A bank offers a 25,000 loan at an interest rate of 7.7%...

A bank offers a $\$ 25,000$ loan at an interest rate of 7.7$\%$ that can be paid back over 2 to 10 years. a. Write the monthly payment formula for this loan situation. Let $t$ represent the number of years from 2 to 10 inclusive. b. Write the total interest formula for this loan situation. Let $t$ represent the number of years from 2 to 10 inclusive. c. Construct a graph. Let the indendent variable represent years and the dependent variable represent the interest paid. d. Use your graph to estimate the interest for a 6$\frac{1}{2}$ -year loan.

A bank offers a $\$ 25,000$ loan at an interest rate of 7.7$\%$ that can be paid back over 2 to 10 years.
a. Write the monthly payment formula for this loan situation. Let $t$ represent the number of years from 2 to 10 inclusive.
b. Write the total interest formula for this loan situation. Let $t$ represent the number of years from 2 to 10 inclusive.
c. Construct a graph. Let the indendent variable represent years and the dependent variable represent the interest paid.
d. Use your graph to estimate the interest for a 6$\frac{1}{2}$ -year loan.

Key Concepts

Total Interest Calculation

The total interest paid on a loan is determined by subtracting the initial principal from the sum of all periodic payments made over the life of the loan. This calculation shows the additional amount paid over and above the borrowed principal, providing insight into the overall cost of the loan.

Graphical Analysis of Loan Costs

Graphing the relationship between the duration of the loan and the total interest paid helps in visualizing how different loan terms affect the cost of borrowing. By plotting the total interest against the number of years, one can easily estimate the financial impact of choosing a particular repayment period and compare scenarios based on variations in loan duration.

Amortization Formula

The amortization formula is used to determine the fixed periodic payment needed to pay off a loan over a specified period at a constant interest rate. It incorporates the loan principal, the periodic interest rate, and the total number of payment periods to ensure that the loan is fully repaid with a zero balance at the end of the term.

Conversion of Annual Interest Rate to Monthly Interest Rate

When loan payments are made monthly, the annual interest rate must be converted to a monthly rate by dividing it by 12. This conversion is crucial because the amortization formula requires the interest rate and the number of periods to be in the same time units, ensuring that the calculation accurately reflects the cost of borrowing.

Transcript

00:01 All right, we got a lot, a lot, a lot going on here.

00:03 And i'm going to do my best to just make this as simple as possible.

00:06 First of all, this equation, 100 % taken from our general equation, just with the parameters plugged in.

00:14 First of all, y could kind of be m to make it even look more like that.

00:18 And see these xs, they could be t's as well, just to make this look a little bit more like our equation, right? and so all i plugged in was 25 ,000.

00:30 For the loan amount.

00:32 7 .7 % is a decimal looks like this, right? so everywhere there's an r i plug in 7 .7%.

00:38 And last, i think there is one more parameter.

00:43 No, actually because i couldn't plug in anything for the t because we don't know if it's 2, 3, 4, 5 .5 years up to 10.

00:51 So t remain t.

00:52 Otherwise, that's something that we're used to plugging in.

00:54 But here is that formula.

00:56 It's the general formula with 25 ,000 plugged in the loan amount and 7 .7 % as our rate.

01:05 Now, similar thing here, right? same thing, and you can change this y to m, you can change the x's to t's, but they wanted to know the amount of interest, right? so in the previous problems, think about how you calculated interest.

01:18 This gave you the monthly payment, and then you took the monthly payment, you multiplied it by 12 for the number of months and t or the number of years...

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