If ratio of time periods of investment of P and Q is `4:5` , profit at the end of the year is 75000 and P's share is Rs 15000,then what is the ratio of Q's and P's investment ?

Options

A. A)`5:16`
B. B)`6:7`
C. C)`12:13`
D. D)`16:5`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We know:
- The ratio of time periods of investment of P and Q is \(4:5\).
- The total profit at the end of the year is Rs 75,000.
- P's share of the profit is Rs 15,000.

### Step 2: Determine the profit ratio
Since the profit is distributed based on the product of investment and time, we can express the profit ratio as:
\[ \text{Profit Ratio} = P \times \text{Time of P} : Q \times \text{Time of Q} \]
Given the time ratio is \(4:5\), we can denote:
- Time of P = 4x
- Time of Q = 5x

Thus, the profit ratio becomes:
\[ \text{Profit Ratio} = P \times 4 : Q \times 5 \]

### Step 3: Set up the equation for P's share
From the problem, we know that P's share of the total profit (Rs 75,000) is Rs 15,000. Therefore, we can set up the equation:
\[ \frac{P \times 4}{P \times 4 + Q \times 5} = \frac{15000}{75000} \]
This simplifies to:
\[ \frac{P \times 4}{P \times 4 + Q \times 5} = \frac{1}{5} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 5 \times (P \times 4) = 1 \times (P \times 4 + Q \times 5) \]
This simplifies to:
\[ 20P = P \times 4 + 5Q \]

### Step 5: Rearrange the equation
Rearranging the equation, we have:
\[ 20P - 4P = 5Q \]
This simplifies to:
\[ 16P = 5Q \]

### Step 6: Find the ratio of Q's investment to P's investment
To find the ratio of Q's investment to P's investment, we can express it as:
\[ \frac{Q}{P} = \frac{16}{5} \]

### Step 7: Write the final ratio of investments
Thus, the ratio of Q's investment to P's investment is:
\[ Q : P = 16 : 5 \]

### Final Answer
The ratio of Q's investment to P's investment is \(16 : 5\).

---