If a sum of ₹ 2000 is lent at 10% p.a. compound interest, what is the interest for the second year?

Options

A. ₹225
B. ₹200
C. ₹220
D. ₹440

Answer

To find the interest for the second year when a sum of ₹2000 is lent at 10% per annum compound interest, we can follow these steps:

### Step 1: Calculate the total amount after 2 years using the compound interest formula.
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( A \) = total amount after \( n \) years
- \( P \) = principal amount (initial sum)
- \( r \) = rate of interest per annum
- \( n \) = number of years

Here, \( P = 2000 \), \( r = 10\% \), and \( n = 2 \).

Substituting the values:
\[ A = 2000 \left(1 + \frac{10}{100}\right)^2 \]
\[ A = 2000 \left(1 + 0.1\right)^2 \]
\[ A = 2000 \left(1.1\right)^2 \]
\[ A = 2000 \times 1.21 \]
\[ A = 2420 \]

### Step 2: Calculate the total interest for 2 years.
The total interest earned over 2 years is given by:
\[ \text{Total Interest} = A - P \]
Substituting the values:
\[ \text{Total Interest} = 2420 - 2000 \]
\[ \text{Total Interest} = 420 \]

### Step 3: Calculate the interest for the first year.
To find the interest for the first year, we will use the same formula for \( n = 1 \):
\[ A_1 = 2000 \left(1 + \frac{10}{100}\right)^1 \]
\[ A_1 = 2000 \left(1.1\right) \]
\[ A_1 = 2000 \times 1.1 \]
\[ A_1 = 2200 \]

Now, calculating the interest for the first year:
\[ \text{Interest for 1st year} = A_1 - P \]
\[ \text{Interest for 1st year} = 2200 - 2000 \]
\[ \text{Interest for 1st year} = 200 \]

### Step 4: Calculate the interest for the second year.
The interest for the second year can be found by subtracting the interest for the first year from the total interest for 2 years:
\[ \text{Interest for 2nd year} = \text{Total Interest} - \text{Interest for 1st year} \]
\[ \text{Interest for 2nd year} = 420 - 200 \]
\[ \text{Interest for 2nd year} = 220 \]

### Final Answer:
The interest for the second year is ₹220.

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