To determine how many years it will take to double your money in a compound interest account with a 7.2% annual interest rate, follow these steps:
1. Understanding Compound Interest:
- Compound interest means that the interest earned each year is added to the principal, so that the balance doesn’t merely grow, it grows at an increasing rate. The formula for compound interest is:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
where:
- [tex]\(A\)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\(P\)[/tex] is the principal amount (the initial amount of money).
- [tex]\(r\)[/tex] is the annual interest rate (decimal).
- [tex]\(n\)[/tex] is the number of times that interest is compounded per year.
- [tex]\(t\)[/tex] is the time the money is invested or borrowed for, in years.
2. Problem Simplification:
- For this problem, [tex]\(n = 1\)[/tex] because the interest is compounded yearly.
- The interest rate [tex]\(r = 7.2\% = 0.072\)[/tex].
- We want to double our initial investment, so [tex]\(A = 2P\)[/tex].
3. Rearranging the Formula to Solve for [tex]\(t\)[/tex]:
- Starting from [tex]\(A = 2P = P \left(1 + \frac{r}{n}\right)^{nt}\)[/tex].
- Dividing both sides by [tex]\(P\)[/tex] gives: [tex]\(2 = \left(1 + \frac{r}{n}\right)^{nt}\)[/tex].
- Taking the natural logarithm (ln) of both sides to solve for [tex]\(t\)[/tex]:
[tex]\[
\ln(2) = nt \cdot \ln\left(1 + \frac{r}{n}\right)
\][/tex]
- Since [tex]\(n = 1\)[/tex]:
[tex]\[
\ln(2) = t \cdot \ln(1 + r)
\][/tex]
- Solving for [tex]\(t\)[/tex]:
[tex]\[
t = \frac{\ln(2)}{\ln(1 + r)}
\][/tex]
4. Calculating the Exact Value:
- We can use the values [tex]\(\ln(2)\)[/tex] and [tex]\(\ln(1 + 0.072)\)[/tex] to find [tex]\(t\)[/tex]:
- Using these values, calculate:
[tex]\[
t = \frac{\ln(2)}{\ln(1 + 0.072)}
\][/tex]
By following these calculations, you find that:
The time it will take to double your money, considering a 7.2% annual interest rate compounded yearly, is approximately 10 years.
