– WELCOME TO A LESSON ON HOW TO DETERMINE THE

REMAINING LOAN BALANCE AFTER A CERTAIN NUMBER OF YEARS

USING THE LOAN FORMULA. WITH LOANS IT IS OFTEN DESIRABLE

TO DETERMINE WHAT THE REMAINING LOAN BALANCE

WILL BE AFTER SOME NUMBER OF YEARS. FOR EXAMPLE,

IF YOU PURCHASE A HOME AND PLAN TO SELL IT

IN FIVE YEARS, YOU MIGHT WANT TO KNOW HOW MUCH

OF THE LOAN BALANCE YOU WILL HAVE PAID OFF AND HOW MUCH YOU HAVE TO PAY

FROM THE SALE. TO DETERMINE THE REMAINING LOAN

BALANCE AFTER A NUMBER OF YEARS WE FIRST NEED TO KNOW

THE LOAN PAYMENT IF WE DON’T ALREADY KNOW. REMEMBER THAT ONLY A PORTION

OF A LOAN PAYMENT GOES TOWARDS THE LOAN BALANCE. A PORTION ALSO GOES TOWARD

THE LOAN INTEREST. FOR EXAMPLE, IF YOU’RE PAYMENTS

WERE $1,000 A MONTH, AFTER A YEAR YOU WILL NOT HAVE

PAID OFF $12,000 OF THE LOAN BALANCE. TO DETERMINE THE REAMING LOAN

BALANCE WE CAN THINK HOW MUCH LOAN

WILL THESE LOAN PAYMENTS BE ABLE TO PAY OFF IN THE

REMAINING TIME OF THE LOAN? SO HERE’S OUR LOAN FORMULA, WHICH WE SHOULD ALREADY BE

FAMILIAR WITH, P SUB ZERO IS THE LOAN AMOUNT, D IS THE LOAN PAYMENT

PER UNIT OF TIME, R IS THE ANNUAL INTEREST RATE

AS A DECIMAL. K IS THE NUMBER OF COMPOUNDS

PER YEAR, AND N IS THE LENGTH OF THE LOAN

IN YEARS. LET’S CONSIDER A CAR LOAN

EXAMPLE. IF YOU PURCHASE A NEW CAR

FOR $20,000 USING A LOAN WITH MONTHLY PAYMENTS

FOR 5 YEARS AT 4% INTEREST. WHAT WOULD THE REMAINING BALANCE

BE AFTER THREE YEARS? SO OFTEN WHEN ANSWERING

THIS TYPE OF PROBLEM THERE ARE TWO STEPS. NUMBER ONE, WE CALCULATE THE

MONTHLY PAYMENTS ON THE LOAN, AND THEN, TWO, CALCULATES

THE REMAINING OF THE BALANCE BASED ON THE REMAINING TIME

OF LOAN. SO IT DOES SEEM A BIT STRANGE

THAT IF THIS WAS YOUR NEW CAR YOU WOULDN’T KNOW

THE CAR PAYMENTS. BUT IN EITHER CASE, WE’LL BEGIN BY DETERMINING WHAT

THE MONTHLY PAYMENT WOULD BE. LET’S BEGIN BY IDENTIFYING

THE GIVEN INFORMATION. THE ORIGINAL BALANCE IS $20,000,

SO P SUB 0=20,000. THE PAYMENTS ARE MONTHLY, SO WE’LL ASSUME MONTHLY

COMPOUNDED INTEREST, AND THEREFORE K=12. THE LOAN IS FOR FIVE YEARS

SO N=5. THE INTEREST IS 4%

SO R AS A DECIMAL WOULD BE 0.04. AND WE’RE TRYING TO FIND MONTHLY

PAYMENT WHICH WOULD BE D. NOW WE’LL SUB THESE VALUES

INTO THE FORMULA. SO, AGAIN, WE’VE DONE THIS

BEFORE, BUT P SUB 0=20,000, R=0.04, N=5, AND K=12. K APPEARS THREE TIMES. NOW, WE WANT TO SOLVE THIS

FOR D, SO WE’LL SIMPLIFY INSIDE THE

PARENTHESIS IN THE NUMERATOR, THEN THE DENOMINATOR,

AND THEN SOLVE FOR D. SO IN THE NUMERATOR WE HAVE

1 – THE QUANTITY 1 + 0.04 DIVIDED BY 12. I’M GOING TO RAISE TO THIS

