Question

you decide to put $150 in a savings account to save for a $3,000 down payment on a new car. If the account has an interest rate of 2.5% per year and is compounded monthly, how long does it take you to earn $3000 without depositing any additional funds

Answer 1

(Look at the attached graphic)
Years = log (3,000 / 150) / 12 * log (1 + .025 / 12)
Years = log (20) / 12 * log ( 1.0020833333 )
Years = 1.3010299957 / 12 * .0009038389838
Years = 1.3010299957 / 0.0108460678
Years = 119.954
That's a LONG time

Answer 2

The time is 119.973 years.

Explanation:

In the starting, we put $150, to accumulate to $3,000 so that we can put a down payment on our car.

An interest rate of 2.5% compounded monthly.

So, we earn
`2.5/12` = 0.2083% per month.

Appying the formula :
`A=P(1+r/n)^(nt)`

Now at time 't' (in months), the total amount will be
`150(1.002083)^(t)`

Given is that you need to get $3000 at the end of t years.

So, the equation becomes:

`150(1.002083)^(t)=3000`

=>
`(1.002083)^(t)=20`

Taking log on both sides,

ln
`(1.002083^(t))` = ln(20)

`t=ln(20)/ln(1.002083)`

t = 1439.68 months

or
`1439.68/12=119.973` years

Note: This value of years is unrealistic.