- 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.
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.