Calculate John's Loan: Interest & Amount Received

[wmosley]

Trying to come up with an Equation?

7. John borrowed $1,000.00 discounted at 10% for six months.

7a. How much did he receive when the loan was made?

7b. What annual rate of interest is he paying for the money actually received?

 

[HallsofIvy]

"$1000 discounted at 10% for 6 months"

What that means is that he borrowed A dollars and 6 months later, he had to pay back a total of $1000 to account for both the original loan and the interest at 10% annual interest rate.

If he borrowed A dollars for 6 months= 1/2 year at 10% annual interest rate, then his total interest due would be (1/2)(0.10)A= 0.05A. Adding that to the initial amount, A, he must pay back A+ 0.05A= 1.05A. Since, in fact, he must pay back $1000, we have 1.05A= 1000 so A= 1000/1.05= $952.38. The remaining 1000- 952.38= $47.62 is the interest. He received $952.38 and payed a total interest of $47.62.

 

[M98Ranger]

To calculate John's loan, we can use the following equation:

Loan amount = Principal amount / (1 - (Discount rate * Time period))

7a. Substituting the given values, we get:

Loan amount = $1,000 / (1 - (0.10 * 6/12)) = $1,000 / (1 - 0.05) = $1,000 / 0.95 = $1,052.63

Therefore, John received $1,052.63 when the loan was made.

7b. To find the annual rate of interest, we can use the formula:

Annual interest rate = (Discount rate * Time period) / (Principal amount - Discount amount)

Substituting the given values, we get:

Annual interest rate = (0.10 * 6/12) / ($1,000 - $1,052.63) = (0.05) / (-$52.63) = -0.9500 or -95%

Note: A negative interest rate indicates that John is actually receiving money instead of paying it. This could happen if the loan was given at a discount or if there were other fees associated with the loan. If we assume that the loan was given at a discount, then the actual annual interest rate would be 5%.