馃攳
Ex: Compounded Interest Formula - Determine Deposit Needed (Present Value) - YouTube
Channel: Mathispower4u
[0]
- HOW MUCH WOULD YOU NEED
TO DEPOSIT IN AN ACCOUNT
[3]
NOW IN ORDER TO HAVE $3,000
IN THE ACCOUNT IN FIVE YEARS?
[9]
ASSUME THE ACCOUNT EARNS 3%
INTEREST COMPOUNDED MONTHLY.
[13]
WE'LL BE USING THE COMPOUNDED
INTEREST FORMULA
[16]
TO SOLVE THIS PROBLEM.
[17]
IF WE TAKE A LOOK AT THESE TWO
FORMULAS HERE BELOW,
[20]
THEY ARE EQUIVALENT BUT USE
DIFFERENT VARIABLES
[23]
TO REPRESENT
THE SAME QUANTITIES.
[26]
THIS FIRST FORMULA IS PROBABLY
[27]
A MORE COMMON COMPOUNDED
INTEREST FORMULA,
[30]
BUT OUR TEXT BOOK DOES USE THIS
FORM OF THE EQUATION HERE.
[34]
WHERE THE ACCOUNT BALANCE AFTER
A CERTAIN AMOUNT OF TIME
[36]
IN THIS FORMULA IS "A,"
IN THIS FORMULA IT'S P SUB N,
[42]
P IN THIS FORMULA AND P SUB ZERO
IN THIS FORMULA
[46]
REPRESENT THE STARTING AMOUNT
OR THE PRESENT VALUE
[49]
OR AS THE ANNUAL INTEREST RATE
EXPRESSES
[51]
A DECIMAL IN BOTH FORMULAS.
[54]
IN THIS FIRST FORMULA, N IS THE
NUMBER OF COMPOUNDS PER YEAR.
[58]
NOTICE HOW N OCCURS
HERE AND HERE.
[61]
IN OUR SECOND FORMULA, K IS THE
NUMBER OF COMPOUNDS PER YEAR
[65]
WHICH OCCURS HERE AND HERE.
[68]
AND THEN FINALLY,
IN THIS FIRST FORMULA,
[71]
LOWER CASE T IS TIME IN YEARS
[74]
AND THE SECOND FORMULA CAPITAL
N, IS TIME IN YEARS.
[78]
TO BE CONSISTENT WITH OUR TEXT,
[81]
LET'S GO AHEAD AND USE THIS
FORMULA HERE.
[83]
BECAUSE WE WANT $3,000
IN THE ACCOUNT IN FIVE YEARS,
[87]
P SUB CAPITAL N WOULD BE P SUB 5
WHICH IS EQUAL TO $3,000,
[95]
CAPITAL N, THE NUMBER OF YEARS
IS EQUAL TO 5.
[100]
THE INTEREST IS 3%
SO R IS EQUAL TO 3%.
[106]
AS A DECIMAL THIS WOULD BE 0.03.
[110]
SINCE THE INTEREST
IS COMPOUNDED MONTHLY,
[113]
K THE NUMBER OF COMPOUNDS
IS EQUAL TO 12.
[118]
WE WANT TO FIND THE INITIAL
DEPOSIT OF THE PRESENT VALUE
[121]
WHICH WOULD BE P SUB ZERO.
[124]
SO NOW TO SET UP OUR EQUATION,
[126]
WE'D HAVE P SUB 5 OR 3,000 = P
SUB ZERO
[133]
WHICH WE'RE TRYING TO FIND x THE
QUANTITY 1 + R DIVIDED BY K
[140]
THAT WOULD BE 0.03 DIVIDED
BY 12,
[145]
RAISED TO THE POWER OF N x K
WHICH WOULD BE 5 x 12.
[152]
NOTICE HOW HERE WE'RE DIVIDING
BY 12
[154]
TO GET A MONTHLY INTEREST RATE
[156]
AND HERE WE'RE MULTIPLYING
THE NUMBER OF YEARS x 12
[159]
TO GET THE TOTAL NUMBER OF
MONTHS.
[161]
5 x 12 WOULD BE 60,
[163]
SO WE HAVE 3,000 = P SUB ZERO
x THE QUANTITY 1 + 0.03
[170]
DIVIDED BY 12
RAISED TO THE POWER OF 60.
[174]
WE WANT TO SOLVE THIS EQUATION
FOR P SUB ZERO.
[177]
WE DO HAVE TO BE CAREFUL
ABOUT ROUNDING ERRORS
[179]
WHEN SOLVING EQUATIONS
LIKE THIS.
[181]
WHAT WE DON'T WANT TO DO IS GET
A DECIMAL APPROXIMATION
[184]
FOR THE SUM AND THEN RAISE IT TO
THE SIXTIETH POWER
[187]
BECAUSE IT WOULD PRODUCE
A ROUNDING ERROR.
[189]
LET'S GO AHEAD AND LEAVE IT
IN THIS FORM
[191]
AND SOLVE FOR P SUB ZERO
BY DIVIDING BOTH SIDES
[194]
OF THE EQUATION BY THE SUM
RAISED TO THE SIXTIETH POWER.
[213]
NOTICE ON THE RIGHT SIDE THIS
WOULD SIMPLIFY TO ONE
[217]
SO WE HAVE P SUB ZERO IS EQUAL
TO THIS QUOTIENT HERE
[222]
WHICH WE'LL ROUND TO THE NEAREST
CENT.
[225]
SO NOW WE'LL GO
TO THE CALCULATOR.
[228]
AND WE HAVE 3,000 DIVIDED BY THE
QUANTITY 1 + .03 DIVIDED BY 12
[239]
RAISED TO THE 60th POWER.
[245]
BUT UP TO THE NEAREST CENT
WE WOULD HAVE $2,582.61.
[260]
THIS IS THE AMOUNT THAT WOULD
HAVE TO BE DEPOSITED TODAY
[267]
TO HAVE $3,000 IN THE ACCOUNT
IN FIVE YEARS IF THE ACCOUNT
[271]
EARNED 3% INTEREST COMPOUNDED
MONTHLY.
[274]
THIS CAN ALSO BE CONSIDERED
THE PRESENT VALUE
[277]
OF THIS $3,000 FIVE YEARS
INTO THE FUTURE.
[282]
I HOPE YOU FOUND THIS HELPFUL.
Most Recent Videos:
You can go back to the homepage right here: Homepage