Find the compound rate of interest of 10% rate for 2 years?

Options

A. `20%`
B. `21% `
C. `23%`
D. `22%`

Answer

To find the compound rate of interest for a rate of 10% over 2 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Principal Amount (P)**:
Let's assume the principal amount is \( P \).

2. **Calculate Interest for the First Year**:
The interest for the first year at a rate of 10% is calculated as:
\[ \text{Interest for Year 1} = P \times \frac{10}{100} = \frac{P}{10} \]

3. **Calculate the Total Amount After the First Year**:
The total amount after the first year will be the principal plus the interest earned:
\[ \text{Total Amount After Year 1} = P + \frac{P}{10} = P \left(1 + \frac{1}{10}\right) = P \left(\frac{11}{10}\right) \]

4. **Calculate Interest for the Second Year**:
The interest for the second year is calculated on the new total amount (which includes the interest from the first year):
\[ \text{Interest for Year 2} = \left(P \times \frac{11}{10}\right) \times \frac{10}{100} = \frac{11P}{100} \]

5. **Calculate the Total Amount After the Second Year**:
The total amount after the second year will be the amount after the first year plus the interest earned in the second year:
\[ \text{Total Amount After Year 2} = P \left(\frac{11}{10}\right) + \frac{11P}{100} \]
To combine these, we need a common denominator:
\[ = \frac{110P}{100} + \frac{11P}{100} = \frac{121P}{100} \]

6. **Calculate the Total Compound Interest Earned**:
The total compound interest earned over the 2 years is:
\[ \text{Total Compound Interest} = \text{Total Amount After Year 2} - P = \frac{121P}{100} - P = \frac{121P}{100} - \frac{100P}{100} = \frac{21P}{100} \]

7. **Calculate the Compound Rate of Interest**:
The compound rate of interest is calculated as the total interest earned divided by the principal, expressed as a percentage:
\[ \text{Compound Rate of Interest} = \left(\frac{\text{Total Compound Interest}}{P}\right) \times 100 = \left(\frac{\frac{21P}{100}}{P}\right) \times 100 = 21\% \]

### Final Answer:
The compound rate of interest for a rate of 10% over 2 years is **21%**.

---