Question 106429: A dishwasher costs $875. You pay $100 down and finance the rest at 11.5% APR for 24 months. How much more did you pay for the dishwasher by financing it instead of paying cash?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! To start, we need to make some assumptions about how often the interest is compounded, and how frequently you would be making payments. For example, the typical way things would go would be that you would make a series of 24 equal monthly payments starting the month after you made the down payment and received the merchandise. That will result in one answer. On the other hand, you might let the interest accrue at some compounding rate (quarterly, monthly, daily) and make one large payment at the end of the 24 months. This would result in a much different answer. I'm going to show you the solution for the typical scenario.
The first thing to determine is the principal amount of the loan. In this case, the cost of the item is $875, but you paid a $100 down payment. Therefore you are financing $875 - $100 = $775. So P = $775.
The next thing to determine is the interest rate per payment period. You are given the ANNUAL rate of interest, 11.5%, but the payment period is monthly, therefore the rate per payment period is 11.5%/12 = 0.95833%. You will need to express this percentage as a decimal number, so 0.95833% means r = 0.0095833.
The formula for the payment amount for a loan with constant payments and a constant interest rate is:
Where r is the periodic rate, n is the number of periods, and P is the principal amount of the loan.
I'll leave the substitution and the arithmetic to you, but you should arrive at Payment = $36.30
Since you are making 24 payments of $36.30, at the end of the 24 months, you will have paid 24 X $36.20 = $871.20 in monthly payments plus the $100 down payment for a total of $971.20.
Had you paid the full $875 when you bought the item you would have paid 971.20 - $875 = $96.20 less.
Depending on how and where you round off when you do the arithmetic, your answer might be 2 or 3 cents different than mine, but for this sort of problem that shouldn't matter. 3 cents doesn't mean much when you are talking about $100.
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Just to show the difference, had you opted to let the interest accrue for the entire 24 months and make a single large payment at the end, the amount of that payment would be given by:
Substituting $775 for P, .0095833 for r, and 24 for n and using a calculator (even better an Excel spreadsheet), you get Payment = $974.35. That plus your $100 down is $1074.35, or about $103 more than making the 24 periodic payments and nearly $200 more than paying the entire cost at the beginning.
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