If a sum increases by 21% after 2 years, then the rate of compound interset per annum , when compounded annually must be.

Options

A. `10.5 %`
B. `11.5 %`
C. `10%`
D. `11%`

Answer

To find the rate of compound interest per annum when a sum increases by 21% after 2 years, we can follow these steps:

### Step 1: Understand the problem
We know that the sum increases by 21% over 2 years. This means if we start with a principal amount (let's denote it as P), after 2 years, the total amount (A) will be:
\[ A = P + 0.21P = 1.21P \]

### Step 2: Use the formula for compound interest
The formula for compound interest when compounded annually is given by:
\[ A = P(1 + r)^n \]
where:
- \( A \) is the amount after n years,
- \( P \) is the principal amount,
- \( r \) is the rate of interest (in decimal),
- \( n \) is the number of years.

In our case, we have:
- \( A = 1.21P \)
- \( n = 2 \)

### Step 3: Substitute the values into the formula
Substituting the known values into the compound interest formula:
\[ 1.21P = P(1 + r)^2 \]

### Step 4: Simplify the equation
We can divide both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 1.21 = (1 + r)^2 \]

### Step 5: Take the square root of both sides
To solve for \( r \), we take the square root of both sides:
\[ \sqrt{1.21} = 1 + r \]

Calculating the square root:
\[ 1.1 = 1 + r \]

### Step 6: Solve for r
Now, we isolate \( r \):
\[ r = 1.1 - 1 = 0.1 \]

### Step 7: Convert r to percentage
To convert \( r \) into a percentage, we multiply by 100:
\[ r = 0.1 \times 100 = 10\% \]

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

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