Question

trent is 25 years old and works for a company that matches his 401(k) contribution up to 5%. the interest rate for his 401(k) is 7.3%. if he puts away 10% of his $32,000 salary every year, what would the total of his 401(k) be in 10 years? round your answer to the nearest cent. a. $67,266.16 b. $72,176.59 c. $33,250.60 d. $35,677.89

Explanation:

Step1: Calculate Annual Contribution

Trent's salary is $32,000. He contributes 10%, and the company matches up to 5%. So total contribution rate is \(10\% + 5\% = 15\%\).
Annual contribution: \(0.15\times32000 = 4800\) dollars.

Step2: Use Future Value of Annuity Formula

The formula for the future value of an ordinary annuity is \(FV = P\times\frac{(1 + r)^n - 1}{r}\), where \(P = 4800\) (annual payment), \(r=\frac{0.073}{1}\) (annual interest rate), \(n = 10\) (number of years).

First, calculate \((1 + 0.073)^{10}\):
\((1.073)^{10}\approx1.9837\)

Then, \((1.9837 - 1)=0.9837\)

Next, \(\frac{0.9837}{0.073}\approx13.4753\)

Finally, \(FV = 4800\times13.4753\approx64681.44\)? Wait, no, maybe I made a mistake. Wait, wait, no—wait, maybe the company's match: wait, Trent contributes 10%, company matches up to 5%, so his own contribution is \(0.10\times32000 = 3200\), company matches \(0.05\times32000 = 1600\), so total annual contribution is \(3200 + 1600 = 4800\), that's correct. Wait, but maybe the formula is for compound interest with annual deposits. Wait, let's recalculate \((1.073)^{10}\):

Using a calculator, \(1.073^{10}\):
\(1.073^1 = 1.073\)
\(1.073^2 = 1.073\times1.073\approx1.1513\)
\(1.073^3\approx1.1513\times1.073\approx1.2354\)
\(1.073^4\approx1.2354\times1.073\approx1.3256\)
\(1.073^5\approx1.3256\times1.073\approx1.4224\)
\(1.073^6\approx1.4224\times1.073\approx1.5262\)
\(1.073^7\approx1.5262\times1.073\approx1.6376\)
\(1.073^8\approx1.6376\times1.073\approx1.7571\)
\(1.073^9\approx1.7571\times1.073\approx1.8854\)
\(1.073^{10}\approx1.8854\times1.073\approx2.0230\)

Now, \((2.0230 - 1)=1.0230\)

\(\frac{1.0230}{0.073}\approx14.0137\)

Then \(FV = 4800\times14.0137\approx67265.76\), which is close to option a (\$67,266.16, slight difference due to more precise calculation of \((1.073)^{10}\)).

Answer:

a. $67,266.16