[well]

This is not really hard, did you try it yourself?

$(t): amount of dollars after t months, $(0)=7000
g: growrate, (100%+9%)/100% = 1.09

Now $(t)=$(0)*g^t, the standard exponential growth formula.
We want to know when we can buy the car, or: $(t)=9000

9000 = 7000 * 1.09 ^ t
{ division by 7000 }
9/7 = 1.09 ^ t
{ e^log(x) = x }
9/7 = ( e ^ log(1.09) ) ^ t
{ ( x ^ y ) ^ z = x ^ zy }
9/7 = e ^ t*log(1.09)
{ log, log(e ^ x) = x }
log(9/7) = t*log(1.09)
{ division by log(1.09) }
log(9/7) / log(1.09) = t

Hence, t = 2.916...

The rent is compounded monthly, so after 3 months we can buy the car.
We even have about $65 left!

I hope now you can explain it to her.

Next Time