Lauren and Serena each have 56-week loans. At the end of their terms, they each repay $2,875. 

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 8.9%. 

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

To solve this problem, we need to find the annual simple discount rate on Lauren's loan under the assumption that she and Serena borrowed the same amount of money. The information we have is:

- Both Lauren and Serena repay $2,875 at the end of their 56-week loan term.
- Serena's loan uses simple interest with an annual rate of 8.9%.
- Lauren's loan uses simple discount, and we need to determine the simple discount rate.

### Step 1: Calculate the amount Serena borrowed

Using the formula for the simple interest model, we can determine how much Serena borrowed:

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

Where:
- $ A = 2875 $ (the amount repaid)
- $ P $ is the principal (the amount borrowed)
- $ r = 8.9\% = 0.089 $ (annual simple interest rate)
- $ t = \frac{56}{52} $ (since 56 weeks is slightly more than a year, the time factor is $ \frac{56}{52} $).

We can now solve for $ P $:

\[
2875 = P(1 + 0.089 \times \frac{56}{52})
\]

\[
P = \frac{2875}{1 + 0.089 \times \frac{56}{52}}
\]

Now, let's calculate the value of $ P $.The amount Serena borrowed was approximately $ 2623.54 $.

### Step 2: Calculate the simple discount rate for Lauren's loan

Now, we know that Lauren also repays $ 2875 $ at the end of the 56-week loan term, but her loan follows the simple discount model. The formula for the simple discount model is:

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

Where:
- $ A = 2875 $
- $ P = 2623.54 $ (the amount borrowed, same as Serena's loan)
- $ d $ is the annual simple discount rate (what we're trying to find)
- $ t = \frac{56}{52} $

Rearranging the formula to solve for $ d $:

\[
2875 = 2623.54(1 - d \times \frac{56}{52})
\]

\[
1 - d \times \frac{56}{52} = \frac{2875}{2623.54}
\]

\[
d = \frac{1 - \frac{2875}{2623.54}}{\frac{56}{52}}
\]

Now, let's calculate $ d $.The annual simple discount rate on Lauren's loan would have to be approximately $ -8.9\% $, which suggests that in this case, the discount rate is equivalent in magnitude but opposite in sign to the interest rate used in Serena's loan.

Would you like further details or clarification on any part of the solution?

Here are 5 related questions to expand on this topic:

1. What is the difference between the simple interest model and the simple discount model?
2. How would the repayment amount change if Lauren borrowed less than Serena?
3. Can the simple discount rate ever be positive in practical scenarios?
4. How is simple interest affected by different loan terms (e.g., monthly, quarterly)?
5. What happens to both loans if the interest and discount rates change midway through the loan?

**Tip:** Always check the loan agreement to see which model (simple interest or simple discount) is being used, as it affects the amount borrowed and repaid.

Lauren and Serena each have 56-week loans. At the end of their terms, they each repay $2,875. 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 8.9%. What would the annual simple discount rate have to be on Lauren's loan if they borrowed the same amount of money?

Learn how to calculate the annual simple discount rate for Lauren's loan, which uses the simple discount model, based on Serena's simple interest loan at an 8.9% rate. Both repay $2,875 after 56 weeks. This problem involves applying key algebraic concepts and financial formulas, suitable for Grades 10-12.

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