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QuestionA 20-year-old deposits \55monthlyinanIRAat55 monthly in an IRA at 4\%$ APR. What is the amount at age 65 compared to total deposits?

Studdy Solution

STEP 1

Assumptions1. The person starts the IRA at age20 and retires at age65. . The APR is4%.
3. The person deposits $55 at the end of each month.
4. The interest is compounded monthly.
5. The total amount in the IRA when the person retires is the sum of the compounded monthly deposits.

STEP 2

First, we need to calculate the number of months the person will be making deposits. This is the difference in years between the retirement age and the start age, multiplied by12 (the number of months in a year).
Number of months=(Retirement age−Start age)times12Number\, of\, months = (Retirement\, age - Start\, age) \\times12

STEP 3

Now, plug in the given values for the retirement age and start age to calculate the number of months.
Number of months=(65−20)times12Number\, of\, months = (65 -20) \\times12

STEP 4

Calculate the number of months.
Number of months=(65−20)times12=540Number\, of\, months = (65 -20) \\times12 =540

STEP 5

Next, we need to calculate the total amount in the IRA when the person retires. This is a future value of a series problem, and the formula isV=times((1+r)n−1r)V = \\times \left( \frac{(1 + r)^n -1}{r} \right)where- VV is the future value of the account- $$ is the monthly deposit- $r$ is the monthly interest rate- $n$ is the number of months

STEP 6

Before we can use this formula, we need to convert the APR to a monthly interest rate. This is done by dividing the APR by12.
r=APR/12r = APR /12

STEP 7

Now, plug in the given value for the APR to calculate the monthly interest rate.
r=4%/12r =4\% /12

STEP 8

Convert the percentage to a decimal value.
4%=0.044\% =0.04r=0.04/12r =0.04 /12

STEP 9

Calculate the monthly interest rate.
r=.04/12=.003333r =.04 /12 =.003333

STEP 10

Now, plug in the values for $$, $r$, and $n$ into the future value formula to calculate the total amount in the IRA when the person retires.
V=$55times((+0.003333)540−0.003333)V = \$55 \\times \left( \frac{( +0.003333)^{540} -}{0.003333} \right)

STEP 11

Calculate the future value.
V=$55times((+0.003333)540−0.003333)=$165,432.17V = \$55 \\times \left( \frac{( +0.003333)^{540} -}{0.003333} \right) = \$165,432.17

STEP 12

Next, we need to calculate the total deposits made over the time period. This is simply the monthly deposit amount multiplied by the number of months.
Total deposits=Monthly deposittimesNumber of monthsTotal\, deposits = Monthly\, deposit \\times Number\, of\, months

STEP 13

Now, plug in the values for the monthly deposit and the number of months to calculate the total deposits.
Total deposits=$55times540Total\, deposits = \$55 \\times540

STEP 14

Calculate the total deposits.
Total deposits=$55times540=$29,700Total\, deposits = \$55 \\times540 = \$29,700The IRA will contain 165,432.17whenthepersonretires,comparedtothetotaldepositsof165,432.17 when the person retires, compared to the total deposits of 29,700 over the time period.

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