A sum of Rs. 4.000 is lent at `10%` p.a., interest compounded annually. What is the difference between the compound interest for the `2^(nd)` year and the `3^(rd)` year?

Options

A. Rs. 45.00
B. Rs 46.50
C. Rs. 45.40
D. Rs. 44.00

Answer

To find the difference between the compound interest for the 2nd year and the 3rd year on a sum of Rs. 4000 lent at 10% per annum, compounded annually, we can follow these steps:

### Step 1: Calculate the amount at the end of the first year
The formula for compound interest is:
\[ A = P(1 + r)^n \]
Where:
- \( A \) = Amount after n years
- \( P \) = Principal amount (initial sum)
- \( r \) = Rate of interest (in decimal)
- \( n \) = Number of years

For the first year:
- \( P = 4000 \)
- \( r = \frac{10}{100} = 0.10 \)
- \( n = 1 \)

Calculating the amount:
\[ A_1 = 4000(1 + 0.10)^1 = 4000 \times 1.10 = 4400 \]

### Step 2: Calculate the compound interest for the first year
The compound interest for the first year (CI1) is:
\[ CI_1 = A_1 - P = 4400 - 4000 = 400 \]

### Step 3: Calculate the amount at the end of the second year
Using the amount from the first year as the principal for the second year:
\[ A_2 = 4400(1 + 0.10)^1 = 4400 \times 1.10 = 4840 \]

### Step 4: Calculate the compound interest for the second year
The compound interest for the second year (CI2) is:
\[ CI_2 = A_2 - A_1 = 4840 - 4400 = 440 \]

### Step 5: Calculate the amount at the end of the third year
Using the amount from the second year as the principal for the third year:
\[ A_3 = 4840(1 + 0.10)^1 = 4840 \times 1.10 = 5324 \]

### Step 6: Calculate the compound interest for the third year
The compound interest for the third year (CI3) is:
\[ CI_3 = A_3 - A_2 = 5324 - 4840 = 484 \]

### Step 7: Calculate the difference between the compound interest for the second and third years
Now, we find the difference:
\[ \text{Difference} = CI_3 - CI_2 = 484 - 440 = 44 \]

### Final Answer
The difference between the compound interest for the 2nd year and the 3rd year is Rs. 44.

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