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Math Help - Future Values of Annuities

  1. #1
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    Future Values of Annuities

    Suppose you want yo buy a new car. You have $5000 in cash and can afford monthly payments of $250. a) What price car can you buy if you can obtain a 36-month loan and 6% compounded monthly? b) If you get the loan, what is the unpaid balance after 2 years?
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  2. #2
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    K, this should be right for ya... sorry if the format is somewhat elementary. I'm a sophmore in college, but this is my first helpful post on here.

    "Suppose you want to buy a new car. You have $5000 in cash and can afford monthly payments of $250. a) What price car can you buy if you can obtain a 36-month loan and 6% compounded monthly? b) If you get the loan, what is the unpaid balance after 2 years? "


    I actually remember deriving this equation once.. haha

    Allright, THIS is your "Compounding Interest Equation". A is the total amount of money you have (that will be charged interest). ONLY MONEY that is assosciated with interest should go in this equation! Moving on... r is your interest rate, k is the number of times compounded per year, t is the time in years, and P is the amount before interest.

    Your values are r=.06, k=12, t=3, and A= (250*36)=9000

    Knowing to use A instead of P is the tricky part here, as well as knowing how to calculate it. Realize that you need to find how much money you have available AFTER the interest has accrued for 3 years!

    So, we need to find out how much money (call it P) will equal $9000 after 36 months has gone by. This will allow us to spend every dollar when we add the $5000 cash we have NOW onto this number "P".

    So, plug in you're numbers and solve for P (the Price of the car).

    9000=P(1+.06/12)^(12*3)

    1+.06/12=1.005-> 1.005^36 = 1.19668 -> 9000/1.19668 = $7,520.804

    Now, this means that $7,520.80 will come to a total of $9,000 after 36 months have passed by with monthly compounding at a 6% interest rate.

    Since you have the $5,000 you started with and the $9,000 you will deposit over the next 3 years (which will really only be able to pay off $7520.80 towards the car since the rest will go to interest), you can buy a car worth:

    $5,000 + $7520.80 = $12,520.80

    I hope this makes sense.. it's my 'help' first post. Thanks is appeciated if it helped!

    -Andrew
    Last edited by drewkx152; March 24th 2009 at 06:19 PM.
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  3. #3
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    b) If you get the loan, what is the unpaid balance after 2 years?

    My bad! Didn't see part b!

    Does she want the unpaid balance on the LOAN or the CAR? The unpaid loan balance tells you how much you still have to pay if you continue with the loan for the next year. The unpaid balance on the CAR tells you how much you would need to have in cash to buy the rest of the car "now" and avoid interest for the third year.

    I assume your teacher didn't clarify, so perhaps you can impress her by solving both. Here's how, pay close attention to the logic.

    LOAN BALANCE:

    The total loan amount remember is $250*36= $9,000

    After two years, you can deduct whatever you have put in from that.
    $9,000 - $250*24 = $3,000 LEFT TO PAY on the loan.

    If your teacher wants the balance on the CAR instead of the loan (which might be the case since the loan is so easy to find out)

    CAR BALANCE:

    We know we have spent $6,000 paying off the loan so far. The car is worth $12,520.80. If we learn how much $ went to interest, then we know how much went to the car (since the money only goes to one of those two places).

    Find the ratio of money that goes to your car vs how much you spend total, so use the VALUE of your CAR in the loan divided by TOTAL PRICE of your LOAN.

    AKA: $7,520.80 / $9,000 = .835644

    This means 83.56 cents of every dollar went to the CAR, the rest went to interest so far.
    The amount of the car that is still unpaid (if you were to buy the rest of the car IN CASH right now for instance to avoid having to pay interest for the next year), is $7,520.80 - ($6,000 *.835644) = $2506.93, that is the unpaid balance on the car if you wanted to BUY IT IN CASH now.

    Hope it helps bud!
    Last edited by drewkx152; March 24th 2009 at 06:30 PM. Reason: Fixed error in logic (included cash in interest)
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