Ex: Determine a Monthly Payment Needed to Pay Off a Credit Card

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Video Transcript:

– YOU HAVE $3,000
ON A CREDIT CARD THAT CHARGES A 17%
INTEREST RATE. IF YOU WANT TO PAY OFF THE
CREDIT CARD IN TWO YEARS, HOW MUCH WILL YOU NEED TO PAY
EACH MONTH? THIS ASSUMES YOU DON'T CHARGE
ANYTHING NEW TO THE CARD. SO TO SOLVE THIS PROBLEM
WE'LL USE THE LOAN FORMULA GIVEN HERE BELOW. LET'S REVIEW
WHAT THE VARIABLES REPRESENT. P SUB ZERO IS THE STARTING LOAN
AMOUNT. D IS THE PAYMENT AMOUNT, OR, IN
THIS CASE, THE MONTHLY PAYMENT. R IS THE ANNUAL INTEREST RATE
EXPRESSED AS A DECIMAL. K IS THE NUMBER OF COMPOUNDS
PER YEAR. SINCE WE'RE PAYING MONTHLY WE'LL ASSUME THE INTEREST
IS COMPOUNDED MONTHLY. AND N IS THE NUMBER OF YEARS. SO FROM THE GIVEN INFORMATION
P SUB 0 = $3,000. THE CREDIT CARD CHARGES
17% INTEREST, SO R = 17% WHICH AS A DECIMAL
WOULD BE 0.17. YOU WANT TO PAY OFF
THE CREDIT CARD IN TWO YEARS, SO N = 2 YEARS. AND, AGAIN, YOU'RE MAKING
MONTHLY PAYMENTS, SO K = 12. AND WE'RE TRYING TO FIND
THE MONTHLY PAYMENT WHICH WOULD BE D. SO TO SET OUR EQUATION UP
WE WOULD HAVE 3,000 = THIS FRACTION HERE ON THE RIGHT. WE'RE LOOKING AT THE NUMERATOR, YOU WOULD HAVE E
x THE QUANTITY 1 MINUS THE QUANTITY + R
DIVIDED BY K, WHICH IS 0.17 DIVIDED BY 12
RAISED TO THE -2 x 12 POWER.

THIS IS ALL DIVIDED
BY THE FRACTION R/K OR 0.17 DIVIDED BY 12. WHEN SOLVING THIS FOR D WE DO HAVE TO BE CAREFUL
ABOUT ROUNDING. FOR EXAMPLE,
IF WE ROUND THIS SUM, THEN RAISE IT TO THIS POWER, AND THEN SOLVE FOR D WE'LL HAVE
MORE OF A ROUNDING ERROR THAN IF WE LEAVE THIS
IN THE EXACT FORM AND ROUND AT THE VERY END. LET'S TRY TO LEAVE THIS IN THE
EXACT FORM AND SOLVE FOR D. LET'S BEGIN BY CLEARING
THIS FRACTION HERE BY MULTIPLYING BOTH SIDES OF THE
EQUATION BY 0.17 DIVIDED BY 12.

Ex: Determine a Monthly Payment Needed to Pay Off a Credit Card

IF IT'S HELPFUL, WE CAN THINK OF
THIS AS BEING OVER 1, THEREFORE THIS SIMPLIFIES TO 1. SO WE HAVE 3,000 X 0.17/12 = ON THE RIGHT WE HAVE D
X THE QUANTITY 1 MINUS THE QUANTITY 1 + 0.17
DIVIDED BY 12 RAISED TO THE POWER -24. AND NOW TO SOLVE FOR D WE'LL DIVIDE BOTH SIDES
BY THIS QUANTITY HERE. SO WE'LL DIVIDE BOTH SIDES BY
1 – THE QUANTITY 1 + 0.17 DIVIDED BY 12
RAISED TO THE POWER -24. NOTICE ON THE RIGHT SIDE
THIS SIMPLIFIES TO 1, SO WE HAVE D =
THIS QUOTIENT HERE WHICH WE'LL ROUND
TO THE NEAREST CENT. WHEN ENTERING THIS
INTO THE CALCULATOR WE WILL INCLUDE ANOTHER SET OF
PARENTHESIS FOR THE NUMERATOR. SO WE HAVE OPEN PARENTHESIS
(3,000 x 0.17 DIVIDED BY 12), CLOSE PARENTHESIS
FOR THE FRACTION, CLOSE PARENTHESIS
FOR THE NUMERATOR DIVIDED BY OPEN PARENTHESIS
FOR DENOMINATOR. AND THEN WE HAVE 1 – THE
QUANTITY 1 + 0.17 DIVIDED BY 12, CLOSE PARENTHESIS. THIS IS RAISED
TO THE -24TH POWER, AND CLOSE PARENTHESIS
FOR THE DENOMINATOR.

SO THE MONTHLY PAYMENT TO
NEAREST CENT WOULD BE $148.33. SO THIS IS THE MONTHLY PAYMENT
REQUIRED TO PAY OFF THIS CREDIT CARD
IN TWO YEARS. NOW LET'S GO BACK TO THE
CALCULATOR JUST FOR A MOMENT. THIS IS THE MONTHLY PAYMENT
WE WOULD MAKE EACH MONTH FOR TWO YEARS, WHICH MEANS YOU'D MAKE
24 PAYMENTS OVER THE TWO YEAR PERIOD. SO WE MULTIPLY THIS BY 24, THIS WOULD BE THE TOTAL AMOUNT
THAT YOU WOULD PAY OVER THE TWO YEAR PERIOD. NOTICE HOW SINCE THE CREDIT CARD
BALANCE WAS $3,000, NOTICE HOW YOU WOULD BE PAYING
ALMOST $560 IN INTEREST OVER THE TWO YEAR PERIOD. THIS IS BECAUSE THE INTEREST
RATE ON MOST CREDIT CARDS IS SO HIGH. NOTICE HOW THE INTEREST RATE
WAS 17%. I HOPE YOU FOUND THIS HELPFUL..

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License: Creative Commons