Break-Even: When Are Health Club Costs the Same?

What You Will Learn

• Setting up linear equations from real-world cost scenarios
• Solving systems of equations by substitution and elimination
• Interpreting break-even points in business contexts
• Understanding the relationship between fixed and variable costs
• Verifying solutions by checking both original equations

What This Looks Like

Let's visualize how the costs grow over time for each club:

Solution: Method 1 — The Equation Setup

The most natural approach is to write cost equations for each club and find where they're equal.

Step 1 — Define the variable

Let m = number of months of membership.

Step 2 — Write cost equations for each club

Each club has a one-time initial fee plus monthly fees that accumulate:

Step 3 — Set the costs equal

We want to find when the total costs are the same, so:

Step 4 — Solve for m

Subtract 10m from both sides:

Subtract 11 from both sides:

Divide by 7:

Step 5 — Find the total cost at break-even

Substitute m = 2 into either cost equation:

Solution: Method 2 — The Cost Difference Analysis

Instead of setting equations equal, we can think about how the cost difference changes over time.

Step 1 — Analyze the initial difference

At the start (month 0), Club B costs more:

Club B starts $14 ahead in cost.

Step 2 — Find how the gap closes each month

Each month, Club A charges $17 while Club B charges $10:

Club A gains $7 on Club B each month.

Step 3 — Calculate when the gap closes

Club A needs to make up the $14 initial disadvantage at $7 per month:

Step 4 — Find the total cost

After 2 months, both clubs will cost:

Verification

Let's check our answer by substituting m = 2 back into both original cost equations:

Perfect! Both clubs cost exactly $45 after 2 months.

We can also check our logic by testing nearby values:

• At 1 month: Club A costs $28, Club B costs $35 (Club A is cheaper)
• At 3 months: Club A costs $62, Club B costs $55 (Club B is cheaper)

The costs cross over exactly at 2 months, confirming our answer.

Watch Out For These

Does This Seem Reasonable?

Our answer says the clubs cost the same after 2 months. Let's see if this makes intuitive sense:

Initial advantage: Club A starts $14 cheaper ($11 vs $25 initial fee).

Monthly disadvantage: Club A costs $7 more per month ($17 vs $10 monthly fee).

Break-even timing: It should take about $14 ÷ $7 = 2 months for Club A's higher monthly fees to catch up to Club B's higher initial fee.

This matches our calculated answer perfectly. For short-term memberships (under 2 months), choose Club A. For long-term memberships (over 2 months), choose Club B.

The General Pattern

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

Where:

• a₁, a₂ are the initial costs
• b₁, b₂ are the rates per time period
• t is the break-even time

This formula only works when b₁ ≠ b₂ (different rates). If the monthly fees were the same, the cheaper initial option would always remain cheaper.

Real Applications

• Cell phone plans: Comparing unlimited vs pay-per-use plans with different activation fees and monthly rates
• Software subscriptions: Annual vs monthly billing with different upfront discounts
• Equipment leasing: Lease vs buy decisions where lease has lower upfront cost but higher monthly payments

What If?