Comparing Two Class-Pass Plans and a Misread Break-Even
A fitness studio sells two class-pass options. Plan A charges a $45 enrollment fee plus $14 per class, with no expiration. Plan B charges a $120 enrollment fee plus $8 per class. A new member expects to attend 9 classes per month and intends to stay for 5 months, for 45 classes total. She has paid nothing yet and wants a single recommendation. A friend tells her, “The break-even is 12 classes, so after your second month Plan B is cheaper forever.” Meanwhile, a promotional flyer graphs both plans with total cost on the vertical axis and number of classes on the horizontal axis; on the flyer the two lines appear to cross at exactly 10 classes. The member wants the total cost of each plan at 18 classes and at 45 classes, a check on the friend's break-even figure, and an explanation of what the flyer's crossing point would imply if it were accurate.
1. Solve for the break-even number of classes and evaluate the friend's claim that it is 12.
Read the worked answer
Set the total costs equal, with n = number of classes:
45 + 14n = 120 + 8n 14n − 8n = 120 − 45 6n = 75 n = 12.5
The break-even is 12.5 classes, not 12. Since classes are whole units, Plan A is cheaper through 12 classes and Plan B is cheaper from the 13th class onward. Check: at n = 12, A = 45 + 168 = $213 and B = 120 + 96 = $216, so B is still $3 more expensive. At n = 13, A = $227 and B = $224.
The friend likely rounded the crossing point down and then described it as the point where B “wins.” At the crossing itself the plans cost the same; the cheaper plan is determined by which side of 12.5 you are on.
2. Compute total cost under each plan for 18 classes and for 45 classes, and make the recommendation with justification.
Read the worked answer
At 18 classes:
- Plan A: 45 + 14(18) = 45 + 252 = $297
- Plan B: 120 + 8(18) = 120 + 144 = $264 Plan B saves $33.
At 45 classes:
- Plan A: 45 + 14(45) = 45 + 630 = $675
- Plan B: 120 + 8(45) = 120 + 360 = $480 Plan B saves $195.
Both attendance targets exceed 12.5, so Plan B is cheaper in both cases, and the advantage widens by $6 per additional class (the difference in per-class rates). The recommendation depends on the attendance assumption, not on the enrollment fees alone: if she dropped to 8 classes, A = $157 and B = $184, and Plan A would be the better buy. The honest answer is “Plan B at her stated attendance, with the caveat that the choice flips below 13 classes.”
3. The flyer shows the lines crossing at 10 classes. Assuming the printed per-class rates and fees are correct, what would a genuine crossing at 10 classes require, and what does the mismatch suggest?
Read the worked answer
At a crossing, the fee difference must equal the rate difference multiplied by the crossing quantity: 120 − 45 = $75 must equal 10 × (14 − 8) = $60. These are not equal, so a crossing at 10 is inconsistent with the stated prices.
Working backward, a true crossing at 10 classes with a $75 fee gap would require a per-class gap of $7.50 (for example, $14.50 versus $7.00, or $15 versus $7.50). Equivalently, with the stated $6 rate gap, the crossing must sit at 75 ÷ 6 = 12.5.
A crossing drawn at 10 therefore signals a plotting or labeling error on the flyer - perhaps the axes were scaled unevenly or one plan's enrollment fee was entered incorrectly. The algebra is the authority here; a graph read off a promotional image should be treated as a check, not as the source of the break-even value.