Ram and Shyam invest in the ratio of `3:4` in a partnership business and Ram got Rs. 2700 as profit share out of total profit of Rs.5100 , then find the ratio of period of investment of Shyam to that of Ram ?

Options

A. `3:4`
B. `3:5`
C. `2:3`
D. `1:2`

Answer

To solve the problem step by step, let's break down the information given and apply the concepts of ratio, proportion, and partnership.

### Step 1: Understand the Investment Ratio
Ram and Shyam invest in the ratio of 3:4.
- Let Ram's investment be \(3X\) and Shyam's investment be \(4X\), where \(X\) is a common multiplier.

### Step 2: Determine Profit Shares
The total profit from the business is Rs. 5100. Ram's share of the profit is Rs. 2700.
- Therefore, Shyam's share of the profit can be calculated as:
\[ \text{Shyam's Profit} = \text{Total Profit} - \text{Ram's Profit} = 5100 - 2700 = 2400 \]

### Step 3: Set Up the Profit Sharing Equation
The profit share in a partnership is proportional to the product of the investment and the time for which the investment is made.
- Let \(T_1\) be the time for which Ram invested and \(T_2\) be the time for which Shyam invested.
- The profit share can be expressed as:
\[ \frac{\text{Ram's Profit}}{\text{Shyam's Profit}} = \frac{3X \cdot T_1}{4X \cdot T_2} \]

### Step 4: Substitute Known Values
Substituting the known profit values into the equation:
\[ \frac{2700}{2400} = \frac{3X \cdot T_1}{4X \cdot T_2} \]
- The \(X\) cancels out:
\[ \frac{2700}{2400} = \frac{3T_1}{4T_2} \]

### Step 5: Simplify the Profit Ratio
Now simplify the left side:
\[ \frac{2700}{2400} = \frac{27}{24} = \frac{9}{8} \]
- So we have:
\[ \frac{9}{8} = \frac{3T_1}{4T_2} \]

### Step 6: Cross Multiply to Find the Ratio
Cross-multiplying gives:
\[ 9 \cdot 4T_2 = 8 \cdot 3T_1 \]
\[ 36T_2 = 24T_1 \]

### Step 7: Find the Ratio of Time Periods
Now, dividing both sides by 12:
\[ 3T_2 = 2T_1 \]
- Rearranging gives:
\[ \frac{T_2}{T_1} = \frac{2}{3} \]

### Step 8: Conclusion
Thus, the ratio of the period of investment of Shyam to that of Ram is:
\[ \frac{T_2}{T_1} = \frac{2}{3} \]

### Final Answer
The ratio of the period of investment of Shyam to that of Ram is \(2:3\).

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