A man derives his income from the investment of Rs. 4150 at a certain rate of interest and Rs. 3500 at 1 per cent higher. This whole income for 4 years is Rs. 1211. Find the rates of interest..

Options

A. `3 1/2%` , `4 1/2%`
B. `2 1/2%`, `3 1/2%`
C. `4 1/2%`, `5 1/2%`
D. None of these

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI):

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

where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

### Step 1: Set up the problem
We have two investments:
1. Investment of Rs. 4150 at a certain rate \( R \).
2. Investment of Rs. 3500 at a rate \( R + 1\% \).

The total income from both investments over 4 years is Rs. 1211.

### Step 2: Write the equations for Simple Interest
For the first investment:
\[ \text{SI}_1 = \frac{4150 \times R \times 4}{100} \]

For the second investment:
\[ \text{SI}_2 = \frac{3500 \times (R + 1) \times 4}{100} \]

### Step 3: Combine the equations
The total interest from both investments is given as Rs. 1211:
\[ \text{SI}_1 + \text{SI}_2 = 1211 \]

Substituting the expressions for SI:
\[ \frac{4150 \times R \times 4}{100} + \frac{3500 \times (R + 1) \times 4}{100} = 1211 \]

### Step 4: Simplify the equation
Multiply through by 100 to eliminate the denominator:
\[ 4150 \times R \times 4 + 3500 \times (R + 1) \times 4 = 121100 \]

Now divide everything by 4:
\[ 4150R + 3500(R + 1) = 30275 \]

### Step 5: Expand and combine like terms
Expanding the second term:
\[ 4150R + 3500R + 3500 = 30275 \]

Combine like terms:
\[ 7650R + 3500 = 30275 \]

### Step 6: Isolate \( R \)
Subtract 3500 from both sides:
\[ 7650R = 30275 - 3500 \]
\[ 7650R = 26775 \]

### Step 7: Solve for \( R \)
Now divide by 7650:
\[ R = \frac{26775}{7650} \]

### Step 8: Simplify the fraction
Calculating the division:
\[ R = 3.5 \]

### Step 9: Find the second rate
The second investment's rate is:
\[ R + 1 = 3.5 + 1 = 4.5 \]

### Conclusion
The rates of interest are:
- First investment: **3.5%**
- Second investment: **4.5%**

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