A person lends 40% of his sum of money at 15% per annum, 50% of rest at 10% per annum and the rest at 18% per annum rate of interset. What would be the annual rate of interset. If the interset is calculated on the whole sum?

Options

A. `13.4%`
B. `14.33%`
C. `14.4%`
D. `13.33%`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Total Sum of Money
Let the total sum of money be \( x \).

### Step 2: Calculate the Amount Lent at Different Rates
1. **Amount lent at 15%:**
- 40% of \( x \) is lent at 15%.
- Amount = \( 0.40x \)

2. **Remaining Amount after lending at 15%:**
- Remaining = \( x - 0.40x = 0.60x \)

3. **Amount lent at 10%:**
- 50% of the remaining amount (which is \( 0.60x \)) is lent at 10%.
- Amount = \( 0.50 \times 0.60x = 0.30x \)

4. **Remaining Amount after lending at 10%:**
- Remaining = \( 0.60x - 0.30x = 0.30x \)

5. **Amount lent at 18%:**
- The remaining amount (which is \( 0.30x \)) is lent at 18%.
- Amount = \( 0.30x \)

### Step 3: Calculate the Interest Earned from Each Amount
1. **Interest from the amount lent at 15%:**
- Interest = Principal × Rate × Time
- Interest = \( 0.40x \times \frac{15}{100} \times 1 = 0.06x \)

2. **Interest from the amount lent at 10%:**
- Interest = \( 0.30x \times \frac{10}{100} \times 1 = 0.03x \)

3. **Interest from the amount lent at 18%:**
- Interest = \( 0.30x \times \frac{18}{100} \times 1 = 0.054x \)

### Step 4: Calculate the Total Interest Earned
Total Interest = Interest from 15% + Interest from 10% + Interest from 18%
\[ \text{Total Interest} = 0.06x + 0.03x + 0.054x = 0.144x \]

### Step 5: Calculate the Annual Rate of Interest on the Whole Sum
The annual rate of interest on the whole sum is calculated by dividing the total interest by the total sum and then multiplying by 100 to get the percentage.
\[ \text{Annual Rate of Interest} = \left( \frac{\text{Total Interest}}{x} \right) \times 100 = \left( \frac{0.144x}{x} \right) \times 100 = 14.4\% \]

### Final Answer
The annual rate of interest on the whole sum is **14.4%**.

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