To determine the rate of interest at which a sum of money triples itself in 25 years under compound interest, we can use the compound interest formula, which is given by:
[tex]\[ A = P \left(1 + \frac{R}{100}\right)^T \][/tex]
Here,
- [tex]\( A \)[/tex] is the final amount,
- [tex]\( P \)[/tex] is the principal amount,
- [tex]\( R \)[/tex] is the annual interest rate (in percent),
- [tex]\( T \)[/tex] is the time in years.
Given that the amount triples itself in 25 years, we have:
[tex]\[ A = 3P \][/tex]
[tex]\[ P \text{ is the principal amount (which cancels out in the equations)}, \][/tex]
[tex]\[ T = 25 \][/tex] (years).
Substituting these values into the compound interest formula:
[tex]\[ 3P = P \left(1 + \frac{R}{100}\right)^{25} \][/tex]
Next, we can simplify by dividing both sides by [tex]\( P \)[/tex]:
[tex]\[ 3 = \left(1 + \frac{R}{100}\right)^{25} \][/tex]
Our goal is to solve for [tex]\( R \)[/tex]. We take the 25th root of both sides to remove the exponent:
[tex]\[ \left(1 + \frac{R}{100}\right) = 3^{1/25} \][/tex]
Now, subtract 1 from both sides:
[tex]\[ \frac{R}{100} = 3^{1/25} - 1 \][/tex]
Finally, multiply both sides by 100 to express [tex]\( R \)[/tex] as a percentage:
[tex]\[ R = \left(3^{1/25} - 1\right) \times 100 \][/tex]
After completing this calculation, the rate [tex]\( R \)[/tex] is approximately:
[tex]\[ R \approx 4.49\% \][/tex]
Therefore, the annual interest rate needed for a sum of money to triple itself in 25 years is approximately [tex]\( 4.49\% \)[/tex] per annum.
