is in the ratio of 1:2. We can check if this holds true with our calculated value of T.

### Conclusion
The value of T is 4 months.