The simple and compound interest that can be earned in two years at the same rate is `Rs. 1,000` and `Rs. 1,040` respectively. What is the rate (per cent per annum) of interest?

Options

A. 9
B. 10
C. 8
D. 11

Answer

To find the rate of interest per annum, we can follow these steps:

### Step 1: Understand the given information
We know that:
- Simple Interest (SI) for 2 years = Rs. 1,000
- Compound Interest (CI) for 2 years = Rs. 1,040

### Step 2: Calculate the Simple Interest for 1 year
Since the Simple Interest for 2 years is Rs. 1,000, the Simple Interest for 1 year can be calculated as:
\[ \text{SI for 1 year} = \frac{\text{SI for 2 years}}{2} = \frac{1000}{2} = Rs. 500 \]

### Step 3: Calculate the Compound Interest for 1 year
The Compound Interest for 2 years is Rs. 1,040. The Compound Interest for 1 year can be calculated as follows:
Let the amount after 1 year be \( A_1 \).
The interest for the first year is the same as the Simple Interest for 1 year, which is Rs. 500.
Thus, the amount after 1 year is:
\[ A_1 = P + \text{SI for 1 year} = P + 500 \]
Where \( P \) is the principal amount.

### Step 4: Calculate the Compound Interest for the second year
The Compound Interest for the second year is calculated on the amount after the first year:
\[ \text{CI for 2 years} = A_1 \times \text{rate} + \text{SI for 1 year} \]
Given that the total Compound Interest for 2 years is Rs. 1,040, we can write:
\[ \text{CI for 2 years} = 500 + \text{CI for 1 year} \]
Let the interest for the second year be \( x \). Therefore:
\[ 1000 + x = 1040 \]
This implies:
\[ x = 1040 - 1000 = Rs. 40 \]

### Step 5: Relate the interest to the principal to find the rate
The interest for the second year (Rs. 40) is calculated on the principal amount (P) at the rate of interest (r%):
\[ \text{CI for 1 year} = P \times \frac{r}{100} \]
From the first year, we know:
\[ 500 = P \times \frac{r}{100} \]
From the second year:
\[ 40 = (P + 500) \times \frac{r}{100} \]

### Step 6: Solve the equations
From the first equation:
\[ P = \frac{500 \times 100}{r} = \frac{50000}{r} \]
Substituting \( P \) in the second equation:
\[ 40 = \left(\frac{50000}{r} + 500\right) \times \frac{r}{100} \]
Multiplying through by 100:
\[ 4000 = (50000 + 500r) \times \frac{1}{r} \]
This simplifies to:
\[ 4000r = 50000 + 500r \]
Rearranging gives:
\[ 4000r - 500r = 50000 \]
\[ 3500r = 50000 \]
Thus:
\[ r = \frac{50000}{3500} = \frac{5000}{350} = \frac{500}{35} = \frac{100}{7} \approx 14.29\% \]

### Final Answer
The rate of interest is approximately **14.29% per annum**.

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