Question - Calculating Annual Interest Rate for Compound Interest Formula
mathcompound interest formulainterest rate calculationfinancial mathematicscalculating annual interest rate
Solution:
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.
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