Bernice invests $700 at a rate of r % per year compound interest, compounded every 3 months. At the end of 5 years the value of her investment is $854.13, correct to 2 decimal places. Calculate the value of r, correct to 3 significant figures. Answer r = ___

1. Identify the number of compounding periods and set up the compound interest equation.

Compounded every 3 months means $4$ times per year.

In $5$ years, number of periods $=5\times 4=20$.

The rate per period is $\dfrac{r}{4}\% = \dfrac{r}{400}$ as a multiplier increase.

$700\left(1+\frac{r}{400}\right)^{20}=854.13$

2. Isolate the quarterly growth factor.

$\left(1+\frac{r}{400}\right)^{20}=\frac{854.13}{700}$

$1+\frac{r}{400}=\left(\frac{854.13}{700}\right)^{\frac{1}{20}}$

3. Solve for $r$ and round to 3 significant figures.

$\frac{r}{400}=\left(\frac{854.13}{700}\right)^{\frac{1}{20}}-1$

$r=400\left[\left(\frac{854.13}{700}\right)^{\frac{1}{20}}-1\right]$

$r=3.99998\ldots$

Correct to 3 significant figures,

$r=4.00$