A part of 38,800 is lent out at 6% per six months. The rest of the amount is lent out at 5% per annum after one year. The ratio of interest after 3 years from the time when first amount was lent out is 5 : 4. Find the second part that was lent out at 5%

Options

A. 26, 600
B. 28800
C. 27500
D. 28000

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let:
- \( x \) = the amount lent out at 6% per six months
- \( 38800 - x \) = the amount lent out at 5% per annum

### Step 2: Calculate the interest for each part
1. **Interest for the amount lent at 6% per six months**:
- The interest rate for 6 months is 6%, which is equivalent to 12% per annum.
- The amount is lent for 3 years, which is 6 periods of 6 months.
- The formula for simple interest is:
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]
- Therefore, the interest for the first part is:
\[ I_1 = x \times \frac{12}{100} \times 3 = x \times 0.36 \]

2. **Interest for the amount lent at 5% per annum**:
- This amount is lent out after one year, so it will earn interest for 2 years.
- The interest for this part is:
\[ I_2 = (38800 - x) \times \frac{5}{100} \times 2 = (38800 - x) \times 0.1 \]

### Step 3: Set up the ratio of interests
According to the problem, the ratio of the interest earned from the first part to the second part after 3 years is 5:4. Therefore, we can set up the equation:
\[ \frac{I_1}{I_2} = \frac{5}{4} \]

Substituting the expressions for \( I_1 \) and \( I_2 \):
\[ \frac{x \times 0.36}{(38800 - x) \times 0.1} = \frac{5}{4} \]

### Step 4: Cross-multiply to solve for \( x \)
Cross-multiplying gives:
\[ 4 \times (x \times 0.36) = 5 \times ((38800 - x) \times 0.1) \]
This simplifies to:
\[ 1.44x = 1940 - 0.5x \]

### Step 5: Combine like terms
Bringing all terms involving \( x \) to one side:
\[ 1.44x + 0.5x = 1940 \]
\[ 1.94x = 1940 \]

### Step 6: Solve for \( x \)
Dividing both sides by 1.94:
\[ x = \frac{1940}{1.94} \approx 1000 \]

### Step 7: Find the second part
The second part lent out at 5% is:
\[ 38800 - x = 38800 - 1000 = 37800 \]

### Final Answer
The second part that was lent out at 5% per annum is **37,800**.

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