A man invests `₹2000` for `3` years. He also invests `₹1600` for `3` years at a rate `2%` higher than the first one. He earns an interest of `₹996` at the end of `3` years. Find the rates of interest in both the cases.

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the Problem
We have two investments:
1. ₹2000 for 3 years at an unknown rate \( R \).
2. ₹1600 for 3 years at a rate \( R + 2\% \).

The total interest earned from both investments after 3 years is ₹996.

### Step 2: Calculate Interest from the Second Investment
We can calculate the interest earned from the second investment using the formula for simple interest:

\[ \text{Interest} = \frac{P \times R \times T}{100} \]

For the second investment:
- Principal \( P = 1600 \)
- Rate \( R + 2 \)
- Time \( T = 3 \)

Thus, the interest from the second investment is:

\[ \text{Interest}_{1600} = \frac{1600 \times (R + 2) \times 3}{100} \]

### Step 3: Calculate Interest from the First Investment
Now, we calculate the interest from the first investment:

For the first investment:
- Principal \( P = 2000 \)
- Rate \( R \)
- Time \( T = 3 \)

Thus, the interest from the first investment is:

\[ \text{Interest}_{2000} = \frac{2000 \times R \times 3}{100} \]

### Step 4: Set Up the Equation
According to the problem, the total interest from both investments is ₹996. Therefore, we can set up the equation:

\[ \text{Interest}_{2000} + \text{Interest}_{1600} = 996 \]

Substituting the expressions we derived:

\[ \frac{2000 \times R \times 3}{100} + \frac{1600 \times (R + 2) \times 3}{100} = 996 \]

### Step 5: Simplify the Equation
We can simplify the equation by multiplying through by 100 to eliminate the denominator:

\[ 2000 \times R \times 3 + 1600 \times (R + 2) \times 3 = 99600 \]

This simplifies to:

\[ 6000R + 4800 + 4800R = 99600 \]

Combining like terms gives:

\[ 10800R + 4800 = 99600 \]

### Step 6: Solve for \( R \)
Now, isolate \( R \):

\[ 10800R = 99600 - 4800 \]
\[ 10800R = 94800 \]
\[ R = \frac{94800}{10800} = 8.33\% \]

### Step 7: Find the Second Rate of Interest
Since the second investment has a rate that is 2% higher than the first:

\[ R + 2 = 8.33 + 2 = 10.33\% \]

### Conclusion
The rates of interest are:
- For the first investment: **8.33%**
- For the second investment: **10.33%**