The minimum time in which some amount will become three times at 25% rate of compound interest?

Options

A. 3 year
B. 3 year
C. 6 year
D. 5 year

Answer

To find the minimum time in which some amount will become three times at a 25% rate of compound interest, we can use the formula for compound interest.

### Step-by-Step Solution:

1. **Understand the Problem**: We want to find the time (t) it takes for a principal amount (P) to grow to three times its original amount (3P) at a compound interest rate of 25%.

2. **Use the Compound Interest Formula**: The formula for compound interest is:
\[ A = P(1 + r)^t \]
where:
- \( A \) is the final amount (3P in our case),
- \( P \) is the principal amount,
- \( r \) is the rate of interest (25% or 0.25),
- \( t \) is the time in years.

3. **Set Up the Equation**: Since we want the amount to be three times the principal, we can set up the equation as follows:
\[ 3P = P(1 + 0.25)^t \]

4. **Simplify the Equation**: We can divide both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = (1.25)^t \]

5. **Take Logarithm of Both Sides**: To solve for \( t \), we take the logarithm of both sides:
\[ \log(3) = \log((1.25)^t) \]

6. **Use the Power Rule of Logarithms**: According to the power rule, we can bring down the exponent \( t \):
\[ \log(3) = t \cdot \log(1.25) \]

7. **Solve for \( t \)**: Now, we can solve for \( t \):
\[ t = \frac{\log(3)}{\log(1.25)} \]

8. **Calculate the Values**: Using a calculator, we find:
- \( \log(3) \approx 0.4771 \)
- \( \log(1.25) \approx 0.0969 \)

Therefore,
\[ t \approx \frac{0.4771}{0.0969} \approx 4.93 \]

9. **Conclusion**: The minimum time required for the amount to become three times at a 25% compound interest rate is approximately 4.93 years.