When Is the Second Phone Plan Cheaper?

What This Problem Teaches

• Setting up inequalities to compare linear cost functions
• Understanding the relationship between fixed costs and variable rates
• Interpreting break-even analysis in real-world contexts
• Translating "preferable" into mathematical language (less than)
• Working with decimal coefficients and mixed units (dollars and cents)

Solution: Method 1 — The Inequality Approach

The key insight here is that "preferable" means cheaper, so we need to find when Plan 2 costs less than Plan 1.

Step 1 — Define the variable and write cost expressions

Let m = number of minutes used per month

Notice I converted everything to dollars to avoid mixing units. Plan 1 has no fixed fee, just 25¢ per minute. Plan 2 has a $0.95 monthly fee plus 12¢ per minute.

Step 2 — Set up the inequality for when Plan 2 is preferable

Plan 2 is preferable when its cost is less than Plan 1's cost:

Step 3 — Solve the inequality

Subtract 0.12m from both sides:

Step 4 — Isolate the variable

Divide both sides by 0.13:

Step 5 — Interpret the result

Since we can't use a fraction of a minute in this context, Plan 2 becomes preferable when you use more than 7.31 minutes, or practically speaking, 8 or more minutes per month.

Solution: Method 2 — Finding the Break-Even Point First

Sometimes it's clearer to find where the plans cost exactly the same, then determine which side of that point favors Plan 2.

Step 1 — Set the costs equal

Find where both plans cost the same amount:

Step 2 — Solve for the break-even point

Subtract 0.12m from both sides:

Step 3 — Test which plan is cheaper above the break-even point

Let's test with 10 minutes (above 7.31):

Step 4 — State the conclusion

Since Plan 2 costs less at 10 minutes, Plan 2 is preferable for any usage above 7.31 minutes per month.

Verification

Let's verify by testing values just below and above our break-even point of 7.31 minutes.

At exactly 7.31 minutes:

Just below break-even (7 minutes):

Plan 1 is cheaper, as expected.

Above break-even (8 minutes):

Plan 2 is cheaper, confirming our answer. ✓

Watch Out For These

Does This Seem Reasonable?

Let's think about this intuitively. Plan 2 has a higher upfront cost ($0.95) but saves 13 cents per minute compared to Plan 1. So it makes sense that you'd need some minimum usage to overcome that upfront investment.

Also notice how low the break-even point is—just over 7 minutes! This suggests Plan 2 is designed to attract customers, since most people use far more than 7 minutes per month.

The Pattern Behind This

This is a classic break-even analysis between a fixed-cost plan and a variable-cost plan. The general structure is:

In our case: Usage > $0.95 ÷ ($0.25 - $0.12) = $0.95 ÷ $0.13 = 7.31 minutes

This pattern appears everywhere: gym memberships (monthly fee + per-class vs. per-class only), subscription services, bulk purchasing, and utility rate structures. The plan with higher fixed costs but lower variable rates always becomes preferable at some usage level.

Real Applications

Subscription services: Netflix vs. renting individual movies, Spotify vs. buying songs, or cloud storage plans all follow this same mathematical structure.

Business cost analysis: Companies regularly face "lease vs. buy" decisions, whether for equipment, software, or facilities. The break-even calculation determines the optimal choice.

Utility rate structures: Many electric companies offer plans with high fixed fees but lower per-kWh rates. High-usage customers benefit from these plans using exactly this calculation.

What If?