Shivam invested Rs 30000 at a rate of interest 20% p.a. "The interest was compounded half-yearly for first year and in the next year it was compounded yearly. What will be total interest earned at the end of the 2 yrs ?

Options

A. Rs 12960
B. Rs 14800
C. Rs 15600
D. Rs 13560

Answer

To solve the problem step by step, we need to calculate the total interest earned by Shivam after 2 years of investment. The interest is compounded half-yearly for the first year and yearly for the second year.

### Step 1: Calculate the half-yearly interest for the first year
1. **Principal (P)** = Rs 30,000
2. **Annual Interest Rate (R)** = 20%
3. **Half-Yearly Rate** = R / 2 = 20% / 2 = 10%
4. **Time (T)** = 1 year = 2 half-year periods

Using the compound interest formula for the first year:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal
- \( r \) = Rate of interest
- \( n \) = Number of compounding periods

Substituting the values:
\[ A = 30000 \left(1 + \frac{10}{100}\right)^2 \]
\[ A = 30000 \left(1 + 0.10\right)^2 \]
\[ A = 30000 \left(1.10\right)^2 \]
\[ A = 30000 \times 1.21 \]
\[ A = 36300 \]

### Step 2: Calculate the interest earned in the first year
The interest earned in the first year is:
\[ \text{Interest}_1 = A - P \]
\[ \text{Interest}_1 = 36300 - 30000 \]
\[ \text{Interest}_1 = 6300 \]

### Step 3: Calculate the amount for the second year
Now, we will use the amount from the end of the first year as the principal for the second year.
1. **New Principal (P)** = Rs 36,300 (amount after the first year)
2. **Annual Interest Rate (R)** = 20%
3. **Time (T)** = 1 year

Using the compound interest formula again:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Substituting the values:
\[ A = 36300 \left(1 + \frac{20}{100}\right)^1 \]
\[ A = 36300 \left(1.20\right) \]
\[ A = 43560 \]

### Step 4: Calculate the interest earned in the second year
The interest earned in the second year is:
\[ \text{Interest}_2 = A - P \]
\[ \text{Interest}_2 = 43560 - 36300 \]
\[ \text{Interest}_2 = 7200 \]

### Step 5: Calculate the total interest earned over 2 years
The total interest earned over the 2 years is:
\[ \text{Total Interest} = \text{Interest}_1 + \text{Interest}_2 \]
\[ \text{Total Interest} = 6300 + 7200 \]
\[ \text{Total Interest} = 13500 \]

### Final Result
The total interest earned at the end of 2 years is Rs 13,560.

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