Question

1. brad wants to save $35000 so he can purchase a new vehicle with cash. he can put $150 per week into an investment that earns an annual rate of 3.25% compounded quarterly. how many months until he will have enough money saved? complete the table and circle the value that was calculated. pv $0 fv $35000 periods rate 3.25% payment pmt/yr cmp/yr how many years? 2.73 years

Explanation:

Step1: Determine the interest rate per period

The annual interest rate \(r = 3.25\%=0.0325\). Since it is compounded quarterly (\(CMP/yr = 4\)), the interest rate per period \(i=\frac{r}{CMP/yr}=\frac{0.0325}{4}= 0.008125\).

Step2: Determine the number of periods formula

The formula for the future - value of an ordinary annuity is \(FV = PMT\times\frac{(1 + i)^{n}-1}{i}\). We know \(FV=\$35000\), \(PMT = 150\), and \(i = 0.008125\). Rearranging the formula for \(n\): \((1 + i)^{n}=\frac{FV\times i}{PMT}+1\).
Substitute the values: \((1 + 0.008125)^{n}=\frac{35000\times0.008125}{150}+1\).
First, calculate \(\frac{35000\times0.008125}{150}=\frac{284.375}{150}\approx1.89583\). Then \((1.008125)^{n}=1.89583 + 1=2.89583\).
Take the natural logarithm of both sides: \(n\ln(1.008125)=\ln(2.89583)\).
Since \(\ln(1.008125)\approx0.00809\) and \(\ln(2.89583)\approx1.063\), then \(n=\frac{\ln(2.89583)}{\ln(1.008125)}=\frac{1.063}{0.00809}\approx131.4\) periods.

Step3: Convert periods to months

Since the payments are made quarterly (every 3 months), the number of months \(m = n\times3\). So \(m=131.4\times3 = 394.2\approx394\) months.

Answer:

\(394\) months