At what rate per cent compound interest does a sum of money become four-fold in 2 years?

Answer

To solve the problem of finding the rate per cent at which a sum of money becomes four-fold in 2 years under compound interest, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find the rate of compound interest (R) at which a principal amount (P) becomes four times itself in 2 years. This means the final amount (A) will be 4P.

2. **Use the Compound Interest Formula**: The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^t \]
where:
- A = final amount
- P = principal amount
- R = rate of interest per annum
- t = time in years

3. **Substitute the Known Values**: In our case, we know that:
- A = 4P
- t = 2 years
Thus, we can substitute these values into the formula:
\[ 4P = P \left(1 + \frac{R}{100}\right)^2 \]

4. **Cancel Out P**: Since P is common on both sides and P ≠ 0, we can cancel it:
\[ 4 = \left(1 + \frac{R}{100}\right)^2 \]

5. **Take the Square Root**: To eliminate the square, we take the square root of both sides:
\[ \sqrt{4} = 1 + \frac{R}{100} \]
This simplifies to:
\[ 2 = 1 + \frac{R}{100} \]

6. **Solve for R**: Now, we can isolate R:
\[ \frac{R}{100} = 2 - 1 \]
\[ \frac{R}{100} = 1 \]
Multiply both sides by 100:
\[ R = 100 \]

7. **Conclusion**: The rate of compound interest required for the sum of money to become four-fold in 2 years is:
\[ R = 100\% \text{ per annum} \]