Bill Casler bought a $8000, 9-month certificate of deposit (CD) that would earn 8.4% annual simple interest.
(a) What is the value of the CD when it matures?
$  

(b) Three months before the CD was due to mature, Bill needed his CD money, so a friend agreed to lend Bill an amount of money for three months, at the end of which Bill would give his friend the full maturity value of the CD. If their agreement allowed the friend to earn a 10% annual simple interest return on his loan to Bill, how much did Bill receive from his friend? (Round your answer to the nearest cent.)
$  

(c) What annual simple interest rate did Bill Casler end up making on his investment? Round your answer to two decimal places.
  %

To solve this, let's address each part of the question step-by-step.

### Given Information:
- Initial investment (principal) in CD, $ P = 8000 $
- CD duration: 9 months
- Annual interest rate on CD: $ 8.4\% $ (simple interest)
- Interest rate on friend's loan: $ 10\% $ (simple interest for 3 months)

### Part (a): Value of the CD at Maturity
Since the interest is simple, we use the formula:
\[
A = P \left(1 + \frac{r \cdot t}{100}\right)
\]
where:
- $ A $ is the amount at maturity,
- $ r $ is the annual interest rate,
- $ t $ is the time in years.

#### Step 1: Convert 9 months to years
\[
t = \frac{9}{12} = 0.75 \text{ years}
\]

#### Step 2: Plug in values
\[
A = 8000 \left(1 + \frac{8.4 \times 0.75}{100}\right)
\]
\[
A = 8000 \left(1 + \frac{6.3}{100}\right)
\]
\[
A = 8000 \left(1 + 0.063\right)
\]
\[
A = 8000 \times 1.063 = 8504
\]
Thus, the value of the CD at maturity is **$8504.00**.

---

### Part (b): Amount Bill Received from Friend
Bill's friend lent him an amount that would, after 3 months at 10% interest, grow to the maturity value of the CD, which is $8504.

Let $ L $ represent the amount loaned by the friend. Using the simple interest formula for a 3-month period (or $ \frac{3}{12} = 0.25 $ years), we have:
\[
8504 = L \left(1 + \frac{10 \times 0.25}{100}\right)
\]
\[
8504 = L \left(1 + 0.025\right)
\]
\[
8504 = L \times 1.025
\]
\[
L = \frac{8504}{1.025}
\]
\[
L \approx 8296.10
\]
So, Bill received approximately **$8296.10** from his friend.

---

### Part (c): Effective Annual Simple Interest Rate for Bill
To find Bill's effective annual interest rate, we look at the total return he earned over his original investment.

1. **Total Interest Earned**: $ 8504 - 8000 = 504 $
2. **Investment Period**: 9 months (or $ \frac{9}{12} = 0.75 $ years)

Using the formula for simple interest rate:
\[
\text{Effective rate} = \frac{\text{Total Interest}}{\text{Principal} \times \text{Time}} \times 100
\]
\[
\text{Effective rate} = \frac{504}{8000 \times 0.75} \times 100
\]
\[
\text{Effective rate} = \frac{504}{6000} \times 100
\]
\[
\text{Effective rate} = 8.4\%
\]

So, Bill's effective annual interest rate on his investment is **8.40%**.

---

### Summary of Answers:
- **(a)** The value of the CD when it matures: **$8504.00**
- **(b)** Amount Bill received from his friend: **$8296.10**
- **(c)** Annual simple interest rate Bill made: **8.40%**

Would you like further explanation on any part of the solution?

---

#### Further Questions:
1. How is the simple interest formula different from compound interest?
2. How would Bill's return change if the CD offered compound interest instead?
3. What if the loan from the friend had been for a different interest rate?
4. How could Bill have calculated the future value if he used an investment with monthly compounding?
5. What would the effective rate be if Bill reinvested his earnings?

#### Tip:
When calculating interest, always convert months to years if the rate is annual to avoid errors.

Bill Casler bought a $8000, 9-month certificate of deposit (CD) that would earn 8.4% annual simple interest. (a) What is the value of the CD when it matures? $ (b) Three months before the CD was due to mature, Bill needed his CD money, so a friend agreed to lend Bill an amount of money for three months, at the end of which Bill would give his friend the full maturity value of the CD. If their agreement allowed the friend to earn a 10% annual simple interest return on his loan to Bill, how much did Bill receive from his friend? (Round your answer to the nearest cent.) $ (c) What annual simple interest rate did Bill Casler end up making on his investment? Round your answer to two decimal places. %

Learn how to calculate the maturity value of an $8000 certificate of deposit (CD) with 8.4% annual simple interest over 9 months, along with effective interest rate analysis. Discover how much Bill Casler received from a friend who loaned him funds for the final 3 months at 10% annual interest. Suitable for high school students in Grades 10-12.

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