What is the difference (in Rs) in Compound interest earned in 1 year on a sum of Rs 10,000 at 40% per annum compounded quarterly and annually?

Options

A. 461
B. 346
C. 463
D. 641

Answer

To find the difference in compound interest earned in one year on a sum of Rs 10,000 at 40% per annum compounded quarterly and annually, we will follow these steps:

### Step 1: Calculate Compound Interest when Compounded Annually
The formula for compound interest (CI) when compounded annually is given by:
\[ CI = P \left(1 + \frac{r}{100}\right)^t - P \]
Where:
- \( P = 10,000 \) (Principal)
- \( r = 40 \) (Rate of interest per annum)
- \( t = 1 \) (Time in years)

Substituting the values:
\[ CI_{annual} = 10,000 \left(1 + \frac{40}{100}\right)^1 - 10,000 \]
\[ = 10,000 \left(1 + 0.4\right) - 10,000 \]
\[ = 10,000 \times 1.4 - 10,000 \]
\[ = 14,000 - 10,000 = 4,000 \]

### Step 2: Calculate Compound Interest when Compounded Quarterly
When interest is compounded quarterly, the rate and time need to be adjusted:
- Quarterly rate \( r_{quarterly} = \frac{40}{4} = 10\% \)
- Number of quarters in one year \( n = 4 \)

Using the same formula for compound interest:
\[ CI_{quarterly} = P \left(1 + \frac{r_{quarterly}}{100}\right)^n - P \]
Substituting the values:
\[ CI_{quarterly} = 10,000 \left(1 + \frac{10}{100}\right)^4 - 10,000 \]
\[ = 10,000 \left(1 + 0.1\right)^4 - 10,000 \]
\[ = 10,000 \left(1.1\right)^4 - 10,000 \]

Calculating \( (1.1)^4 \):
\[ (1.1)^4 = 1.4641 \]
Now substituting back:
\[ CI_{quarterly} = 10,000 \times 1.4641 - 10,000 \]
\[ = 14,641 - 10,000 = 4,641 \]

### Step 3: Calculate the Difference in Compound Interest
Now, we find the difference between the compound interest earned quarterly and annually:
\[ \text{Difference} = CI_{quarterly} - CI_{annual} \]
\[ = 4,641 - 4,000 = 641 \]

### Final Answer
The difference in compound interest earned in one year on a sum of Rs 10,000 at 40% per annum compounded quarterly and annually is **Rs 641**.

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