A loan payment of ​$1000.00 was due 30 days ago and another payment of ​$900.00 is due 60 days from now. What single payment 120 days from now will pay off the two obligations if interest is to be 4​% and the agreed focal date is 120 days from​ now? 

To solve this problem, we can apply the concept of **equivalent payments** with a **focal date** and **simple interest**.

### Step-by-Step Explanation:

1. **Identify the values:**
   - The first payment of $1,000 was due **30 days ago** (i.e., 150 days from the focal date of 120 days in the future).
   - The second payment of $900 is due **60 days from now** (i.e., 60 days before the focal date).
   - The interest rate is **4% per year**, which translates to a **daily rate** of $ \frac{0.04}{365} $.

2. **Formula for Equivalent Payment:**

   The formula for finding the equivalent payment at a future date $ P_f $ based on a payment $ P $ due at a different date is:

   \[
   P_f = P \times \left( 1 + r \times t \right)
   \]
   Where:
   - $ P_f $ = payment at the focal date (120 days from now),
   - $ P $ = original payment amount,
   - $ r $ = interest rate per day $ \left( \frac{0.04}{365} \right) $,
   - $ t $ = number of days from the original payment date to the focal date (positive if after, negative if before).

3. **Adjust each payment to the focal date (120 days from now):**

   - **First payment of $1,000:**
     This payment was due 30 days ago, which means it's 150 days before the focal date.
     \[
     P_1 = 1000 \times \left( 1 + \frac{0.04}{365} \times (-150) \right)
     \]

   - **Second payment of $900:**
     This payment is due 60 days from now, which is 60 days before the focal date.
     \[
     P_2 = 900 \times \left( 1 + \frac{0.04}{365} \times (-60) \right)
     \]

4. **Calculate the total payment:**
   The equivalent single payment is the sum of the adjusted payments at the focal date.

Let's compute the values.The equivalent single payment to settle both obligations 120 days from now is approximately **$1,877.64**.

Would you like more details on the process or calculations? Here are some related questions to explore further:

1. How would the result change if the interest rate increased to 5%?
2. What happens if the focal date is moved to 90 days instead of 120 days?
3. Can the same method be applied if the payments were compounded monthly instead of using simple interest?
4. How would an earlier payment (e.g., 60 days ago) affect the final amount?
5. What if one of the payments had a different interest rate?

**Tip:** Always confirm the time units (days, months, years) used for the interest rate to ensure accurate calculations.

A loan payment of $1000.00 was due 30 days ago and another payment of $900.00 is due 60 days from now. What single payment 120 days from now will pay off the two obligations if interest is to be 4% and the agreed focal date is 120 days from now?

Learn how to calculate the equivalent single loan payment 120 days from now, given two loan payments of $1000 and $900, using a 4% interest rate. This problem involves time value of money and simple interest formulas. Suitable for students in Grades 10-12.

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