For services​ rendered, an attorney accepts a 120​-day note for ​$6 comma 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 ​$6 comma 780 after 30 days. What annual interest rate will the third party receive for the​ investment?
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Part 1
The third party will receive an annual interest rate of

enter your response here​%.
​(Type an integer or decimal rounded to two decimal places as​ needed.)

To determine the annual interest rate that the third party receives, we need to use the formula for simple interest and account for the fact that the note is sold before maturity. Here's the breakdown:

### Step 1: Calculate the maturity value of the note
The note has a face value of $6,700, and the interest rate is 7% per annum for 120 days. We can calculate the maturity value (the total amount to be paid at the end of the note's term) using the simple interest formula:

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

Where:
- $ P = 6700 $ (the principal)
- $ r = 7\% = 0.07 $ (the annual interest rate)
- $ t = \frac{120}{360} = \frac{1}{3} $ (the time in years, assuming a 360-day year)

\[
\text{Interest} = 6700 \times 0.07 \times \frac{1}{3} = 156.33
\]

Thus, the maturity value is:

\[
\text{Maturity Value} = 6700 + 156.33 = 6856.33
\]

### Step 2: Determine the interest earned by the third party
The attorney sells the note for $6,780 after 30 days, meaning there are 90 days remaining. The third party will collect the maturity value of $6,856.33 in 90 days. Their investment is $6,780, and the interest they earn is:

\[
\text{Interest Earned} = 6856.33 - 6780 = 76.33
\]

### Step 3: Calculate the annual interest rate for the third party
Now, we use the simple interest formula again to find the annual interest rate, $ r $:

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

Where:
- $ P = 6780 $ (the amount the third party invested)
- $ t = \frac{90}{360} = \frac{1}{4} $ (the time in years)
- $ \text{Interest Earned} = 76.33 $

So,

\[
76.33 = 6780 \times r \times \frac{1}{4}
\]

Solving for $ r $:

\[
r = \frac{76.33 \times 4}{6780} = \frac{305.32}{6780} \approx 0.045
\]

Converting this to a percentage:

\[
r \approx 4.50\%
\]

### Final Answer:
The third party will receive an annual interest rate of approximately **4.50%**.

---

Would you like further details or clarification? Here are some related questions to deepen your understanding:

1. How is the simple interest formula derived?
2. What is the difference between simple and compound interest?
3. How would the interest rate change if the note were sold earlier or later?
4. How is the effective annual interest rate different from the nominal rate?
5. What happens if the attorney holds the note until maturity?

**Tip:** Always consider the time frame and interest rate type (simple vs. compound) when calculating returns on investments!

For services rendered, an attorney accepts a 120-day note for $6,700 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 $6,780 after 30 days. What annual interest rate will the third party receive for the investment?

Learn how to calculate the annual interest rate for a third party purchasing a 120-day note after 30 days, involving simple interest. In this problem, the note is initially $6,700 at 7% interest, and the note is sold for $6,780. The third party earns interest over the remaining 90 days. Suitable for college-level students or advanced high school learners.

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