If the ratio of time periods of investment of Ayush and Nikhil is `4:5` , profit at the end of the year is Rs 100000 and Ayush's share in it is Rs 20000, then what is the ratio of Ayush's and Nikhil's investment ?

Options

A. `14:5`
B. `8:7`
C. `5:16`
D. `16:5`

Answer

To find the ratio of Ayush's and Nikhil's investments based on the given information, we can follow these steps:

### Step 1: Understand the given data
- The ratio of time periods of investment of Ayush and Nikhil is \(4:5\).
- The total profit at the end of the year is Rs. 100,000.
- Ayush's share in the profit is Rs. 20,000.

### Step 2: Calculate Nikhil's share of the profit
Since the total profit is Rs. 100,000 and Ayush's share is Rs. 20,000, we can find Nikhil's share:
\[ \text{Nikhil's profit} = \text{Total profit} - \text{Ayush's profit} = 100,000 - 20,000 = 80,000 \]

### Step 3: Set up the profit-sharing equation
The profit earned by each partner is proportional to the product of their investment and the time period for which they invested. Let:
- Ayush's investment = \(I_1\)
- Nikhil's investment = \(I_2\)

The profit-sharing can be expressed as:
\[ \frac{\text{Ayush's profit}}{\text{Nikhil's profit}} = \frac{I_1 \times \text{Time of Ayush}}{I_2 \times \text{Time of Nikhil}} \]

### Step 4: Substitute known values
From the problem, we know:
- Time of Ayush = \(4x\)
- Time of Nikhil = \(5x\)

Substituting the values into the equation:
\[ \frac{20,000}{80,000} = \frac{I_1 \times 4x}{I_2 \times 5x} \]

### Step 5: Simplify the equation
We can simplify the left side:
\[ \frac{20,000}{80,000} = \frac{1}{4} \]

Now, substituting this into the equation gives:
\[ \frac{1}{4} = \frac{I_1 \times 4}{I_2 \times 5} \]

### Step 6: Cross-multiply to find the ratio of investments
Cross-multiplying gives:
\[ I_2 \times 5 = I_1 \times 16 \]

Rearranging this gives:
\[ \frac{I_1}{I_2} = \frac{5}{16} \]

### Step 7: State the final ratio
Thus, the ratio of Ayush's investment to Nikhil's investment is:
\[ \text{Ratio of Ayush's investment to Nikhil's investment} = 5:16 \]