Lauren and Serena each have 56-week loans. At the end of their terms, they each repay $3,955. 

Lauren's loan uses the simple discount model while Serena's loan uses the simple interest model. The annual simple interest rate on Serena's loan is 10.3%. 

What would the annual simple discount rate have to be on Lauren's loan if they borrowed the same amount of money? 

Round your answer to the nearest tenth of a percent. 

We need to determine the annual simple discount rate on Lauren's loan, given that both Lauren and Serena repay the same amount of $3,955 at the end of their 56-week loans. Serena's loan uses the simple interest model with an annual interest rate of 10.3%, and Lauren's loan uses the simple discount model.

### Step 1: Determine the amount Serena borrowed.

For Serena's loan, the formula for simple interest is:

\[
A = P(1 + rt)
\]

where:
- $ A $ is the final amount ($3,955),
- $ P $ is the principal (the amount borrowed),
- $ r $ is the annual interest rate (10.3% or 0.103),
- $ t $ is the time in years (56 weeks is $\frac{56}{52} \approx 1.077$ years).

We can rearrange this formula to solve for $ P $ (the amount borrowed by Serena):

\[
P = \frac{A}{1 + rt}
\]

Substituting the known values:

\[
P = \frac{3,955}{1 + 0.103 \times 1.077}
\]

Let’s calculate this value.

### Step 2: Calculate the principal amount for Serena.

\[
P = \frac{3,955}{1 + 0.103 \times 1.077} = \frac{3,955}{1 + 0.110931} = \frac{3,955}{1.110931}
\]

Now, calculating the principal:

\[
P \approx \frac{3,955}{1.110931} \approx 3,558.60
\]

### Step 3: Determine the discount rate for Lauren's loan.

For Lauren's loan, we use the formula for simple discount:

\[
A = P(1 - dt)
\]

where:
- $ A $ is the final amount ($3,955),
- $ P $ is the principal (which we assume is the same as Serena’s principal, $ P = 3,558.60 $),
- $ d $ is the annual discount rate (what we need to find),
- $ t $ is the time in years (1.077 years).

Rearranging the formula to solve for $ d $:

\[
d = \frac{1 - \frac{A}{P}}{t}
\]

Substituting the known values:

\[
d = \frac{1 - \frac{3,955}{3,558.60}}{1.077}
\]

Now let’s calculate $ d $.

### Step 4: Calculate the discount rate.

\[
d = \frac{1 - \frac{3,955}{3,558.60}}{1.077} = \frac{1 - 1.1115}{1.077} = \frac{-0.1115}{1.077}
\]

Now, calculating $ d $:

\[
d \approx -0.1035
\]

Thus, the annual simple discount rate is approximately **10.4%** (rounded to the nearest tenth of a percent).

### Conclusion:
The annual simple discount rate for Lauren’s loan would have to be **10.4%**.

---

Do you want further details or explanations on any of the steps? Here are five questions that can extend your understanding:

1. How does simple interest differ from simple discount?
2. Why do we use $\frac{56}{52}$ to convert weeks to years?
3. What happens if the interest or discount rates were compounded rather than simple?
4. How would the problem change if Lauren and Serena had different repayment amounts?
5. Can you derive the formula for simple interest or discount from first principles?

**Tip:** Always check if the rates given are annual or for the loan's term to avoid incorrect calculations.

Lauren and Serena each have 56-week loans. At the end of their terms, they each repay $3,955. Lauren's loan uses the simple discount model while Serena's loan uses the simple interest model. The annual simple interest rate on Serena's loan is 10.3%. What would the annual simple discount rate have to be on Lauren's loan if they borrowed the same amount of money? Round your answer to the nearest tenth of a percent.

Learn how to calculate the annual simple discount rate for a 56-week loan where Lauren and Serena repay the same amount of $3,955. Explore key finance concepts like simple interest and simple discount, along with the formulas involved. This problem is suitable for high school students in Grades 11-12.

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