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Math  /  Algebra

QuestionDecide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. If interest rates stay at 6%6 \% APR and I continue to make my monthly $50\$ 50 deposits into my retirement plan, I should have at least $40,000\$ 40,000 saved when I retire in 35 years.
The statement □\square does not make sense because I will have \ \square$ in my retirement account when I retire in 35 years. (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

1. The interest rate is compounded monthly.
2. The monthly deposit is \$50.
3. The interest rate is 6% APR.
4. The time period is 35 years.

STEP 2

1. Convert the annual interest rate to a monthly interest rate.
2. Calculate the total number of deposits.
3. Use the future value of an annuity formula to calculate the total savings.
4. Compare the calculated savings to \$40,000.

STEP 3

Convert the annual interest rate to a monthly interest rate by dividing by 12.
Monthly interest rate=6%12=0.5%=0.005\text{Monthly interest rate} = \frac{6\%}{12} = 0.5\% = 0.005

STEP 4

Calculate the total number of monthly deposits over 35 years.
Total deposits=35×12=420\text{Total deposits} = 35 \times 12 = 420

STEP 5

Use the future value of an annuity formula:
FV=P×(1+r)n−1rFV = P \times \frac{(1 + r)^n - 1}{r}
where P=50 P = 50 , r=0.005 r = 0.005 , and n=420 n = 420 .
FV=50×(1+0.005)420−10.005FV = 50 \times \frac{(1 + 0.005)^{420} - 1}{0.005}
Calculate FV FV .

STEP 6

Calculate (1+0.005)420 (1 + 0.005)^{420} .
(1+0.005)420≈6.022575(1 + 0.005)^{420} \approx 6.022575
Substitute back into the formula:
FV=50×6.022575−10.005≈50×1004.515=50225.75FV = 50 \times \frac{6.022575 - 1}{0.005} \approx 50 \times 1004.515 = 50225.75

STEP 7

Compare the calculated savings to \$40,000.
Since \$50,225.75 is greater than \$40,000, the statement does make sense.
The statement does make sense because I will have \$50,225.75 in my retirement account when I retire in 35 years.

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