In this question we will check if a discount is actually a discount. Say you are to borrow $100 for a year at a simple interest rate of 10%. Determine the discount D and the proceeds P for this loan. Next take another look at the situation. You take away P dollars and at the end of the year you pay $100 what rate of simple interest did you actually pay and is this rate more or less than the 10% of the loan with the discount. Please show work!!

Question

In this question we will check if a discount is actually a discount. Say you are to borrow $100 for a year at a simple interest rate of 10%. Determine the discount D and the proceeds P for this loan. Next take another look at the situation. You take away P dollars and at the end of the year you pay $100 what rate of simple interest did you actually pay and is this rate more or less than the 10% of the loan with the discount. Please show work!!

Answer 1

To determine the discount D and the proceeds P for a loan, we need to calculate the interest charged on the principal amount borrowed.

1. Calculate the interest charged for a year at a simple interest rate of 10%:
- Principal amount = $100
- Interest rate = 10%
- Simple interest = Principal * Interest rate = $100 * 0.10 = $10

The discount D can be calculated as the difference between the principal amount borrowed ($100) and the interest charged ($10):
- Discount = Principal - Simple interest = $100 - $10 = $90

The proceeds P can be calculated as the principal amount borrowed ($100) minus the discount D:
- Proceeds = Principal - Discount = $100 - $90 = $10

So, the discount is $90 and the proceeds are $10.

Now, let's consider the second scenario where you take away P dollars (which is $10 in this case) and at the end of the year, you pay $100. We need to calculate the effective interest rate you paid and compare it with the 10% discount rate.

1. Calculate the interest charged for this scenario:
- Principal amount = $100
- Amount taken away = $10
- Amount to be paid at the end of the year = $100
- Interest charged = Amount to be paid - Amount taken away = $100 - $10 = $90

To find the rate of simple interest for this scenario, we can use the formula:
- Rate = (Interest / (Principal - Amount taken away)) * 100

Substituting the values, we have:
- Rate = ($90 / ($100 - $10)) * 100 = (90/90) * 100 = 100

So, the rate of simple interest you actually paid in this scenario is 100%.

To determine if this rate is more or less than the 10% discount rate, we compare them.

The rate of 100% is higher than the 10% discount rate. Therefore, in this scenario, the rate of simple interest paid is more than the 10% discount rate.

Note: In the initial calculation, the discount was determined based on the assumption that the interest is charged on the full principal amount ($100). However, in the second scenario, you took away $10, resulting in an effective principal amount of $90. Hence, the two scenarios are different and result in different interest rates.