Copyright © 2008, The McGraw-Hill Companies, Inc.
The Rule of 78 reflects the idea (which we also saw with amortization) that interest
should constitute a higher proportion of the payments on a loan early in the term. For a
12-month loan, we assume that 12/78 of the total interest is paid with the first payment,
11/78 is paid with the second, 10/78 is paid with the third, and so on until the last payment
takes care of 1/78 of the total interest. The reason that we are dividing by 78 is that 12
11 10 9 ... 2 1 78, and so in total over the 12 months, 12/78 11/78
10/78 9/78 ... 2/78 1/78 78/78 (i.e., all) of the interest is paid.
The Rule of 78 gets its name from the fact that the total of the interest fractions for
12 months is 78/78, and 12 months is a very common term for an installment plan. How-
ever, if the term is different than 1 year, we use whatever the total of all the whole numbers
from 1 to the number of payments adds up to. For Bill’s dining room set in Example 10.3.1,
the number to use would be 9 8 7 ... 2 1 45, so his first payment would take
care of 9/45 of the total interest, and so on.
It would be helpful to have a convenient formula to calculate this total of the whole
numbers up to a given number. Fortunately, a surprisingly simple formula can do this.^6FORMULA 10.3.1
The Sum of Whole Numbers FormulaThe sum of all the whole numbers from 1 to n is:1 2 3 ... n n(n 1)
________ 2We could have used this formula to find the total 9 8 7 ... 2 1 above by using
n 9 in this formula to get (9)(10)/2 90/2 45. Having this formula is especially help-
ful with longer-term loans.Example 10.3.3 How much interest does Tonya (from Example 10.3.2) pay with her
fi rst monthly payment? How much interest does she pay with her second payment?
What about her last payment?Tonya’s loan is for 24 months, so the denominator we need is 1 2 3 ... 24
(24)(25)/2 300. The total interest she is paying is $288. So, with her fi rst payment she is
paying (24/300)($288) $23.04. With her second payment, she pays (23/300)($288)
$22.08. With her last payment, she pays (1/300)($288) $0.96 in interest.If Tonya decides to pay off her loan early, the Rule of 78 can be used to determine the
payoff amount. To do this, we calculate the remaining interest that she would pay over the
remaining term of the loan, and then subtract that from her remaining payments.Example 10.3.4 Suppose that after 10 months, Tonya decides to pay off her piano
in full. Assuming the Rule of 78 is used for his payment plan, how much would she
need to pay it off at that point?After 10 months, she has 24 10 14 payments left. At $87.00 a month, this would
mean a total of 14($87.00) $1,218. If she pays early, though, we would reduce this by the
amount of the interest that she would have paid over these last 14 payments.Over the course of those 14 remaining payments, Tonya would have paid 14/300 13/300
12/300 ... 1/300 of the interest. We can use our sum formula to fi nd that 14 13 12
... 1 (14)(15)/2 105. So in total her last 14 payments work out to 105/300 of the total
interest, or (105/300)($288) $100.80. By paying early she avoids this interest, so her loan
payoff would be $1,218.00 $100.80 $1,117.20.(^6) This formula has a story behind it. According to legend, one day in school the boy who grew up to be the great
mathematician Carl Friedrich Gauss was bored and causing trouble in class. The teacher, knowing that Gauss
liked arithmetic, gave him the assignment of adding up all the whole numbers from 1 to 1,000, thinking it would
keep him busy for a while. Moments later, Gauss supplied the correct total, having come up with the formula
on his own.
10.3 Installment Plans 451