# SOLUTION: A car dealer offers you a choice of 0% financing for 60 months or \$2500 cash back on a new vehicle. You have a pre-approved 60-month loan you can use from your credit union at a 4

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 Question 321853: A car dealer offers you a choice of 0% financing for 60 months or \$2500 cash back on a new vehicle. You have a pre-approved 60-month loan you can use from your credit union at a 4% interest rate. If the monthly payments at 0% are \$16.67 per \$1000 financed, and the monthly payments at 4% are \$18.41 per \$1000 financed, what is the range of new car prices for which the cash back option will cost you less? For what range of car prices should you take the 0% financing? This is what I have.. \$16.67 * \$1000 = \$16,670. \$16,670. /60 = \$277.83 \$18.41 * \$1000 = \$18,410. - \$2500 = \$15,910. \$15,910 /60 = \$265.17 Answer by Theo(3458)   (Show Source): You can put this solution on YOUR website!The problem states that if you take the 0% financing, then you don't get \$2500 cash back, but if you take the 4% financing, then you do get \$2500 cash back. A basic assumption is that you use the \$2500 cash back as a down payment on the car. This means that if you take the 4% financing with the \$2500 cash back, then the amount you have to finance is \$2500 less. Let x = the price of the car. With 0% financing, you have to finance the full price of the car which is equal to x. With 4% financing, you have to finance \$2500 less because you are given \$2500 Cash Back which you use as a down payment on the car. this means you have to finance (x - 2500). You have 2 equations to work with. Your first equation with the 0% financing is: total payments = 60 * (16.67 / 1000) * x Your second equation with the 4% financing is: total payments = 60 * (18.41 / 1000) * (x - 2500) You want to know when the 0% financing plan is better than the 4% financing plan. It is better when the total payments under the 0% financing plan are less than the total payments under the 4% financing plan. You find this by setting up an equation that compares the total payments under the 0% financing plan with the total payments under the 4% financing plan. You get: Total Payments with 0% financing plan < Total Payments with 4% financing plan. Replace the above statement with the equations for each of those options to get: 60 * (16.67 / 1000) * x < 60 * (18.41 / 1000) * (x - 2500) You solve this equation in the following manner: Divide both sides of this equation by 60 and multiply both sides of this equation by 1000 to get: 16.67 * x < 18.41 * (x-2500) Simplify this equation by multiplying out the right hand side of the equation to get: (16.67 * x) < (18.41 * x) - (18.41 * 2500) Simplify this further to get: (16.67 * x) < (18.41 * x) - 46025. Add 46025 to both sides of this equation and subtract 16.67 * x from both sides of this equation to get: 46025 < (18.41 * x) - (16.67 * x) Simplify this further to get: 46025 < 1.75 * x Divide both sides of this equation by 1.74 to get: 26451.14943 < x This is the same as: x > 26451.14943 Since x is the price of the car, what this means is that you will be better off with the 0% financing option when the price of the car is greater than \$26,451.14943. In order to confirm that this value is good, we need to put in some values for the price of the car to see when the 0% financing plan is better and when the 4% financing plan is better. If the price of the car is less than \$26,451.14943, the 0% financing plan should cost more (total payments will be higher). If the price of the car is greater than \$26,451.14943, the 0% financing plan should cost less (total payments will be lower). Your break even point should be when the price of the car is exactly \$26,451.14943. I will create a table assuming the price of the car is either: \$20,000 (less than \$26,451.14943) \$26,451.14943 (EQUAL TO \$26,451.14943) \$30,000 (greater than \$26,451.14943) The table is shown below: ```POC = Price of car. TP0 = Total Payments with the 0% financing plan. TP4 = Total Payments with the 4% financing plan. POC TP0 TP4 Result \$20,000.00 \$20,004.00 \$19,330.50 TP0 > TP4 \$26,451.15 \$26,456.44 \$26,456.44 TP0 = TP4 \$30,000.00 \$30,006.00 \$30,376.50 TP0 < TP4 ``` From the above Table: When the price of the car is less than \$26,451.15, the 0% financing plan is more expensive. When the price of the car is equal to \$26,451.15, the 0% financing plan and the 4% financing plan cost the same. This is the break even point. When the price of the car is greater than \$26,451.15, the 0% financing plan is less expensive. These numbers confirm the results of the equation analysis. The formulas used are: Total Payments with the 0% financing plan = (60 * 16.67)/1000 * x Total Payments with the 4% financing plan = (60 * 18.41)/1000 * (x-2500) x is the price of the car. With the 0% financing plan, you have to finance x. With the 4% financing plan, you have to finance (x - 2500). This is because you got \$2500 cash back with the 4% financing plan and you used that \$2500 as a down payment on the car which meant that you had to finance \$2500 less.