Question 983869
The best way to see questions like this is to break up time. 

Let's first look at time 0.


At time 0: We put $150 in hopes to accumulate to $3,000 so that we can put a down payment on our car.

We earn 2.5% interest compounded monthly.

So effectively we earn 2.5/12% or .2083% per month.


So at time t (in months): we have accumulated to  150(1.002083)^t


And we want to figure out at what t value do we accumulate to 3000?

So if we set the two equal:

150(1.002083)^t = 3000

(1.002083)^t = 3000/150 = 20

(1.002083)^t = 20

Take the natural log of both sides.

ln(1.002083^t) = ln(20)

Use properties of logs to finish this:

t * ln(1.002083) = ln(20)

t = ln(20)/ln(1.002083) = 1439.68 months  or as an integer:  1440 months.

1440 months is kind of a nasty answer, so let's divide that by 12 to get 120 years.
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NOTE:

I find 120 years to be an unreasonable amount of time to save for a down payment, so maybe there is a typo in your question. If so, let me attempt to address that.

Let's assume you meant to say $1500 in a savings account (that would make much more sense).

Then all the math would be the same except 3000/1500 = 2 not 20.

So our answer would be t = ln(2)/ln(1.002083) = 28 months

In terms of years, this is 2 1/3 years. Much more reasonable.

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If you have additional questions, please contact me at swincher4391@yahoo.com.

Thanks,

Devin