Question

Aileen deposited money into an account in which interest is compounded semiannually at a rate of 2.1%.

How much did she deposit if the total amount in her account after 3 years was $3952.08, and she made no other deposits or withdrawals?
(Points : 3)
$3489
$3562
$3712
$3751

Answer 1

please help @amistre64

Answer 2

what have you tried?

Answer 3

thats the thing i dont know what to try ive nevr done something like this because i have been behind in school and wasnt there to see the lesson, i dont know what to start with :(

Answer 4

start with the basics, what does interest do?

Answer 5

adds more?

Answer 6

yep, we we add 2.1% more to the account each period.

Answer 7

how do you think we would calculate that increase?

Answer 8

hmmm do we add it to the big number>

Answer 9

if i have $100, and interest is 2%, what is the amount of interest that was earned in one interest period?

100(.02) = 2, the interest earned is $2 right?

Answer 10

the 3952.08 number

Answer 11

lets work on the process, since you missed a lesson on it

Answer 12

yes its 2 % of your original number

Answer 13

so, when we compound, we add the interest back to our account balance.

100 + 100(.02)

100(1 + .02)

100(1.02) = 102

does this make sense?

Answer 14

yes i think so, your plugging in numbers to get the 102

Answer 15

yeah

now, the next period we have interest that is calculated on 102 because thats our NEW balance in the bank

102 + 102(.02)

102(1.02)

but we really want to view this from our original balance,
recall that 102 = 100(1.02) soo

100(1.02)(1.02 ) = 104.04

or written more traditionally

100(1.02)^2 = 104.04

Answer 16

if we compound it again for another period, what do you think our calculation will be in terms of the original balance? do you see a pattern?

Answer 17

um i do but i dont think its relevant to the problem

Answer 18

why would i speak of things that arent relevant to the problem?

Answer 19

what is the pattern that you see?

Answer 20

well i see that there is an increase of 2. firt was 102.2 then 104.4, its like 2 4 ..6 ...8 and so on

Answer 21

no i wasnt saying u would saying it was irrelevant i was saying my answer was

Answer 22

oh, lol ... at least your trying, thats importnat, ill guide you back on track with any luck

Answer 23

the pattern is that we have a starting balance of 100, and an interest that compounds at (1.02) and the exponent on the interest is equal to the period we are trying to find.

say the 5th period, what is our balance?

100 (1.02)^5 = 110.408... or 110.41

theres only one other thing we need to do to solve your problem, maybe 2 things

Answer 24

lets see first if you understand this concept

what is our 7th period balance? what would we setup?

Answer 25

gaaaawwwww im so confused lol sorry im trying to understand but im lost

Answer 26

the pattern is in changing the exponent ...

the 3rd period, what is our balance?
^^^^^^^

100 (1.02)^(3) = 106.120.. or 106.12
^^
------------------------------

the 5th period, what is our balance?
^^^^^^^

100 (1.02)^(5) = 110.408... or 110.41
^^
------------------------------

the 27th period, what is our balance?
^^^^^^^

100 (1.02)^(27) = 170.688... or 170.69
^^
-----------------------------

all that changes is an exponent

Answer 27

now try to tell me, show me your work, on how we find the 7th period balance.

Answer 28

okay 100(1.02)^(7)

Answer 29

perfect ..... thats all there is to the calculation

let put this into a formula, so that all we have to do is plug in values,

\[A=P(1+r)^n\]

P is, Principle amount (starting balance)
A is, the balance Amount we have at a given period
r is the interest rate for that period

does this make sense?

Answer 30

yes i think lol

Answer 31

its exposure at least, we will use this formula to solve your problem, but we need to define r and n to do it.

your problem says it compounds semiannually - which simply means it occurs 2 times a year.

if something happens 2 times a year, for 3 years. how many times has it occured?

Answer 32

6

Answer 33

good, then n=6

now we need the proper rate adjustment.

we are given a yearly interest rate of 2.1%, we need to DIVIDE this over the year in order to determine the 'period' interest rate.

2 times a year, wouldnt we divide 2.1 in half? that way in 1 year we have used the full 2.1% value.

tell me what im saying for the interest rate

Answer 34

well would you 2.1 to 2.1 cause its 2 times a year?

Answer 35

lets say we have $2.10 to spread out over 2 people.

how much does each person get?

Answer 36

1.05

Answer 37

perfect

you agree that its fair to give each person $1.05.

(2.10)/2 = 1.05

same concept applies here, we want to spread out 2.10% over 2 periods in the year.

each period therefore gets 1.05%

r = 1.05%, or written in decimal form:

r = .0105

Answer 38

we are given an ending balance of: A = 3952.08

n = 6, and r = .0105

\[A=P(1+r)^n\]
\[3952.08=P(1+\frac{.0210}{2})^{2*3}\]
\[3952.08=P(1+.0105)^{6}\]
\[3952.08=P(1.0105)^{6}\]
how do we solve for P?

Answer 39

do the (1.0105)^6

Answer 40

we could, but eventually what do we need to do? divide right?

n = jm, then to solve for j

j = n/m

Answer 41

the numbers are going to be what they are, but the process remains true regardless

Answer 42

\[P=\frac{A}{(1+r)^n}\]

Answer 43

P = 3952.08/(1.0105)^6

all that background, just for that lol

Answer 44

ok okay lol i see hahah sorry

Answer 45

thank you

Answer 46

$3712 is the answer

Answer 47

yes, yes it is