# How much interest did she earn? (compound interest)

Jul 2, 2018

Total Interest of 4 years is: $195.83 #### Explanation: Given: Annual compound interest is 10% Calculation cycle: 4 times a year. Final sum in account:$600.00 after 4 years.

Let the initial unknown principle sum be $P$
Let the total interest in dollars be $x$
Let the count of years be $n = 4$
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P(1+10/(nxx100))^(4n) =" total in the account"=$600 As the is a cycle of 4 calculations per year we have the indices of $4 n$This is over 4 years so $n = 4$giving: P(1+10/(400))^(16) =$600

P(410/(400))^(16) =$600 P=$600-:(410/400)^(16) = $404.17496...  Rounding gives: P=$404.17

Thus the interest is the difference.

$600.00 ul($404.17 larr" Subtract")
$195.83 Jul 2, 2018 color(blue)($195.83)

#### Explanation:

Compound interest is given by the formula:

$\text{FV"="PV} {\left(1 + \frac{r}{n}\right)}^{n t}$

Where:

$F V = \text{ future value}$

$P V = \text{ present value}$

$\boldsymbol{r} = \text{ interest rate}$ ( Given as a decimal )

$\boldsymbol{n} = \text{ compounding period}$

$\boldsymbol{t} = \text{ time in years}$

We are give interest rate of 10% so.

$r = \frac{10}{100} = 0.1$

compounding period is quarterly so,

$n = 4$

Time period is 4 years so:

$t = 4$

$F V = 600$

To find the amount of interest earned, we first need to find the present value:

Using:

$\text{FV"="PV} {\left(1 + \frac{r}{n}\right)}^{n t}$

$600 = P V {\left(1 + \frac{0.1}{4}\right)}^{16}$

$600 = P V {\left(1.025\right)}^{16}$

$P V = \frac{600}{{\left(1.025\right)}^{16}} = 404.1749600$

Interest earned is:

$F V - P V$

$600 - 404.1749600 = 195.8250400$

\$195.83 to nearest cent.