Question 983869: 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 $3,000 without depositing any additional funds?
Answer by swincher4391(1107)   (Show Source):
You can put this solution on YOUR website! 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
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