TACHS: Two pricing plans: set the costs equal
Write each plan as a fixed fee plus a rate, find where two plans cost the same, and decide which plan is cheaper or how much one costs more.
How do you find when two plans with different fees and hourly rates cost the same?
Short answer
Write each plan's cost as fee + rate × hours, set the two expressions equal, and solve. Check by putting your answer back into both plans.
Know first: Writing an expression with a variable, Solving a one-step equation
Step 1 of 6
A plan's cost is a fee plus a rate
A gym that charges $25 to join plus $4 per visit costs 25 + 4v for v visits. The fee is paid once. The rate is multiplied by the number of visits.
Two plans cost the same when their expressions are equal. So most of these questions become one equation. Gather the variable on one side and the numbers on the other.
| The question asks | Write |
|---|---|
| When do the plans cost the same? | Plan A = Plan B, then solve |
| How much less is one plan for 8 hours? | Find each cost at 8, then subtract |
| Which plan is cheaper at a given time? | Compare both costs at that time |
| What fee makes them equal at 5 hours? | Find Plan A at 5, then solve for the fee |
| When does Plan B cost $6 more? | Plan B = Plan A + 6 |
Every question starts with the same two expressions.
Step 2 of 6
Set the expressions equal and solve
Worked example: Two gyms
Gym A charges $25 to join plus $4 per visit. Gym B charges $10 to join plus $7 per visit. After how many visits do the two gyms cost the same?
- A3
- B5
- C15
- D45
- 1
Write each cost
Gym A: 25 + 4v. Gym B: 10 + 7v.
- 2
Set them equal
25 + 4v = 10 + 7v.
- 3
Collect the variable
Subtract 4v from both sides: 25 = 10 + 3v. Subtract 10: 15 = 3v.
- 4
Divide
v = 15 ÷ 3 = 5.
- 5
Check
Gym A: 25 + 4 × 5 = $45. Gym B: 10 + 7 × 5 = $45.
- 6
See why the other choices are there
3 is the difference in rates and 15 is the difference in fees. You need 15 ÷ 3. 45 is the cost at the break-even point, not the number of visits.
Answer
The gyms cost the same after 5 visits.
Wrong: Gym B costs $6 more than Gym A: 10 + 7v + 6 = 25 + 4v.
Right: Gym B costs $6 more than Gym A: 10 + 7v = 25 + 4v + 6.
To make two costs equal, add the $6 to the cheaper one. Here that gives 3v = 21, so v = 7: Gym B costs $59 and Gym A costs $53.
Step 3 of 6
Before and after the break-even point
The plan with the larger rate grows faster. Before the break-even point, it is cheaper because of its smaller fee. After the break-even point, it costs more.
With the gyms: at 3 visits, Gym A costs $37 and Gym B costs $31, so B is cheaper. At 5 visits they tie at $45. At 10 visits, Gym A costs $65 and Gym B costs $80, so A is cheaper.
Step 4 of 6
Write both costs, then compare
Check yourself · Question 1
Kayak Plan A costs $14 plus $5 per hour. Plan B costs $2 plus $8 per hour. At how many hours do the two plans cost the same?
Answer: B
14 + 5h = 2 + 8h. Subtract 5h and 2 from both sides: 12 = 3h, so h = 4. Check: both cost $34.
Trap answer: 12
Stopping one step short
- Why it looks right
- 12 = 3h is the last line before the answer, and 12 is the biggest number on it.
- Why it's wrong
- 12 is how many dollars Plan A's fee is ahead. Plan B closes that gap at $3 an hour, so it takes 12 ÷ 3 hours.
- The right answer
- 4 hours.
Check yourself · Question 2
Plan C costs $10 plus $6 per hour. Plan D costs $4 per hour plus a fixed fee. For a 3-hour rental, the two plans cost the same. What is Plan D's fixed fee?
Answer: B
Plan C for 3 hours: 10 + 6 × 3 = $28. Plan D: fee + 4 × 3 = fee + 12. So fee + 12 = 28, and the fee is $16.
Check yourself · Question 3
Plan A costs $9 plus $4 per hour. Plan B has no fee and costs $7 per hour. For which rental time is Plan B cheaper?
Answer: A
At 2 hours: A costs 9 + 8 = $17 and B costs $14, so B is cheaper. The plans tie at 3 hours ($21 each), and after that B costs more.
Common mistake
"Plan B has no fee, so it is always the cheaper plan."
- Why it's tempting
- A smaller fee is the first thing you notice, and it does make Plan B cheaper for short rentals.
- Do this instead
- Plan B's higher rate catches up. Find the break-even point and compare on each side of it.
Step 5 of 6
What to remember
Remember
- Write each plan as fee + rate × hours.
- Set the expressions equal to find the break-even point; add any extra amount to the cheaper side.
- Check by putting the answer into both plans; the plan with the larger rate is cheaper only before the break-even point.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what equation did you write, and what number do you divide the difference in fees by?