Acton choose from two loans offersS: 12,000 at 8% simple interest for 9 months: or a 12,000 9 month discounted loan at 7% discount. Based on the actual interest paid and the true rate on the discounted loan, which of the two loan offers will Acton choose?
For the simple loan what is the interest rate paid for 9 months?
Interest paid = .08(9/12)(12,000) = 720
r = what you paid/what you got = 720/12,000 = .06
what is the real interest rate on the discounted loan?
Well, how much is the nominal interest paid?
I = .07 * (9/12)(12,000) = 630 looks cool but
what you got = 12,000 - 630 = 11,370
so
r = 630/11370 = .055
so you are half a percent better off with the discounted loan, although you walk out of the bank with less than the 12,000 you needed.
To determine which loan offer Acton would choose, we need to compare the actual interest paid on each loan and the true rate on the discounted loan.
Let's start with the first loan offer:
Loan Offer S: $12,000 at 8% simple interest for 9 months.
To calculate the interest paid on this loan, we can use the formula:
Interest = Principal x Rate x Time
Principal = $12,000
Rate = 8% or 0.08 (in decimal form)
Time = 9 months
Interest = $12,000 x 0.08 x 9 = $8,640
The actual interest paid on this loan would be $8,640.
Now let's move on to the second loan offer:
Loan Offer D: $12,000 9-month discounted loan at 7% discount.
To determine the true rate on the discounted loan, we need to first find the discount amount and then calculate the effective interest rate.
Discount Amount = Principal x Discount Rate
Principal = $12,000
Discount Rate = 7% or 0.07 (in decimal form)
Discount Amount = $12,000 x 0.07 = $840
The discounted amount is $840.
To calculate the effective interest rate, we can use the formula:
Rate = Discounted Amount / Principal x Time
Rate = $840 / ($12,000 x 9) = 0.007
To convert this into a percentage, multiply by 100:
Rate = 0.007 x 100 = 0.7%
The true rate on the discounted loan is 0.7%.
Now, let's compare the actual interest paid on the first loan ($8,640) with the true rate on the discounted loan (0.7%).
Since the actual interest paid on the first loan is significantly higher than the true rate on the discounted loan, Acton would choose the second loan offer (Loan Offer D: $12,000 9-month discounted loan at 7% discount). This is because the second loan option offers a lower interest burden.