Question

Bryce wants to renovate his kitchen. The construction estimate is $28,400. He can finance through his local bank or directly through the construction company. The bank (loan A) offers a maximum 6-year loan at 8% annual interest, resulting in monthly payments of $497.94.The construction company (loan B) offers a maximum 10-year loan at 5.5% annual interest, resulting in monthly payments of $308.21.Which loan requires Bryce to pay less, and by how much?First, calculate the total payback for each loan.

Answer 1

Loan A requires Bryce to pay less by $1,133.52

Step-by-step explanation:

For bank A, we're given;

Length of loan in years(t) = 6 years

Annual interest rate(r) = 8% = 8/100 = 0.08

Loan payment(PMT) = $497.94

Number of compounds per year(n) = 12

We can go ahead and determine the loan amount using the below formula as seen below;

`\begin{gathered} Loan\text{ }amount,P_0=(PMT(1-(1+(r)/(n))^(-nt)))/((r)/(n)) \\ \\ =(497.94(1-(1+(0.08)/(12))^(-12*6))/((0.08)/(12)) \\ \\ =\text{ \$}28399.77 \end{gathered}`

So we can see that Bryce will pay back;

`\begin{gathered} Bank\text{ }A\text{ }Loan\text{ }Payback=497.94(12*6) \\ \\ =497.94\left(72\right) \\ \\ =\text{\$}35,851.68 \end{gathered}`

For bank B, we're given;

Length of loan in years(t) = 10 years

Annual interest rate(r) = 5.5% = 5.5/100 = 0.055

Loan payment(PMT) = $308.21

Number of compounds per year(n) = 12

We can go ahead and determine the loan amount using the below formula as seen below;

`\begin{gathered} Loan\text{ }amount,P_0=(PMT(1-(1+(r)/(n))^(-nt)))/((r)/(n)) \\ \\ =(308.21(1-(1+(0.055)/(12))^(-12*10))/((0.055)/(12)) \\ \\ =\text{ \$}28399.57 \end{gathered}`

So we can see that Bryce will pay back;

`\begin{gathered} Bank\text{ B }Loan\text{ }Payback=308.21(12*10) \\ \\ =308.21\left(120\right) \\ \\ =\text{\$}36985.20 \end{gathered}`

We can see that Bryce will pay back $35,851.68 for Loan A and $36,985.20 for Loan B, so Loan A requires Bryce to pay less and by $1,133.52