When Will Two Gym Memberships Cost the Same?

What This Problem Teaches

• Setting up linear equations from real-world situations with fixed and variable costs
• Solving systems of equations by substitution to find break-even points
• Interpreting solutions in context — understanding what the answer means for decision-making
• Recognizing when two different payment structures will result in equal costs
• Building foundation skills for analyzing competing business models or pricing plans

Visualizing the Problem

Let's create a table to see how costs change with the number of classes. This helps us understand the problem before diving into algebra:

Notice how StrongBody starts more expensive but FitLife's costs grow faster. They cross at exactly 10 classes.

Solution: Method 1 — Setting Equal Cost Equations

Step 1 — Write the cost equation for each gym

Let c represent the number of classes taken per month.

Step 2 — Set the costs equal

We want to find when both gyms cost the same amount:

Step 3 — Solve for c

Subtract 2c from both sides:

Subtract 25 from both sides:

Both gyms will cost the same when you take 10 classes per month.

Solution: Method 2 — Cost Difference Analysis

Step 1 — Find the monthly fee difference

StrongBody has a higher monthly fee than FitLife:

Step 2 — Find the per-class difference

But FitLife charges more per class:

Step 3 — Calculate when differences cancel out

StrongBody's $10 higher monthly fee will be offset when FitLife's extra per-class cost reaches $10:

This reasoning gives us the same answer: 10 classes.

Verification

Let's substitute c = 10 into both cost equations to confirm our answer:

✓ Both gyms cost exactly $55 when you take 10 classes per month.

We can also check boundary cases to make sure our answer makes sense:

• 9 classes: FitLife = $52, StrongBody = $53 (StrongBody still more expensive)
• 11 classes: FitLife = $58, StrongBody = $57 (FitLife now more expensive)

Common Pitfalls

The Pattern Behind This

This is a break-even analysis problem. The general structure is:

This formula only works when the variable rates are different — otherwise the plans never intersect (or are always equal).

How to Spot This Problem Type

Watch for these telltale signs:

• "Plan A charges X plus Y per..." — signals fixed cost + variable cost structure
• "When will the costs be the same?" — asking for intersection point
• "Which is cheaper?" followed by specific quantities — often leads to break-even analysis
• Two different pricing schemes with different fixed/variable cost splits

You'll see this pattern in phone plans, gym memberships, subscription services, shipping options, and rental agreements.

Real Applications

• Business pricing decisions: Companies analyze break-even points to set competitive pricing structures and understand when customers will switch between plans.
• Personal finance: Comparing cell phone plans, streaming services, or transportation options (taxi vs. rideshare vs. car ownership) often involves this exact calculation.
• Project management: Choosing between contractors with different fee structures (one charges higher upfront but lower hourly, another has lower setup but higher hourly rates).