Question

1. suppose mickey deposits $318 into an account earning 2.74% annual interest compounded monthly at the end of every month.

i. how much money will be in the account if he does this for 15 years? show all work. round your answer to the nearest cent. your answer must be accurate before rounding.
2.74% 0.0274 n=15·12=180

Explanation:

Step1: Identify the formula for compound interest (monthly compounding, end - of - month deposits: ordinary annuity formula)

The formula for the future value of an ordinary annuity is $F = P\times\frac{(1 + r)^{n}-1}{r}$, where:

• $P$ is the payment per period (monthly deposit), here $P=\$318$.
• $r$ is the interest rate per period. The annual interest rate $i = 2.74\%=0.0274$. Since it is compounded monthly, the monthly interest rate $r=\frac{0.0274}{12}\approx0.00228333$.
• $n$ is the number of periods. The number of years is $t = 15$, and since compounding is monthly, $n=t\times12=15\times12 = 180$.

Step2: Substitute the values into the formula

First, calculate $(1 + r)^{n}$:
$(1 + 0.00228333)^{180}$. Let's compute $1+0.00228333 = 1.00228333$. Then, $(1.00228333)^{180}$. Using a calculator, $(1.00228333)^{180}\approx1.500744$.

Next, calculate $(1 + r)^{n}-1$: $1.500744-1 = 0.500744$.

Then, calculate $\frac{(1 + r)^{n}-1}{r}$: $\frac{0.500744}{0.00228333}\approx219.303$.

Finally, calculate $F = P\times\frac{(1 + r)^{n}-1}{r}$: $F = 318\times219.303$.

$318\times219.303 = 318\times(200 + 19.303)=318\times200+318\times19.303=63600+6138.354 = 69738.354$.

Answer:

The amount of money in the account after 15 years will be approximately $\$69738.35$ (rounded to the nearest cent).