A sum of Rs. 18,000 is lent at 10% p.a. compound interest. Compounded annually. What is the difference between the compounded interset for `3^(rd)` year and `4^(th)` year ?

Options

A. Rs 217.80
B. Rs 220.60
C. Rs. 215.4
D. Rs 221. 80

Answer

To find the difference between the compounded interest for the 3rd year and the 4th year when Rs. 18,000 is lent at 10% per annum compound interest, we can follow these steps:

### Step 1: Calculate the amount at the end of each year
The formula for the amount \( A \) after \( n \) years at compound interest is given by:
\[ A = P(1 + r)^n \]
where:
- \( P \) = principal amount (Rs. 18,000)
- \( r \) = rate of interest (10% or 0.10)
- \( n \) = number of years

### Step 2: Calculate the amount at the end of the 3rd year
Using the formula:
\[ A_3 = 18000(1 + 0.10)^3 \]
Calculating:
\[ A_3 = 18000(1.10)^3 = 18000 \times 1.331 = 23958 \]

### Step 3: Calculate the amount at the end of the 4th year
Using the formula:
\[ A_4 = 18000(1 + 0.10)^4 \]
Calculating:
\[ A_4 = 18000(1.10)^4 = 18000 \times 1.4641 = 26353.80 \]

### Step 4: Calculate the compound interest for the 3rd year
The compound interest for the 3rd year is the amount at the end of the 3rd year minus the amount at the end of the 2nd year.
First, we need to calculate the amount at the end of the 2nd year:
\[ A_2 = 18000(1 + 0.10)^2 = 18000(1.10)^2 = 18000 \times 1.21 = 21780 \]
Now, the compound interest for the 3rd year \( CI_3 \):
\[ CI_3 = A_3 - A_2 = 23958 - 21780 = 1178 \]

### Step 5: Calculate the compound interest for the 4th year
The compound interest for the 4th year \( CI_4 \):
\[ CI_4 = A_4 - A_3 = 26353.80 - 23958 = 2395.80 \]

### Step 6: Find the difference between the compound interest for the 3rd and 4th year
Now, we can find the difference:
\[ \text{Difference} = CI_4 - CI_3 = 2395.80 - 1178 = 1217.80 \]

### Final Answer
The difference between the compounded interest for the 3rd year and the 4th year is Rs. 1217.80.

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