# MATH 1332 7.5.7 Early Payoff: The Rule of 78 Method for Calculating Unearned Interest

Math 1332 chapter 7 consumer mathematics section 5 installment buying video 7 early payoff the rule of 78 method for calculating unearned interest in the previous video we learned about paying off an installment loan early in doing so we are not responsible for paying off the unearned interest that normally would have accrued during the remainder of the loan one

Approach for estimating the unearned interest was the actuarial approach this approach required the apr from the original loan which you may have to look up yourself using the apr table in fact when we did that problem that’s what took most of the time actually finding the apr a second approach called the rule of 78 can be used to estimate the unearned interest

Without the need for the apr table and just as a note the 78 in the rule of 78 comes from the sum 1 plus 2 plus 3 plus 4 plus 5 plus 6 plus 7 plus 8 plus 9 plus 10 plus 11 plus 12. for more details i recommend reading the following article on wikipedia wikipedia i wish that link were clickable but you can find wikipedia and you can just search rule of 78 it

Does come up as rule of 78 but in that article you will see the formula that we’re about to see and it will kind of tr it will try to explain well it explains why it’s called the rule of 78. so what is the rule of 78 well the rule of 78 is as follows we have a formula for unearned interest and don’t let this collection of letters throw you off it’s no worse than

The one we saw before to find the unknown interest you do a division problem or a fraction the top of the fraction is f times k times k plus one and the bottom of the fraction is n times n plus one i do have extra parentheses around both the numerator and the denominator since both are more than a single term the u of c of course is the unearned interest the f is

The finance charge k is the number of remaining payments excluding the current one just like it was in the actuarial method and n is the original number of payments so we won’t have to use the apr table that’s good just remember that in a in an expression like this this means f times k times k plus 1. i thought about writing it like this so that the multiplication

Was a little bit more explicit but well i guess there’s nothing really wrong with that but i’m going to continue to imply the multiplication with adjacency adjacency is the property of being next to something all right so all we need to calculate the unearned interest is the finance charge which if we don’t have we can calculate the number of remaining payments

Excluding the current one and the number of the original number of payments let’s revisit jason and the truck he financed in the previous video shall we he bought a truck priced at twenty nine thousand two hundred seventy five dollars he met a down payment of ten thousand dollars and paid five hundred and excuse me four hundred fifty dollars monthly for set for

For four years i’ll get the words out right in a minute he wants to pay off the loan at the end of 30 months all the proofing i did before i start this video and i’ve still got some errors how much is the unearned interest and payoff amount use the rule of 78 all right for our benefit let’s bring the rule of 78 to this page and there it is we need the finance

Charge the number of remaining payments excluding the current one and the number of original payments well the number of remaining payments and the number of uh original payments is easy um we are at the 30th payment well let’s start with the number of original payments which was 48 four years times 12 months in a year is 48. the number of original payments if

We are at the 30th payment and about to pay it off we haven’t we haven’t made that 30th payment yet so we’ve made 29 payments so far out of 48. 48 minus 29 is um oh excuse me excluding the current one so minus 30 is 18. so the number of remaining payments excluding the one that we’re currently at my apologies so we’re at the 30th payment that’s the current one

So we have 18 payments remaining 48 minus 30. all right so we got all the numbers we need except the finance charge and let’s just remember where the finance charge came from the finance charge is how much more we pay for the truck than its stated price so it’s the total of everything we pay minus the price of the truck what we pay for this truck is the 10 000

Down payment plus 48 payments of 454 dollars so multiply 48 times 454 and then subtract the cost of the truck 29 275 and i don’t remember what the finance charge was when we did this video before so i’ll just recalculate it quickly 10 000 plus 48 times 454 equals that’s the total that we pay minus 29 275 is equal to 2517 was our finance charge and now we have

All the pieces that we need f equals 2 517 k equals 18. n equals 48 and we only have to plug everything into the formula now our unearned interest in the numerator is the finance charge 2517 times k which is 18 times k plus 1. well if k is 18 and k plus 1 is 19. divided by n 48 times n plus 1 49 being a dollar amount we will round it to the nearest cent

Parentheses 2517 times 18 times 19 close parentheses divided by open parenthesis 48 times 49 close parentheses equals 300 the dollar sign 365 and 99 cents that is our unearned interest using the rule of 78 but we’re also asked what is the payoff amount the payoff amount is all of the remaining payments minus the unearned interest how many payments have we not

Pay off amount how many payments have we not made yet well we’ve made 29 payments we haven’t remember we haven’t made the 30th so 48 minus 29 is how many payments we haven’t made because we haven’t made the 30th yet we’ve only made through the 29 payments that’s going to get multiplied times the 454. so that’s what we still owe for the payments minus the