Austin owes $10,000 to a friend who is charging her simple interest of 3.50% p.a. She is required to settle the loan with two equal payments, one today and the other in eight months. Calculate the size of the equal payments using eight months as the focal date.

We are tasked with calculating two equal payments that will settle a $10,000 loan, given that the interest is charged at a simple annual rate of 3.50%. One payment is made today, and the other is made in 8 months, which is our focal date.

### Key Variables:
- Loan amount: $ P = 10,000 $
- Annual simple interest rate: $ r = 3.50\% = 0.035 $
- Time periods: 
  - The first payment is made today (so its time to the focal date is 8 months).
  - The second payment is made in 8 months (so its time to the focal date is 0 months).
- The time period is based on 12 months in a year.

### Step-by-Step Solution:

#### 1. Present Value of Payments:
We need to calculate the present value of both payments as of the focal date (8 months from now). The first payment made today will accumulate interest over the 8 months until the focal date, while the second payment is made exactly at the focal date.

Let the size of the equal payments be denoted by $ X $.

- The present value of the first payment $ X $, when moved forward to the focal date (8 months ahead), must account for simple interest. The future value of the first payment is given by:

\[
FV_1 = X \times (1 + r \times \frac{8}{12}) = X \times (1 + 0.035 \times \frac{8}{12})
\]

\[
FV_1 = X \times (1 + 0.023333) = X \times 1.023333
\]

- The second payment $ X $, made at the focal date, is simply $ X $ since no time has passed.

#### 2. Total Debt Repayment:
The total value of both payments must equal the loan amount of $10,000 at the focal date. Thus, the equation for the total debt is:

\[
X \times 1.023333 + X = 10,000
\]

#### 3. Solve for $ X $:
Simplifying the equation:

\[
X (1.023333 + 1) = 10,000
\]

\[
X \times 2.023333 = 10,000
\]

\[
X = \frac{10,000}{2.023333} = 4942.88
\]

So, the size of each equal payment is **$4,942.88**.

### Conclusion:
Austin must make two equal payments of **$4,942.88**: one today and one in 8 months.

Would you like further details or clarifications? Here are some related questions you might find helpful:

1. How is simple interest different from compound interest?
2. How would the payment change if the interest rate was compounded monthly?
3. What happens if Austin wants to settle the loan with a single payment in 8 months?
4. How would the payments change if the interest rate increased to 5%?
5. Can this method be used for loans with unequal payments?

**Tip**: Simple interest grows linearly with time, while compound interest grows exponentially, leading to larger amounts over time.

Learn how to calculate two equal payments to settle a $10,000 loan under simple interest of 3.50% p.a., with the focal date set at eight months. The step-by-step guide covers key concepts like simple interest, loan repayment, and time value of money. Suitable for high school students in Grades 10-12.

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