+619 Alex needs to borrow $\$10,\!000$ dollars from the bank. The bank gives him two options.
1. A ten-year loan with an interest rate of $10\%$ compounded quarterly, with the condition that at the end of 5 years Alex must make a payment equal to half of his balance. At the end of the ten years, Alex will pay off the remaining balance.
2. A ten-year loan with a simple annual interest rate of $12\%$, with just one lump-sum payment at the end of the ten years. Find the positive difference between the total amounts Alex has to pay back under the two schemes. Round your answer to the nearest dollar.
1)
I assume the loan to be $10,000. You would use this formula to compound the interest for Option 1 for the first 5 years: FV = PV x [1 + R]^N:
FV = 10,000 x ]1 + 0.10/4]^(5*4)
FV = 10,000 x [1 + 0.025]^20
FV = 10,000 x 1.6386164403........
FV = $ 16,386.16 This is principal + compound interst for the first 5 years.
$16,386.16 / 2 =$8,193.08 This is what Alex must pay for the first 5 years.
The other half continues to earn compound interest for another 5 years.
FV =8,193.08 x [1.025]^20
FV =8,193.08 x 1.6386164403........
FV =$13,425.32 This the balance of the loan for the last 5 years:
$13,425.32 + $8,193.08 = $21,618.40 This is the total cost of the loan of option 1.
2)
$10,000 x 12% =$1,200 simple interest for 1 full year.
$1,200 x 10 years = $12,000 Total simple interest on the loan in option 2
$10,000 + 12,000 = $22,000 Total principal + interest for option 2
As you can see Option 1 is slightly cheaper by:
$22,000 - $21,618.40 =$381.60 The difference between Option 1 and 2.
