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- A BANK FEATURES A SAVINGS ACCOUNT
THAT HAS ANNUAL PERCENTAGE RATE OF 3.4%
WITH INTEREST COMPOUNDED QUARTERLY.
ASHLEY DEPOSITS $2,000 INTO THE ACCOUNT.
WE FIRST WANT TO KNOW WHAT EQUATION CAN BE USED
TO DETERMINE THE ACCOUNT BALANCE AFTER T YEARS,
AND THEN WHAT THE ACCOUNT BALANCE WOULD BE
AFTER FOUR YEARS.
THE COMPOUNDED INTEREST FORMULA IS GIVEN HERE BELOW
WHERE P IS THE PRINCIPLE OR STARTING AMOUNT,
R IS THE ANNUAL INTEREST RATE EXPRESSED AS A DECIMAL,
N IS THE NUMBER OF COMPOUNDS PER YEAR.
NOTICE IT OCCURS HERE AND HERE.
SO IN THIS CASE, SINCE THE INTEREST IS COMPOUNDED QUARTERLY
AND THERE ARE FOUR QUARTERS IN ONE YEAR,
N WOULD BE EQUAL TO FOUR.
IF IT WAS MONTHLY N WOULD BE 12.
T IS TIME IN YEARS.
AND "A" IS THE AMOUNT AFTER TIME T.
SO FOR THIS SITUATION WE'D HAVE THE EQUATION "A"
= THE STARTING AMOUNT,
WHICH IS $2,000 x THE QUANTITY 1
+ THE ANNUAL INTEREST RATE EXPRESSED AS A DECIMAL.
3.4% AS A DECIMAL WOULD BE 0.034.
WE DROP THE PERCENT SYMBOL AND DIVIDE BY 100
OR MOVE THE DECIMAL POINT TO THE LEFT 2 PLACES, DIVIDED BY N,
THE NUMBER OF COMPOUNDS PER YEAR.
IT'S COMPOUNDED QUARTERLY
AND THERE ARE FOUR QUARTERS PER YEAR, SO N IS 4.
THE EXPONENT IS GOING TO BE N x T, AGAIN, N IS 4.
SO WE HAVE 4T AS OUR EXPONENT.
SO NOTICE BY DIVIDING THE ANNUAL INTEREST RATE BY 4
THIS WOULD BE THE QUARTERLY INTEREST RATE.
THEN MULTIPLYING 4 x THE NUMBER OF YEARS
THIS EXPONENT WOULD BE THE NUMBER OF QUARTERS.
NOW LET'S GO AHEAD AND SIMPLIFY THIS.
WE'D HAVE "A' = 2,000 x 1 + 0.034 DIVIDED BY 4.
THIS IS GOING TO BE 1 + 0.0085.
SO WE HAVE 1.0085 RAISED TO THE POWER OF 4T.
SO THIS WOULD BE THE EQUATION
THAT WOULD GIVE US THE ACCOUNT BALANCE AFTER T YEARS.
SO IF WE WANT TO KNOW THE ACCOUNT BALANCE AFTER FOUR YEARS
WE'RE JUST GOING TO SUBSTITUTE FOUR FOR T.
SO THE AMOUNT OF MONEY IN THE ACCOUNT
WOULD BE 2,000 x 1 + 0085 RAISED TO THE 4 x 4 POWER,
THAT WOULD BE THE 16TH POWER.
LET'S GO AHEAD AND DETERMINE THIS VALUE.
WE'LL ROUND TO NEAREST CENT.
SO WE'LL HAVE 2,000 x 1.0085 RAISED TO THE POWER OF 4 x 4
OR 16.
SO THE ACCOUNT BALANCE WOULD BE $2,290.05.