watch the video and then solve the problem given below. click here to watch the video. at age 30, you decide…

watch the video and then solve the problem given below. click here to watch the video. at age 30, you decide to deposit $150 at the end of each month into an ira that pays 4.5% compounded monthly. how much will have you have when you retire at 65? $ (simplify your answer. round to the nearest dollar as needed.)

watch the video and then solve the problem given below. click here to watch the video. at age 30, you decide to deposit $150 at the end of each month into an ira that pays 4.5% compounded monthly. how much will have you have when you retire at 65? $ (simplify your answer. round to the nearest dollar as needed.)

Answer

Explanation:

Step1: Identify the variables

The monthly - payment $P = 150$, the annual interest rate $r=4.5%=0.045$, so the monthly interest rate $i=\frac{0.045}{12}=0.00375$. The number of years from age 30 to age 65 is $n = 65 - 30=35$ years. The number of payments $t=35\times12 = 420$ months.

Step2: Use the future - value of an ordinary annuity formula

The formula for the future value $F$ of an ordinary annuity is $F=P\times\frac{(1 + i)^{t}-1}{i}$. Substitute $P = 150$, $i=0.00375$, and $t = 420$ into the formula: [ \begin{align*} F&=150\times\frac{(1 + 0.00375)^{420}-1}{0.00375}\ \end{align*} ] First, calculate $(1 + 0.00375)^{420}$. Let $x=(1 + 0.00375)^{420}$. Using the formula $a^{b}=e^{b\ln(a)}$, we have $\ln(x)=420\times\ln(1.00375)\approx420\times0.003743=1.57206$. Then $x = e^{1.57206}\approx4.8107$. [ \begin{align*} F&=150\times\frac{4.8107-1}{0.00375}\ &=150\times\frac{3.8107}{0.00375}\ &=150\times1016.1867\ &=152428.005 \end{align*} ]

Answer:

$152428$