Mohit and Sumit start a business with investment of ₹ 74000 and ₹ 96000 respectively. If at the end of the year they earn profit in the ratio of 5 : 8, then what will be ratio of the time period for which they invest their money?

Options

A. `37 : 32 `
B. `13 : 18 `
C. `25 : 32 `
D. `30 : 37 `

Answer

To solve the problem step by step, we will use the formula for profit sharing based on investment and time.

### Step 1: Understand the Profit Sharing
The profit earned by Mohit and Sumit is in the ratio of 5:8. This means that for every 5 parts of profit Mohit earns, Sumit earns 8 parts.

### Step 2: Define the Investments
- Mohit's investment (I1) = ₹74,000
- Sumit's investment (I2) = ₹96,000

### Step 3: Set Up the Profit Formula
Profit is calculated as:
\[ \text{Profit} = \text{Investment} \times \text{Time} \]
Let T1 be the time period for Mohit and T2 be the time period for Sumit.

### Step 4: Write the Profit Equations
From the profit sharing:
1. For Mohit:
\[ \text{Profit}_\text{Mohit} = I1 \times T1 = 74000 \times T1 \]
Since his profit corresponds to 5 parts:
\[ 74000 \times T1 = 5k \] (where k is a constant)

2. For Sumit:
\[ \text{Profit}_\text{Sumit} = I2 \times T2 = 96000 \times T2 \]
Since his profit corresponds to 8 parts:
\[ 96000 \times T2 = 8k \]

### Step 5: Set Up the Ratio of Profits
Now we can set up the ratio of the profits:
\[ \frac{74000 \times T1}{96000 \times T2} = \frac{5}{8} \]

### Step 6: Cross Multiply to Solve for the Ratio
Cross multiplying gives:
\[ 74000 \times T1 \times 8 = 96000 \times T2 \times 5 \]
This simplifies to:
\[ 592000T1 = 480000T2 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ \frac{T1}{T2} = \frac{480000}{592000} \]

### Step 8: Simplifying the Ratio
Now simplify the fraction:
\[ \frac{T1}{T2} = \frac{480}{592} \]
Dividing both the numerator and denominator by 16:
\[ \frac{T1}{T2} = \frac{30}{37} \]

### Final Answer
Thus, the ratio of the time period for which Mohit and Sumit invested their money is:
\[ \text{Ratio of } T1 : T2 = 30 : 37 \]