A savings account has an annual percentage rate of 3.4% with interest compounded quarterly. Ashley deposits 2,000 dollars into the account. What is the account balance after 4 years, rounded to the nearest cent?

Step-by-step solution

1. 1

   Choose the periodic compounding formula. Because interest is added a fixed number of times per year, the balance is $A=P\left(1+\frac{r}{n}\right)^{nt},$ where $P$ is the deposit, $r$ the annual rate as a decimal, $n$ the compoundings per year, and $t$ the number of years. This is not continuous compounding, so $Pe^{rt}$ would be the wrong model here.

2. 2

   Identify each input carefully. $P=2000,\qquad r=0.034,\qquad n=4,\qquad t=4.$ Two things trip people up: the APR must be converted from $3.4\%$ to $0.034$, and "quarterly" means $n=4$ periods per year, so the exponent counts quarters, not years.

3. 3

   Build the periodic rate and the period count. The rate per quarter is $\frac{r}{n}=\frac{0.034}{4}=0.0085\quad (0.85\%\text{ per quarter}),$ and the number of compounding periods is $nt=4\times 4=16\text{ quarters}.$

4. 4

   Evaluate the growth factor to enough digits. $1.0085^{16}=1.14502360738\ldots$ Precision matters here: truncating the factor to $1.1437$ would understate the balance by nearly three dollars, and to $1.145$ would overstate it by a few cents.

5. 5

   Multiply by the principal and round. $A=2000\times 1.14502360738=2290.04721476\ldots$ Rounded to the nearest cent, $A=\$2{,}290.05.$ Of that, \290.05$ is interest earned.

6. 6

   Sanity-check against simple interest. Simple interest would give 2000(1+0.034\times 4)=2000\times 1.136=\2{,}272.00$. Compounding quarterly must beat that, and it does, by$$18.05$- a small but positive gap, exactly what a modest$3.4%$ rate over four years should produce.