P started a business with a intial investment of `₹ 1200`. 'X' month after the start of business, Q joined P with on intial investement of `₹ 1500`. If total profit was 1950 at the end of year and Q's share of profit was 750. Find 'X'.

Options

A. 5month
B. 6month
C. 7month
D. 8month

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the investments and time
P started the business with an investment of ₹1200 for 12 months. Q joined 'X' months after the start with an investment of ₹1500.

### Step 2: Calculate the effective investment of both P and Q
- P's effective investment = Investment × Time = ₹1200 × 12 = ₹14400
- Q's effective investment = Investment × Time = ₹1500 × (12 - X)

### Step 3: Set up the equation for the total profit
The total profit at the end of the year is ₹1950. Q's share of the profit is ₹750. Therefore, P's share of the profit is:
\[ P's \, share = Total \, profit - Q's \, share = 1950 - 750 = ₹1200 \]

### Step 4: Set up the ratio of their effective investments
The ratio of P's and Q's effective investments will be equal to the ratio of their shares in the profit:
\[ \frac{P's \, effective \, investment}{Q's \, effective \, investment} = \frac{P's \, share}{Q's \, share} \]
Substituting the values we have:
\[ \frac{14400}{1500 \times (12 - X)} = \frac{1200}{750} \]

### Step 5: Simplify the ratio
The right side simplifies to:
\[ \frac{1200}{750} = \frac{8}{5} \]
So, we have:
\[ \frac{14400}{1500 \times (12 - X)} = \frac{8}{5} \]

### Step 6: Cross-multiply to solve for X
Cross-multiplying gives:
\[ 14400 \times 5 = 8 \times 1500 \times (12 - X) \]
This simplifies to:
\[ 72000 = 12000 \times (12 - X) \]

### Step 7: Divide both sides by 12000
\[ 6 = 12 - X \]

### Step 8: Solve for X
Rearranging gives:
\[ X = 12 - 6 = 6 \]

Thus, the value of X is **6 months**.

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