Comparing Two Cell Phone Plans: Linear Equation in 1 Variable

Question

Caroline is deciding between two cell phone plans.

• Plan A costs $45 per month plus $0.05 per text message.
• Plan B costs $30 per month plus $0.10 per text message.

For how many text messages will both plans cost the same?

1. 30
2. 75
3. 300
4. 750

This is a classic fixed cost vs. variable cost problem. However, the good news is that we only need one variable to solve it!

1. Step 1: Define the Variable:

   Let n be the number of text messages Caroline sends per month.

2. Step 2: Set Up the Cost Equations:
   • Plan A total cost = $45 + $0.05n
   • Plan B total cost = $30 + $0.10n
3. Step 3: Solve for n:

   Since we need to find when both plans cost the same, we set the equations equal to each other:

   45 + 0.05n = 30 + 0.10n

   Subtract 30 from both sides: 15 + 0.05n = 0.10

   Subtract 0.05n from both sides: 15 = 0.05n

   Divide by 0.05: n = 300

4. Verify the answer: When n = 300 messages, the total cost should be the same for Plan A and Plan B.

   When n = 300, Plan A total cost = 45 + 0.05(300) = 45 + 15 = 60

   When n = 300, Plan B total cost = 30 + 0.10(300) = 30 + 30 = 60

   When n = 300, Plan A total cost = Plan B total cost = $60 ✓

5. Select the correct option: The answer is C).

If Caroline sends 300 text messages per month, both plans will cost the same.