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.