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Let's break down the problem step-by-step to determine the annual interest rate that the third party will receive.

### Problem Summary:
- The attorney accepts a 120-day note for **$5,700** at **7% simple interest** from a client. 
- After 30 days, the attorney sells the note to a third party for **$5,780**.
- We need to find the annual interest rate that the third party will receive for this investment.

### Step 1: Calculate the Amount of Interest on the Original Note
The note is for 120 days with a 7% simple interest rate. Using the simple interest formula:

\[
\text{Interest} = P \times r \times t
\]

Where:
- $ P = 5700 $ (the principal),
- $ r = 0.07 $ (the annual interest rate),
- $ t = \frac{120}{365} $ (time in years, since we have 120 days).

\[
\text{Interest} = 5700 \times 0.07 \times \frac{120}{365} = 131.51
\]

The total amount due at the end of the 120 days is:

\[
\text{Total due} = 5700 + 131.51 = 5831.51
\]

### Step 2: Calculate the Third Party's Investment Return
The third party buys the note for $5,780 and will receive $5,831.51 after 90 days (because the attorney sells the note after 30 days, leaving 90 days until the note matures). The return to the third party is the difference between the maturity value and the purchase price:

\[
\text{Return} = 5831.51 - 5780 = 51.51
\]

### Step 3: Calculate the Interest Rate for the Third Party
Now, we need to calculate the annual interest rate based on the third party’s 90-day investment. Using the simple interest formula again, but solving for the interest rate $ r $:

\[
I = P \times r \times t
\]

Where:
- $ I = 51.51 $ (the interest earned),
- $ P = 5780 $ (the amount the third party invested),
- $ t = \frac{90}{365} $ (time in years).

Rearranging the formula to solve for $ r $:

\[
r = \frac{I}{P \times t} = \frac{51.51}{5780 \times \frac{90}{365}}
\]

Now let's compute this value.The third party will receive an annual interest rate of approximately **3.61%** on the investment.

Would you like more details or have any questions? Here are five related questions that could help expand on the topic:

1. How would the interest rate change if the investment period was shorter?
2. What would happen to the interest rate if the note was sold at a different price?
3. Can the third party’s interest rate exceed the attorney’s original 7% rate? Why or why not?
4. How would compound interest differ from simple interest in this scenario?
5. How would the calculation differ if the time period was in months instead of days?

**Tip:** Simple interest is directly proportional to time, so when calculating it over short periods (like 30 or 90 days), annualizing the rate requires careful attention to the length of the investment period.

For services rendered, an attorney accepts a 120-day note for $5,700 at 7% simple interest from a client. Both interest and principal are repaid at the end of 120 days. Wishing to use her money sooner, the attorney sells the note to a third party for $5,780 after 30 days. What annual interest rate will the third party receive for the investment?

Learn how to calculate the annual interest rate a third-party investor will receive after purchasing a 120-day note for $5,780. This problem uses simple interest formulas and involves calculating returns over short investment periods.

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