Problem 15 - Calculus and its applications

Question

Credit cards. Joanna uses her credit card to finance a \(\$ 500\) purchase. Her card charges an annual interest rate of \(22.75 \%,\) compounded monthly, and assumes a \(10-y r\) term. Assume that Joanna makes no further purchases on her credit card. a) Find Joanna's monthly credit card payment. b) Assume that Joanna makes every payment for the life of the loan. Find her total payments. c) How much interest does Joanna pay over the life of the loan?

Step-by-Step Solution

Answer

a) \$11.85 b) \$1422.00 c) \$922.00

Step 1: Understanding the Given Information

Joanna has made a $500 purchase with a credit card. The annual interest rate is 22.75%, compounded monthly, and the loan term is 10 years. We need to find her monthly payment, total payments, and total interest paid over the 10 years.

Step 2: Set up the Monthly Payment Formula

To find the monthly payment, we'll use the formula for monthly payments on an amortizing loan:\[M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}\]where:- \(M\) is the monthly payment- \(P = \$500\) is the principal loan amount- \(r\) is the monthly interest rate- \(n\) is the total number of payments

Step 3: Calculate the Monthly Interest Rate

The annual interest rate is 22.75%, so the monthly interest rate \(r\) is:\[ r = \frac{22.75\%}{12 \times 100} = \frac{22.75}{1200} = 0.0189583 \]

Step 4: Calculate the Total Number of Payments

The loan term is 10 years, so the total number of monthly payments \(n\) is:\[ n = 10 \times 12 = 120 \]

Step 5: Plug Values into the Formula and Compute Monthly Payment

Substitute \(P = 500\), \(r = 0.0189583\), and \(n = 120\) into the formula and solve for \(M\):\[M = 500 \cdot \frac{0.0189583(1+0.0189583)^{120}}{(1+0.0189583)^{120} - 1}\]Calculate \((1 + 0.0189583)^{120}\) then the rest of the formula to find \(M\). After computing, you get:\[ M \approx \$11.85 \]

Step 6: Calculate Total Payments

Multiply the monthly payment \(M\) by the total number of payments \(n\):\[ \text{Total Payments} = 11.85 \times 120 = \$1422.00 \]

Step 7: Calculate Total Interest Paid

To find the total interest paid, subtract the principal amount from the total payments:\[ \text{Interest Paid} = 1422.00 - 500 = \$922.00 \]

Key Concepts

Amortizing Loan

An amortizing loan is a type of loan where the principal, or original amount borrowed, is paid off through scheduled payments, typically in monthly installments. The great thing about these loans is that each payment gradually reduces the amount owed. In the context of credit card interest calculation, when Joanna uses her credit card, it's like she's receiving a small loan that needs to be paid back over time. Amortizing loans spread out the interest and the principal in such a way that each payment is the same amount. This offers predictability since the payments do not change. With amortizing loans, payments cover both the interest and a portion of the principal. Early in the loan, a larger portion of the payment goes toward paying interest but as time progresses, more of the payment reduces the principal. This is essential for understanding how Joanna's credit card payments work over the 10-year period she's paying off her loan.

Monthly Payment Formula

The monthly payment formula is used to calculate how much Joanna needs to pay each month to pay off her credit card purchase using an amortizing loan over the term chosen, which is 10 years in this situation.The formula for calculating monthly payments is:\[M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}\]Here's how to read it:

• **\(M\):** This is the monthly payment amount, the amount Joanna will need to pay every month.
• **\(P\):** This stands for the principal amount, which is the original \(500 loan value.
• **\(r\):** This is the monthly interest rate, which is derived by dividing the annual rate by 12.
• **\(n\):** The total number of monthly payments, calculated by multiplying the number of years by 12.

By substituting in Joanna's details, the formula predicts that she will need to pay about \)11.85 every month if she doesn't add any more purchases to her account.

Interest Rate Calculation

Interest rate calculation is a critical part of figuring out how loans work since it defines how much extra money Joanna will pay the bank on top of repaying the initial $500.For her credit card, the interest is compounded monthly, meaning that every month, interest is calculated on the amount Joanna still owes. This can make her payoff strategy quite different than if the interest was compounded annually.To find the monthly interest rate Joanna faces, take her annual rate of 22.75% and divide by 12 months:\[ r = \frac{22.75}{1200} = 0.0189583 \]This number indicates how much interest is applied to Joanna's balance each month. Compound this over 10 years, and you can see how making just the minimum payments allows the interest to add up significantly, contributing to the total loan costs she's facing.