At what rate of compund inter est per annum a sum of Rs. 1800 will become `Rs. 2178` in two years?

Options

A. `11%`
B. `12%`
C. `10%`
D. `13% `

Answer

To find the rate of compound interest per annum at which a sum of Rs. 1800 will become Rs. 2178 in two years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Principal (P) = Rs. 1800
- Amount (A) = Rs. 2178
- Time (T) = 2 years

2. **Calculate the Total Interest Earned:**
- Total Interest (I) = Amount (A) - Principal (P)
- I = 2178 - 1800 = Rs. 378

3. **Use the Compound Interest Formula:**
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^T \]
where:
- A = Amount after time T
- P = Principal
- r = Rate of interest per annum
- T = Time in years

4. **Substituting the Known Values:**
\[ 2178 = 1800 \left(1 + \frac{r}{100}\right)^2 \]

5. **Divide Both Sides by 1800:**
\[ \frac{2178}{1800} = \left(1 + \frac{r}{100}\right)^2 \]
\[ 1.21 = \left(1 + \frac{r}{100}\right)^2 \]

6. **Taking the Square Root of Both Sides:**
\[ \sqrt{1.21} = 1 + \frac{r}{100} \]
\[ 1.1 = 1 + \frac{r}{100} \]

7. **Isolate r:**
\[ 1.1 - 1 = \frac{r}{100} \]
\[ 0.1 = \frac{r}{100} \]
\[ r = 0.1 \times 100 = 10 \]

8. **Conclusion:**
The rate of compound interest per annum is **10%**.