Question

i need help. very confusing. pls show me how to work it out.

Luanne wants to have $3275 in her bank account in 10 years. How much money should she deposit if her account earns 7.5% interest which is compounded every month?
A. $1589.01
B. $2159.45
C. $1921.53
D. $1550.62

Answer 1

starting with basic definitions, there are 12 months in a year
also, let x be equal to the amount of money we initially put in

Answer 2

now, after every month, the bank will add an additional 7.5% in interest to the account

any ideas how to represent this as an expression?

Answer 3

is it (7.5% * 12) *10?

Answer 4

or, what the bank account would look like after the first month?

Answer 5

no

Answer 6

what you have written makes no physical sense

10*12 would be the total amount of months but you cant apply interest to the total amount of months, you can only apply interest to a dollar amount

Answer 7

oh ok

Answer 8

ok, to rephrase,
we start at month "0"- this is the initial amount of money we put into the account which would be x

after 1 month, the bank adds an extra 7.5% to your bank account,
so this means you still have x and you add x*(7.5%)
so the total bank amount after 1 month is x+x*(7.5%)

Answer 9

any questions thus far?

Answer 10

no. what's next?

Answer 11

quick quiz, if the initial amount of money i put in the bank is $100
how much money would i have after the first month?

Answer 12

$10.75

Answer 13

care to revise your answer?
we're adding interest to $100
how can the resulting bank account be less than what we put in?
did the bank steal some money?

Answer 14

oh, sorry. I thoght the amount was $10. Anyways, it's $107.50

Answer 15

ok, good, you understand how percentages work

7.5% is the equivalent of multiplying by 0.075

going back to teh expression we made,
after 1 month, the total amount of money in the bank account would be
x+0.075x

Answer 16

adding up like terms, we get the expression

1.075 x
after the first month

Answer 17

any questions?

Answer 18

I'm assuming x still equals zero?

Answer 19

no, x is a variable we use to denote the total amount of money we initially put it

Answer 20

it would be zero is we decided to put zero dollars into the bank account initially
in our example, we initally put in 100 dollars
it could be any of the multiple choice answers

its just like a placeholder since we dont know the actual amount

Answer 21

so we use x, in order to determine a hypothetical situation since we dont know enough information to start from the beginning

Answer 22

got it. what now?

Answer 23

ok, assuming after the first month, we have (1.075 x) dollars in our bank account

after the 2nd month, we add an additional 7.5% to the bank account,
any idea what this expression might look like?

Answer 24

if this doesnt make sense, think back to our example

after the first month, we have $107.5 in the account, and after 1 additional month, we add 7.5% interest of $107.5 to $107.50

Answer 25

so, would it be (1.75x)?

Answer 26

er, highly unlikely

Answer 27

ok, ill walk you through this

Answer 28

after month 1, we have 1.075x

7.5% of 1.075x would be 0.075*( 1.075x)

we add this interest to the amount we have in the account
so it looks like

1.075x +0.075(1.075x)

Answer 29

or in our $100 example
we start with $107.50 in our account after the first month
7.5% of 107.50 is $8.0625

adding the interest to our bank account
107.50 +8.0625
so we have
$115.5625

after the 2nd month

Answer 30

makes sense?

Answer 31

so do i just keep adding 7.5% to the amount of money?

Answer 32

that essentially what the problem means, but there's a trick

Answer 33

before we get into the trick
and before i answer " do i need to do this 120 times"

Answer 34

we look back at the expression after 2 months
1.075x +0.075(1.075x)

now, if you're at all familar with the distribution property, or adding "like terms"
we can simplify this expression

since
1*(1.075x) +0.075(1.075x)
through distribution/factoring property
(1+0.075)(1.075x)

1.075*(1.075x)

now, we could multiply the two 1.075 together, but that wont help you understand the concept

Answer 35

so that means at the end of the 2nd month, we will have
1.075*1.075x in our bank account?

Answer 36

so one last month calculation before i tell you the answers to all your problems

after the 3rd month, how much money would we have in our bank account, or what does the expression look like?

Answer 37

$1.23

Answer 38

er? assuming we started with $100 ?
could you please show your steps since i would like to know at what step you are making a mistake

Answer 39

oh, i thought you were on the original problem, my apologies. is the equation 1.075*1.075+(7.5%+11556.25x)?

Answer 40

107.5*107.5 = 11556.25, and then 7.5%+11556.25x

Answer 41

ok, 1st mistake
its 1.075*1.075, not 107.5 *107.5

Answer 42

and i told you to leave them seperate

Answer 43

also, if you're going to multiply by a percentage, you usually use the decimal equivalent
1%= 0.01
10%=0.1
0.1% = 0.001
7.5% = 0.075

Answer 44

when you want to find the percentage of something, you multiply the value by the decimal equivalent

so, after the 3rd month, assuming we start with (1.075 *1.075 x)
the interest would be
0.075*(1.075 *1.075 x)

adding that interest onto the bank account
(1.075 *1.075 x)+0.75*(1.075 *1.075 x)

Answer 45

any questions?

Answer 46

so far, none

Answer 47

ok, using the method i describe for after determining month 2
can you simplify?

(1.075 *1.075 x)+0.75*(1.075 *1.075 x)

Answer 48

1.075*1.075x^2+0.75?

Answer 49

er no,
FACTORING / distribution

soooo
(1.075 *1.075 x)+0.075*(1.075 *1.075 x)
lets assume that (1.075 *1.075 x)= u
so we have
u+0.075u
1 u + 0.075u
(1+0.075) u
1.075 u

so
1.075*(1.075 *1.075 x)
when simplified

Answer 50

ok so to compare them to each other
initial month
x
1st month
1.075x
2nd month
1.075*1.075 x
3rd month
1.075*1.075*1.075x

if you see the pattern by now
this means
4th would be
1.075*1.075*1.075*1.075 x
...
10th
1.075*1.075*1.075*1.075*1.075........... x

Answer 51

and this is where exponents come in handy
\[1.075 *1.075 = 1.075^2\]
\[1.075*1.075*1.075=1.075^3\]
\[1.075*1.075*1.075*1.075=1.075^4\]

thus meaning at the 120th month
our expression would look like
\[1.075^{120}x\]

Answer 52

does that make sense?

Answer 53

ohhh, i think i made a biggg mistake

Answer 54

instead of 7.5% every month, its 7.5/12 % every month

its 7.5% interest yearly

anyways simple amendment to the expression

7.5 / 12 =0 .625%

hence we would use
(1.00625)^120 x= 3275

Answer 55

and then solve for x

fyi
1.00625 ^ 120 = 2.112

Answer 56

k. thx alot i appreciate it :)

Answer 57

yup no problem