TO THE POWER OF -60. WE DO HAVE TO BE CAREFUL

WHEN ROUNDING. IF WE DO WANT TO ROUND, WE SHOULD INCLUDE AT LEAST

THREE SIGNIFICANT DIGITS. AND NOTICE IN THIS CASE I’M GOING TO INCLUDE ALL DECIMAL

PLACES, WHICH WE SEE HERE. AND OUR DENOMINATOR

IS 0.04 DIVIDED BY 12, WHICH GIVES US

THIS DECIMAL HERE. AGAIN, IF WE DO ROUND, IT IS GOING TO CHANGE THE ANSWER

SLIGHTLY. NOW, BECAUSE WE STILL HAVE THIS

D HERE WE’LL FIND THIS QUOTIENT, THEN MULTIPLY BY D, THEREFORE, THIS QUOTIENT

WILL GIVE US THIS COEFFICIENT, WHICH WE SEE HERE. SO IF WE TAKE 0.1809968963

AND DIVIDE BY 0.0033333333 IT GIVES US THIS VALUE HERE, WHICH, AGAIN,

IS THE COEFFICIENT OF D. AND NOW TO SOLVE FOR D, WE

DIVIDE BOTH SIDES BY THIS VALUE. SO ON THE RIGHT SIDE

WE JUST HAVE D, AND THIS QUOTIENT WILL GIVE US

THE MONTHLY PAYMENT. SO WE HAVE 20,000

DIVIDED BY 54.29906943. ROUNDED TO THE NEAREST CENT, NOTICE HOW THE MONTHLY PAYMENT

IS $368.33. AND NOW WE CAN ANSWER

THE QUESTION WHAT WOULD THE REMAINING BALANCE

BE AFTER THREE YEARS. WELL, AFTER THREE YEARS NOTICE

HOW BECAUSE THIS IS A FIVE YEAR THERE’S ONLY TWO YEARS

REMAINING. WE’LL BE MAKING THESE PAYMENTS

HERE FOR TWO MORE YEARS. WHICH MEANS, WE NOW WANT TO FIND

P SUB ZERO, THE LOAN AMOUNT, AGAIN, WITH TWO YEARS REMAINING. SO R IS STILL 0.04,

N IS NOW 2 YEAR, K IS STILL 12, AND NOW WE KNOW THE MONTHLY

PAYMENT IS $368.33. SO, AGAIN, PERFORMING

SUBSTITUTION INTO OUR LOAN FORMULA, NOTICE R IS 0.04, N IS 2,

K IS 12, AND D IS 368.33. NOTICE HERE WE’RE SOLVING FOR

P SUB ZERO, WHICH MEANS WE JUST NEED TO

SIMPLIFY THE RIGHT SIDE HERE. AND, AGAIN, WE’LL DO THIS

IN STEPS. SO LOOKING AT THE NUMERATOR

IN THE PARENTHESIS WE’D HAVE 1 – THE QUANTITY

1 + 0.04 DIVIDED BY 12. AND NOW WE’RE GOING TO RAISE

THIS TO THE POWER OF -24, WHICH GIVES US

THIS DECIMAL HERE. AND OUR DENOMINATOR IS THE SAME

AS LAST TIME, WHICH WE SEE HERE, SO NOW WE’LL FIND THE PRODUCT

IN THE NUMERATOR AND THEN DIVIDE

BY THE DENOMINATOR. SO LETS PUT THE NUMERATOR

IN PARENTHESIS. WE HAVE (368.33 X 0.0767608361), CLOSE PARENTHESIS AND THEN WE’LL

DIVIDE BY 0.0033333333, ENTER. NOW, NOTICE HERE

IF WE ROUND TO THE NEAREST CENT WE WOULD ACTUALLY ROUND UP, AND THEREFORE 99 CENTS

ROUNDS TO $1, AND THEREFORE THE REMAINING LOAN

BALANCE AFTER 3 YEARS WOULD BE $8,482. WHICH MEANS, IF YOU WANTED TO

SELL YOUR CAR AFTER THREE YEARS YOU’D WANT TO MAKE SURE

YOU SELL THE CAR FOR MORE THAN THIS AMOUNT. OTHERWISE YOU WOULD STILL OWE

ON THE LOAN. I HOPE YOU FOUND THIS HELPFUL.