The profit and loss problem form that this problem fits into is one that has strong DSM’s into mathematics. Here we are tempted to do the math in part because that’s so easy. It’s so available to us.
This is characteristic of a mid-level arithmetic problem where there are some shifts and shimmies but overall it’s a fairly straightforward problem that utilizes no more than the four basic operations. So, on the one hand, this profit and loss problem is pre-algebra or even some sort of grade school math. On the other hand, this makes the solution path much more elusive.
Before we dive into solving this problem, let’s take a look at it:
The total cost of Company X to produce a batch of t-shirts is $5,000 plus $2 per t-shirt. Each t-shirt sells for $12. The gross profit earned from producing and selling these t-shirts is the total income from the sales minus total production costs. If a batch of 20,000 t-shirts is produced and sold, then Company X’s profit per t-shirt is?
A. $9.00
B. $9.50
C. $9.75
D. $10.00
E. $11.75
Solving the Problem Using Math
So of course we can follow the math. We can add up all the costs, five thousand plus two dollars, times twenty thousand. Then contrast that with the revenue that comes in which is 12×20,000. But then we’re left with the ugly division problem that brings us to the profit per t-shirt, this is where the GMAT sticks us.
Instead of handling this in aggregate, it’s strongly preferable to handle it with a higher level solution path. Let’s take a look at a few:
Higher Level Solution Path: Distribution
One way to do this is to distribute the fixed cost over the cost per t-shirt. This is actually a lot easier than it seems. Twenty thousand t-shirts, five thousand dollars, five over twenty is one-quarter.
Therefore, it costs one-quarter per t-shirt in addition to the two dollars in variable cost. So, twelve minus two is equal to ten dollars, minus one quarter is equal to nine dollars and seventy-five cents.
Higher Level Solution Path: Graphical Equalization
We can also use a graphic equalization method in order to get to the same conclusion. If the numbers were more complicated, understanding that that shift is one-quarter down. That is the fixed cost is one-quarter down.
Then we know we’re looking for something that ends in a seventy-five cents. That allows us to eliminate all the answer choices that don’t end in 0.75. Then we can use scale to determine that 9.75 is the correct answer.
Practice Problems
There are more complicated versions of this problem form. In particular, I’d encourage you to explore being told that the t-shirt company is breaking even. Then determining the amount of variable costs or fixed cost that’s there or even the production run. Similarly, you can be given a target profit or loss, the break-even just being the zero, so it’s a bit easier and you have to reverse engineer the relationships.
Once again, this doesn’t have to be done algebraically. As you begin to appreciate the subtlety of the ratio between costs production run and total P&L all of these problems should be simplified and should be very straightforward.
Continue your GMAT practice with the Wedding Guest GMAT problem.