Calculating Annual Interest Rate for Compound Interest Formula
To solve this question, you'll need to use the formula for compound interest, which is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form, so 5% would be 0.05).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for in years.
From the question:
- \( P = $700 \)
- \( A = $854.13 \)
- \( t = 5 \) years
- \( n = 4 \) times per year (compounded every 3 months)
Now we have to find \( r \). Plugging in the values we have:
\[ 854.13 = 700 \left(1 + \frac{r}{4}\right)^{4 \cdot 5} \]
First, divide both sides by 700:
\[ \frac{854.13}{700} = \left(1 + \frac{r}{4}\right)^{20} \]
\[ 1.21932857 = \left(1 + \frac{r}{4}\right)^{20} \]
Now take the 20th root of both sides:
\[ \sqrt[20]{1.21932857} = 1 + \frac{r}{4} \]
\[ 1.0104239684 = 1 + \frac{r}{4} \]
Subtract 1 from both sides:
\[ 0.0104239684 = \frac{r}{4} \]
Multiply both sides by 4 to solve for \( r \):
\[ r = 0.0416958736 \]
Converting this to a percentage:
\[ r = 4.16958736\% \]
And now, rounding to three significant figures:
\[ r \approx 4.17\% \]
So, the annual interest rate is approximately 4.17% to three significant figures.
