To determine how long it would take for an investment to triple in its value at an annual simple interest rate of 10%, we can follow these steps:
1. Identify the given values:
- The rate of interest per annum (r) is 10%, or 0.10 when expressed as a decimal.
- Tripling the value means the future value (A) is three times the principal (P).
2. Use the formula for future value in simple interest:
The formula for simple interest is [tex]\(A = P(1 + rt)\)[/tex], where:
- [tex]\(A\)[/tex] is the future value,
- [tex]\(P\)[/tex] is the principal amount,
- [tex]\(r\)[/tex] is the annual interest rate,
- [tex]\(t\)[/tex] is the time in years.
3. Substitute the values for tripling the investment:
- Let’s assume the principal amount, [tex]\(P\)[/tex], is 1 (this simplification is valid because the time to triple does not depend on the principal amount's specific value).
- Triple the principal means [tex]\(A = 3P = 3 \times 1 = 3\)[/tex].
4. Set up the equation:
Substitute the known values into the formula:
[tex]\[
3 = 1 \times (1 + 0.10t)
\][/tex]
5. Isolate the variable [tex]\(t\)[/tex]:
- Solve the equation for [tex]\(t\)[/tex] as follows:
[tex]\[
3 = 1(1 + 0.10t)
\][/tex]
Simplify the equation by subtracting 1 from both sides:
[tex]\[
3 - 1 = 0.10t
\][/tex]
[tex]\[
2 = 0.10t
\][/tex]
Divide both sides by 0.10:
[tex]\[
t = \frac{2}{0.10}
\][/tex]
[tex]\[
t = 20
\][/tex]
6. Conclusion:
It would take 20 years for the investment to triple in its value at an annual simple interest rate of 10%.
