You haven't been saving your money. You desperately need $100. You decide to go to Payday Loans. Here's how Payday Loans works. You write a check for $116, and they give you $100 cash. In two weeks, they cash your check. You are charged 16% to use the $100 for two weeks. If you don't have that money in the bank in two weeks, they will hold the check for another two weeks for an additional 16%. Every two weeks you don't have the money for them to cash your check, they charge you an extra 16% simple interest. a. A typical car loan rate is 5% for one year. What would you expect to pay for a $100 loan at the end of the year? b. a typical car credit card loan rate is 20% per year. What would you expect to pay for a $100 loan at the end of the year? c. The Payday Loan company's rate is 16% every two weeks. There are 26 two-week periods in a year. What total percent would this be in a year? d. What would you expect to pay the Payday Loan Company for your $100 loan at the end of the year?

Let's break down each part of the problem carefully and solve it step by step:

### Part (a): Car Loan Rate of 5% per year
For a typical car loan with a 5% annual interest rate, you are paying simple interest on the $100.

**Simple interest formula**:  
\[ \text{Interest} = P \times r \times t \]
where:
- $ P $ is the principal ($100),
- $ r $ is the annual interest rate (5% or 0.05),
- $ t $ is the time (1 year).

The interest for the car loan is:
\[
\text{Interest} = 100 \times 0.05 \times 1 = 5
\]

Thus, at the end of the year, you would pay **$5** in interest. The total repayment would be:
\[
100 + 5 = 105
\]
So, you would expect to pay **$105** for the $100 loan at the end of the year.

### Part (b): Credit Card Loan Rate of 20% per year
For a typical credit card loan with a 20% annual interest rate:

\[
\text{Interest} = 100 \times 0.20 \times 1 = 20
\]

At the end of the year, the interest would be **$20**, and the total repayment would be:
\[
100 + 20 = 120
\]
So, you would expect to pay **$120** for the $100 loan at the end of the year.

### Part (c): Payday Loan Rate of 16% every two weeks
The Payday Loan charges 16% every two weeks, and there are 26 two-week periods in a year. To calculate the total interest over the year, we need to consider how much 16% every two weeks adds up over the year.

To find the **effective annual rate**:
- The interest for two weeks is 16%, or 0.16.
- After 1 two-week period, the amount owed becomes $ 100 \times (1 + 0.16) $.
- After 26 two-week periods, the amount owed becomes $ 100 \times (1 + 0.16)^{26} $.

Let's calculate this:
\[
\text{Total after a year} = 100 \times (1 + 0.16)^{26}
\]

I'll compute this.After 26 two-week periods with 16% interest each period, the total amount you would owe is approximately **$4741.41**.

### Part (d): Total Amount Paid to the Payday Loan Company
At the end of the year, you would pay **$4741.41** to the Payday Loan Company for the original $100 loan. This is due to the extremely high effective annual interest rate.

### Summary of Results:
- **(a)** Car loan at 5% per year: You would pay **$105** at the end of the year.
- **(b)** Credit card loan at 20% per year: You would pay **$120** at the end of the year.
- **(c)** Payday Loan at 16% every two weeks results in an effective interest rate of **4741.41%** annually.
- **(d)** You would pay **$4741.41** to the Payday Loan company at the end of the year for a $100 loan.

### Follow-up Questions:
1. What is the formula for calculating compound interest over different time periods?
2. Why does the interest grow so quickly when it's compounded more frequently?
3. What is the difference between simple and compound interest?
4. How would the total be different if the interest were compounded monthly instead of every two weeks?
5. How can someone avoid high-interest payday loans and seek better financial alternatives?

**Tip:** Compound interest grows quickly because the interest is charged not just on the original amount but also on accumulated interest. This leads to exponential growth over time.

Explore how Payday Loan interest rates compare with typical car and credit card loan rates. Learn the differences between simple and compound interest, and understand how a 16% two-week rate leads to an exponential increase over the year, amounting to over $4700 for a $100 loan. Suitable for Grades 10-12.

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