The ratio of investment of Anurag and Ayush is `2:3` and out of total profit of Rs . 76000 shares of Ayush is Rs.36000. Find the ratio of time period for which Anurag and Ayush invested their capital respectively ?

Options

A. `3:2`
B. `6:5`
C. `4:3`
D. None of these

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the given information
We know that:
- The ratio of investment of Anurag (A) to Ayush (Y) is \(2:3\).
- Total profit is Rs. 76,000.
- Ayush's share of the profit is Rs. 36,000.

### Step 2: Calculate Anurag's share of the profit
Since Ayush's share is Rs. 36,000, we can find Anurag's share by subtracting Ayush's share from the total profit:
\[ \text{Anurag's share} = \text{Total profit} - \text{Ayush's share} = 76,000 - 36,000 = 40,000 \]

### Step 3: Set up the profit-sharing ratio
The profit-sharing ratio of Anurag to Ayush can be expressed as:
\[ \text{Profit of Anurag} : \text{Profit of Ayush} = 40,000 : 36,000 \]
This simplifies to:
\[ \frac{40,000}{36,000} = \frac{10}{9} \]

### Step 4: Relate profit to investment and time
The profit is proportional to the product of investment and time. Therefore, we can express the ratio of profits in terms of investments and time:
\[ \frac{\text{Investment of Anurag} \times \text{Time of Anurag}}{\text{Investment of Ayush} \times \text{Time of Ayush}} = \frac{10}{9} \]
Let the investments be:
- Anurag's investment = \(2x\)
- Ayush's investment = \(3x\)

Let:
- Time period for Anurag = \(T_1\)
- Time period for Ayush = \(T_2\)

### Step 5: Substitute the investments into the ratio
Now substituting the investments into the profit ratio:
\[ \frac{2x \cdot T_1}{3x \cdot T_2} = \frac{10}{9} \]
The \(x\) cancels out:
\[ \frac{2T_1}{3T_2} = \frac{10}{9} \]

### Step 6: Cross-multiply to find the relationship between time periods
Cross-multiplying gives:
\[ 2T_1 \cdot 9 = 10 \cdot 3T_2 \]
This simplifies to:
\[ 18T_1 = 30T_2 \]

### Step 7: Solve for the ratio of time periods
Dividing both sides by 6 gives:
\[ 3T_1 = 5T_2 \]
Thus, the ratio of \(T_1\) to \(T_2\) is:
\[ \frac{T_1}{T_2} = \frac{5}{3} \]

### Conclusion
The ratio of the time period for which Anurag and Ayush invested their capital respectively is \(5:3\).

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