A person had ₹ 8400.He lent a part of it at 4% and the remaining at `3 (1)/(3)%` simple interest. His total annual income was ₹ 294. Find the sum he lent at 4%

Options

A. 2310
B. 2110
C. 2500
D. 2100

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
A person has ₹8400. He lends part of it at 4% interest and the remaining at 3⅓% (which is equivalent to 10/3%). His total income from both loans is ₹294. We need to find out how much he lent at 4%.

### Step 2: Calculate the Overall Rate of Interest
The total interest earned is ₹294 from a principal amount of ₹8400. We can calculate the overall rate of interest using the formula:

\[ \text{Overall Rate} = \left(\frac{\text{Total Interest}}{\text{Principal}} \times 100\right) \]

Substituting the values:

\[ \text{Overall Rate} = \left(\frac{294}{8400} \times 100\right) = 3.5\% \]

### Step 3: Set Up the Variables
Let:
- \( x \) = amount lent at 4%
- \( 8400 - x \) = amount lent at 3⅓% (10/3%)

### Step 4: Write the Interest Equations
The interest from both parts can be expressed as follows:

1. Interest from the amount lent at 4%:
\[ \text{Interest}_1 = \frac{x \times 4}{100} = 0.04x \]

2. Interest from the amount lent at 3⅓%:
\[ \text{Interest}_2 = \frac{(8400 - x) \times \frac{10}{3}}{100} = \frac{10(8400 - x)}{300} = \frac{84000 - 10x}{300} = 0.0333(8400 - x) \]

### Step 5: Set Up the Total Interest Equation
The total interest from both parts is equal to ₹294:

\[ 0.04x + 0.0333(8400 - x) = 294 \]

### Step 6: Simplify the Equation
Substituting the values into the equation:

\[ 0.04x + 280 - 0.0333x = 294 \]

Combine like terms:

\[ (0.04 - 0.0333)x + 280 = 294 \]

This simplifies to:

\[ 0.0067x + 280 = 294 \]

### Step 7: Solve for \( x \)
Subtract 280 from both sides:

\[ 0.0067x = 294 - 280 \]
\[ 0.0067x = 14 \]

Now, divide by 0.0067:

\[ x = \frac{14}{0.0067} \approx 2089.55 \]

### Step 8: Round to the Nearest Option
Since the amount lent at 4% must be a whole number, we round it to the nearest option available. The closest option is ₹2100.

### Conclusion
The sum lent at 4% is **₹2100**.

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