Question

*I need HELP*
You decide to put $100 in a savings account to save for a $3,000 down payment on a new car. If the account has an interest rate of 2% per year and is compounded monthly, how long does it take you to earn $3,000 without depositing any additional funds?

170.202 years
14.3129 years
171.755 years
168.354 years

Answer 1

@SolomonZelman

Answer 2

@Hero @butterflydreamer

Answer 3

Use the compound interest formula
\[A=P(1+\frac{ r }{ n })^{nt}\]
A = final amount
P = initial amount
r = interest rate as a decimal
n = number of times interest is compounded in a year
t = number of years → what you're trying to find

Answer 4

Try to fill in the formula

Answer 5

@peachpi can you fill the formula for me? I am having trouble

Answer 6

which variable/number are you having trouble with?

Answer 7

A, P, N

Answer 8

ok. These are all in the question.
For A, what is the amount they're trying to save?
For P, what is the amount that was deposited?
For n, how often is the interest compounded?

Answer 9

A = 3000
P = 100
n = 12
right?

Answer 10

yes

Answer 11

what do you have so far?

Answer 12

\[3000=100(1+\frac{ 1 }{ 12 })^{(12)t}\]

Answer 13

how would i find t

Answer 14

ok. great. The only change is 1/12 should be 0.02/12 because the interest rate is 2%

Answer 15

To solve it, start by dividing by the 100 and then simplifying the stuff in parentheses

Answer 16

\[30= (1.0016)^{(12)t}\]

Answer 17

yes, so now you want to take the natural log (ln) of both sides.
\[\ln 30=\ln (1.0016)^{12t}\]
and apply the rule
\[\ln a^b = b \ln a\]
to get
\[\ln 30 = 12t \ln 1.0016\]

Answer 18

To solve for t, divide by (12 ln 1.0016)

Answer 19

I know you rounded (1 + 0.02/12) to 1.0016 for the sake of writing it down, but you really don't want to round these problems until the end. It can mess up your answer. More digits is better

Answer 20

if you have to round

Answer 21

did you get it?

Answer 22

I get 177.28737576

Answer 23

but it is not in the choices

Answer 24

That's the rounding error I mentioned above. You should use the exact answer for the stuff in parentheses (1 + .02/12) = 601/600 as a fraction, but you can usually get away with 8 to 10 numbers after the decimal point.

Answer 25

Throw a few more 6's on the 1.0016 and you'll come up with the first answer

Answer 26

Wow thank you. this is complicated math.

Answer 27

You're welcome. yeah it is. And it's pretty messed up that 3 of the choices were so close together.