- HOW MUCH WOULD YOU NEED TO DEPOSIT IN AN ACCOUNT
NOW IN ORDER TO HAVE $3,000 IN THE ACCOUNT IN FIVE YEARS?
ASSUME THE ACCOUNT EARNS 3% INTEREST COMPOUNDED MONTHLY.
WE'LL BE USING THE COMPOUNDED INTEREST FORMULA
TO SOLVE THIS PROBLEM.
IF WE TAKE A LOOK AT THESE TWO FORMULAS HERE BELOW,
THEY ARE EQUIVALENT BUT USE DIFFERENT VARIABLES
TO REPRESENT THE SAME QUANTITIES.
THIS FIRST FORMULA IS PROBABLY
A MORE COMMON COMPOUNDED INTEREST FORMULA,
BUT OUR TEXT BOOK DOES USE THIS FORM OF THE EQUATION HERE.
WHERE THE ACCOUNT BALANCE AFTER A CERTAIN AMOUNT OF TIME
IN THIS FORMULA IS "A," IN THIS FORMULA IT'S P SUB N,
P IN THIS FORMULA AND P SUB ZERO IN THIS FORMULA
REPRESENT THE STARTING AMOUNT OR THE PRESENT VALUE
OR AS THE ANNUAL INTEREST RATE EXPRESSES
A DECIMAL IN BOTH FORMULAS.
IN THIS FIRST FORMULA, N IS THE NUMBER OF COMPOUNDS PER YEAR.
NOTICE HOW N OCCURS HERE AND HERE.
IN OUR SECOND FORMULA, K IS THE NUMBER OF COMPOUNDS PER YEAR
WHICH OCCURS HERE AND HERE.
AND THEN FINALLY, IN THIS FIRST FORMULA,
LOWER CASE T IS TIME IN YEARS
AND THE SECOND FORMULA CAPITAL N, IS TIME IN YEARS.
TO BE CONSISTENT WITH OUR TEXT,
LET'S GO AHEAD AND USE THIS FORMULA HERE.
BECAUSE WE WANT $3,000 IN THE ACCOUNT IN FIVE YEARS,
P SUB CAPITAL N WOULD BE P SUB 5 WHICH IS EQUAL TO $3,000,
CAPITAL N, THE NUMBER OF YEARS IS EQUAL TO 5.
THE INTEREST IS 3% SO R IS EQUAL TO 3%.
AS A DECIMAL THIS WOULD BE 0.03.
SINCE THE INTEREST IS COMPOUNDED MONTHLY,
K THE NUMBER OF COMPOUNDS IS EQUAL TO 12.
WE WANT TO FIND THE INITIAL DEPOSIT OF THE PRESENT VALUE
WHICH WOULD BE P SUB ZERO.
SO NOW TO SET UP OUR EQUATION,
WE'D HAVE P SUB 5 OR 3,000 = P SUB ZERO
WHICH WE'RE TRYING TO FIND x THE QUANTITY 1 + R DIVIDED BY K
THAT WOULD BE 0.03 DIVIDED BY 12,
RAISED TO THE POWER OF N x K WHICH WOULD BE 5 x 12.
NOTICE HOW HERE WE'RE DIVIDING BY 12
TO GET A MONTHLY INTEREST RATE
AND HERE WE'RE MULTIPLYING THE NUMBER OF YEARS x 12
TO GET THE TOTAL NUMBER OF MONTHS.
5 x 12 WOULD BE 60,
SO WE HAVE 3,000 = P SUB ZERO x THE QUANTITY 1 + 0.03
DIVIDED BY 12 RAISED TO THE POWER OF 60.
WE WANT TO SOLVE THIS EQUATION FOR P SUB ZERO.
WE DO HAVE TO BE CAREFUL ABOUT ROUNDING ERRORS
WHEN SOLVING EQUATIONS LIKE THIS.
WHAT WE DON'T WANT TO DO IS GET A DECIMAL APPROXIMATION
FOR THE SUM AND THEN RAISE IT TO THE SIXTIETH POWER
BECAUSE IT WOULD PRODUCE A ROUNDING ERROR.
LET'S GO AHEAD AND LEAVE IT IN THIS FORM
AND SOLVE FOR P SUB ZERO BY DIVIDING BOTH SIDES
OF THE EQUATION BY THE SUM RAISED TO THE SIXTIETH POWER.
NOTICE ON THE RIGHT SIDE THIS WOULD SIMPLIFY TO ONE
SO WE HAVE P SUB ZERO IS EQUAL TO THIS QUOTIENT HERE
WHICH WE'LL ROUND TO THE NEAREST CENT.
SO NOW WE'LL GO TO THE CALCULATOR.
AND WE HAVE 3,000 DIVIDED BY THE QUANTITY 1 + .03 DIVIDED BY 12
RAISED TO THE 60th POWER.
BUT UP TO THE NEAREST CENT WE WOULD HAVE $2,582.61.
THIS IS THE AMOUNT THAT WOULD HAVE TO BE DEPOSITED TODAY
TO HAVE $3,000 IN THE ACCOUNT IN FIVE YEARS IF THE ACCOUNT
EARNED 3% INTEREST COMPOUNDED MONTHLY.
THIS CAN ALSO BE CONSIDERED THE PRESENT VALUE
OF THIS $3,000 FIVE YEARS INTO THE FUTURE.
I HOPE YOU FOUND THIS HELPFUL.