Algebraic Cost Comparison: TEAS Practice Question with Answer & Explanation

Understanding Cost Comparison Word Problems

In nursing school and on the TEAS, you will frequently encounter "break-even" or comparison problems. These require you to set up an algebraic equation where two different scenarios are equal to each other. Here, we are looking for the number of guests ($x$) where the total cost of Caterer A equals the total cost of Caterer B.

Step 1: Translate the Word Problem into Algebra

First, define your variable: Let $x$ represent the number of guests.

Next, write an expression for each caterer's total cost:

• Caterer A: $450 + 25x$ (The flat fee plus $25 per person)
• Caterer B: $250 + 35x$ (The flat fee plus $35 per person)

Step 2: Set the Expressions Equal to Each Other

Since we want to find out when the costs are the same, we set the equations equal: $$450 + 25x = 250 + 35x$$

Step 3: Solve for $x$

1. Group the variables: Subtract $25x$ from both sides to keep the $x$ value positive. $$450 = 250 + 10x$$
2. Isolate the variable term: Subtract 250 from both sides. $$200 = 10x$$
3. Solve for $x$: Divide both sides by 10. $$x = 20$$

At 20 guests, both companies will charge exactly the same amount.

Why the Other Options are Incorrect

• A) 12 guests: If you plug 12 into the equations: Caterer A ($450 + 300 = 750$) and Caterer B ($250 + 420 = 670$). The costs are not equal.
• B) 16 guests: Plugging in 16 gives: Caterer A ($450 + 400 = 850$) and Caterer B ($250 + 560 = 810$). Still not equal.
• D) 25 guests: At 25 guests, Caterer A ($1,075$) becomes cheaper than Caterer B ($1,125$). This is the "tipping point" where the company with the lower per-person rate but higher flat fee becomes the better financial choice.

TEAS Math Concept: Linear Equations

This problem tests your ability to create and solve linear equations. In a clinical setting, you might use similar logic to compare medication costs or calculate dosage equivalents across different concentrations.

Memory Tip

When setting up these equations, think: "Fixed + (Rate $\times$ Variable) = Fixed + (Rate $\times$ Variable)". Always move the smaller variable to the side of the larger variable to avoid dealing with negative numbers!