A sum of Rs. 825 is lent in the beginning of a year at a certain rate of interest. After 8 months, a sum of Rs. 412.50 more is lent but at the rate twice the former. At the ned of the year, Rs. 35.75 is earned as interest from the loans. What was the original rate of interest ?

Options

A. `3%`
B. `3.25%`
C. `3.50%`
D. `3.15%`

Answer

To solve the problem step by step, we will break down the information given and calculate the interest earned from both loans to find the original rate of interest.

### Step 1: Define the Variables
Let the original rate of interest be \( x\% \) per annum.

### Step 2: Calculate Interest from the First Loan
The first loan of Rs. 825 is lent for 12 months at the rate of \( x\% \). The interest earned from this loan can be calculated using the formula for simple interest:
\[ \text{Interest} = \frac{P \times R \times T}{100} \]
Where:
- \( P = 825 \)
- \( R = x \)
- \( T = 1 \) year (12 months)

So, the interest from the first loan is:
\[ \text{Interest}_1 = \frac{825 \times x \times 1}{100} = \frac{825x}{100} \]

### Step 3: Calculate Interest from the Second Loan
After 8 months, an additional sum of Rs. 412.50 is lent at a rate of \( 2x\% \) for the remaining 4 months. The interest from this loan is calculated similarly:
\[ \text{Interest}_2 = \frac{P \times R \times T}{100} \]
Where:
- \( P = 412.50 \)
- \( R = 2x \)
- \( T = \frac{4}{12} = \frac{1}{3} \) year (4 months)

So, the interest from the second loan is:
\[ \text{Interest}_2 = \frac{412.50 \times 2x \times \frac{1}{3}}{100} = \frac{412.50 \times 2x}{300} = \frac{825x}{300} \]

### Step 4: Total Interest Earned
According to the problem, the total interest earned from both loans at the end of the year is Rs. 35.75. Therefore, we can set up the equation:
\[ \text{Interest}_1 + \text{Interest}_2 = 35.75 \]
Substituting the expressions we found:
\[ \frac{825x}{100} + \frac{825x}{300} = 35.75 \]

### Step 5: Simplify the Equation
To simplify, we need a common denominator. The least common multiple of 100 and 300 is 300. Thus, we rewrite the equation:
\[ \frac{825x \times 3}{300} + \frac{825x}{300} = 35.75 \]
This simplifies to:
\[ \frac{2475x + 825x}{300} = 35.75 \]
\[ \frac{3300x}{300} = 35.75 \]

### Step 6: Solve for \( x \)
Multiplying both sides by 300:
\[ 3300x = 35.75 \times 300 \]
Calculating the right side:
\[ 3300x = 10725 \]
Now, divide both sides by 3300:
\[ x = \frac{10725}{3300} = 3.25 \]

### Conclusion
The original rate of interest is \( 3.25\% \) per annum.