A sum of 725 is lent in the beginning of a year at a certain rate of interest. After 8 months, a sum of 362.50 more is lent but at the rate twice the former. At the end of the year, 33.50 is earned as interest from both the loans. What was the original rate of interest?

Options

A. `3.6%`
B. `4.5%`
C. `5%`
D. `3.46%`

Answer

To solve the problem step by step, we need to find the original rate of interest (R) based on the information provided.

### Step 1: Understand the Problem
We have two loans:
1. The first loan of ₹725 is lent for 12 months at an interest rate of R%.
2. The second loan of ₹362.50 is lent after 8 months, which means it is lent for 4 months at an interest rate of 2R%.

### Step 2: Calculate the Interest from the First Loan
The formula for calculating simple interest is:
\[ \text{Interest} = \frac{P \times R \times T}{100} \]
Where:
- P = Principal amount
- R = Rate of interest
- T = Time in years

For the first loan:
- P = 725
- R = R (unknown)
- T = 1 year (12 months)

So, the interest from the first loan (I1) is:
\[ I_1 = \frac{725 \times R \times 1}{100} = \frac{725R}{100} \]

### Step 3: Calculate the Interest from the Second Loan
For the second loan:
- P = 362.50
- R = 2R (twice the original rate)
- T = 4 months = \(\frac{4}{12} = \frac{1}{3}\) years

So, the interest from the second loan (I2) is:
\[ I_2 = \frac{362.50 \times 2R \times \frac{1}{3}}{100} = \frac{725R}{300} \]

### Step 4: Set Up the Equation for Total Interest
The total interest earned from both loans is given as ₹33.50. Therefore, we can set up the equation:
\[ I_1 + I_2 = 33.50 \]
Substituting the values we calculated:
\[ \frac{725R}{100} + \frac{725R}{300} = 33.50 \]

### Step 5: Find a Common Denominator
To solve the equation, we need a common denominator. The least common multiple of 100 and 300 is 300. We can rewrite the equation:
\[ \frac{725R \times 3}{300} + \frac{725R}{300} = 33.50 \]
This simplifies to:
\[ \frac{2175R}{300} = 33.50 \]

### Step 6: Solve for R
Now, we can multiply both sides by 300 to eliminate the fraction:
\[ 2175R = 33.50 \times 300 \]
Calculating the right side:
\[ 2175R = 10050 \]
Now, divide both sides by 2175:
\[ R = \frac{10050}{2175} \]
Calculating this gives:
\[ R = 4.62 \]

### Step 7: Conclusion
The original rate of interest is approximately **4.62%**.

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