Julie borrowed ​$4 comma 750 for 8 months from a bank. The bank discounted the loan at 3.5 %.
a. How much interest did Julie pay the bank for the use of its​ money?
b. How much did she receive from the​ bank?
c. What was the actual rate of interest she​ paid?
Question content area bottom
Part 1
a. The interest is ​$
  
enter your response here.
​(Round to the nearest cent as​ needed.)

This type of loan is known as a **discounted loan**, meaning the bank takes the interest from the principal before giving the remainder to the borrower. Here's how to solve each part.

---

### Given Information
- **Principal (P)** = $4,750
- **Discount Rate (r)** = 3.5%
- **Time (t)** = 8 months

Let's break down each part of the question:

---

### Part (a): Calculate the Interest Paid

The formula for calculating the **discount interest** on a loan is:

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

To use this formula, we need **time in years**. Since 8 months is $\frac{8}{12}$ years:

1. **Convert months to years**:  
   \[
   t = \frac{8}{12} = \frac{2}{3} \approx 0.6667 \text{ years}
   \]

2. **Substitute values into the formula**:
   \[
   I = 4750 \times 0.035 \times 0.6667
   \]

3. **Calculate**:
   \[
   I \approx 110.83
   \]

Thus, **the interest Julie paid is approximately $110.83**.

---

### Part (b): Calculate Amount Received by Julie

In a discounted loan, the bank subtracts the interest from the principal before giving the money to the borrower.

\[
\text{Amount received} = P - I
\]

Substitute in the values:

\[
\text{Amount received} = 4750 - 110.83 = 4639.17
\]

So, **Julie received approximately $4,639.17** from the bank.

---

### Part (c): Calculate the Actual Rate of Interest Paid

The actual rate of interest is based on the **amount received** (not the original principal). To find this rate:

1. **Use the formula**:
   \[
   \text{Actual Rate} = \frac{\text{Interest}}{\text{Amount received}} \times 100\%
   \]

2. **Substitute values**:
   \[
   \text{Actual Rate} = \frac{110.83}{4639.17} \times 100\%
   \]

3. **Calculate**:
   \[
   \text{Actual Rate} \approx 2.39\%
   \]

Thus, **the actual rate of interest Julie paid is approximately 2.39%**.

---

### Summary of Answers:
1. **Interest paid**: $110.83
2. **Amount received**: $4,639.17
3. **Actual interest rate**: 2.39%

---

Would you like further details on any part or have questions?

---

### Further Questions:
1. How would the results change if the loan duration was extended to a year?
2. What if the interest rate was 5% instead of 3.5%?
3. How would this calculation vary if the loan used simple interest instead of a discounted method?
4. What are other types of loans with similar interest deductions?
5. How does compounding affect loan payments compared to simple interest?

---

**Tip:** When calculating interest for partial years, always convert months to a fraction of a year to ensure accuracy.

Julie borrowed $4,750 for 8 months from a bank. The bank discounted the loan at 3.5%. a. How much interest did Julie pay the bank for the use of its money? b. How much did she receive from the bank? c. What was the actual rate of interest she paid?

Solve a real-world problem involving a discounted loan of $4,750 at 3.5% for 8 months. Learn how to calculate the interest paid, the amount received, and the actual rate of interest using simple interest formulas. Suitable for high school students in Grades 9-12.

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