Question

Suppose an annuity pays 4% annual interest, compounded annually. If you invest $4,500 in this annuity annually for 10 years, what percentage of the total balance is interest earned? Round your answer to the nearest hundredth of a percent. Do NOT round until you have calculated the final answer.

Answer 1

16.71%

Explanation:

To answer this question, we will calculate the total deposits into the account and then use the savings annuity formula to calculate the balance in the account after 10 years. The sum of all deposits D10 is

D10=10⋅$4,500=$45,000.

We will use the formula

A(t)=d((1+r/n)n⋅t−1)r/n

to find the value of A(10). We know that r=0.04, d=$4,500, n=1 compounding period per year and, t=10 years. Substitute these values into the savings annuity formula to give

A(10)=$4,500⋅((1+0.04/1)10⋅1−1)/(0.04/1).

Simplifying gives A(10)=$4,500⋅((1.04)10−1)/(0.04) and therefore A(10)=$54,027.48.

Now, substitute our numbers into

A(10)=D10+I10

to give

$54,027.48=$45,000+I10.

We now solve for I10 to yield the total interest paid as $9,027.48. Now compute I10/A(10) to get

I10A(10)=$9,027.48$54,027.48=0.16709

(to five decimals). Converting to a percentage we get 16.71%.

Answer 2

`~~~~~~~~~~~~\stackrel{\textit{payments at the beginning of the period}}{\textit{Future Value of an annuity due}} \\\\ A=pmt\left[ \cfrac{\left( 1+(r)/(n) \right)^(nt)-1}{(r)/(n)} \right]\left(1+(r)/(n)\right)`

`\qquad \begin{cases} A=\textit{accumulated amount} \\ pmt=\textit{periodic payments}\dotfill & 4500\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &10 \end{cases}`

`A=4500\left[ \cfrac{\left( 1+(0.04)/(1) \right)^(1 \cdot 10)-1}{(0.04)/(1)} \right]\left(1+(0.04)/(1)\right) \\\\\\ A=4500\left[ \cfrac{(1.04)^(10)-1}{0.04} \right](1.04) \implies A \approx 56188.58`

so every year you were putting in 4500 bucks, so for 10 years that'd be a total deposits for 4500*10 = 45000, so let's squeeze out the 45000 from the the total, that gives us 56188.58 - 45000 ≈ 11188.58.

so, if we take 56188.58 to be the 100%, what's 11188.58 off of it in percentage?

`\begin{array}{ccll} amount&\%\\ \cline{1-2} 56188.58 & 100\\ 11188.58& x \end{array} \implies \cfrac{56188.58}{11188.58}~~=~~\cfrac{100}{x} \implies 56188.58x=1118858 \\\\\\ x=\cfrac{1118858}{56188.58}\implies x\approx \stackrel{\%}{19.91}`