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Complete Question
Consider a three-year loan (so we'll assume the numbers 1 through 36) for $5,000 with interest at 10% per year. Using standard amortization, the monthly payment is $161.33. In this example, we will not worry about exact or ordinary interest because the total interest to be paid is $808.13. After the fifth month the borrower decides to prepay the whole loan. Under a standard amortization plan the borrower would have paid $198.28 in cumulative interest. However, using the Rule of 78 a lender would calculate the fraction of the total interest based on two series:
[tex]\dfrac{(n+35)+(n+34)+(n+33)+(n+32)+(n+31)} {(n)+(n+1)+...+(n+35)}[/tex]
Answer:
See below
Step-by-step explanation:
36+35+34+33+32=170
- If you add 36, 35, 34, 33, and 32, the sum is 170.
Now, 1,2,3,...36 forms an arithmetic series whose first and last term are 1 and 36 respectively. Its sum is determined using the formula: [tex]S_{n}=\frac{n}{2}(a+l) \\[/tex]
[tex]S_{36}=\frac{36}{2}(1+36) =18*37=666[/tex]
- If you sum the numbers from 1 to 36, the sum is 666.
[tex]=\dfrac{170}{666}= 0.255=25.5\% $(to the nearest tenth)[/tex]
- The fraction (the first sum / the total sum) to the nearest tenth = 25.5%.
The lender will multiply this fraction by the total interest.
- The cumulative interest [tex]= 25.5\% \times \$808.13 = \$206.07[/tex]
The difference between the amount paid under a standard amortization plan and the amount paid under a Rule of 78 plan is:
$206.07-198.28=$7.79
