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Math problem set Grade 8 Math

Denton Ridge Water Bills: Comparing Two Rate Plans with Linear Systems

A grade 8 math set where students write linear equations for two water-billing plans and solve the system to find the exact usage level where both plans cost the same.

Objectives

  1. Students will write linear equations that model two water-billing plans in terms of gallons used.
  2. Students will solve a system of two linear equations to find the usage level at which both plans cost the same.
  3. Students will interpret the solution to the system in terms of the real-world billing scenario.

Teacher overview

Water utilities sometimes let customers choose between different monthly billing plans for the same water: one plan with a flat fee plus a per-gallon charge, and another with no flat fee but a higher rate per gallon. Each plan can be written as a linear equation in gallons used, and the point where the two lines cross is the usage level at which both plans cost exactly the same, a direct application of solving a system of equations. A common misconception here is treating the flat monthly fee as though it were a rate rather than a fixed starting value, which produces an equation like y = 15x instead of y = 15 + 0.006x; watch for that error in problem 1, since it would make a plan's cost rise far too quickly as usage grows.

Denton Ridge Water Bills: Comparing Two Rate Plans with Linear Systems — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

Denton Ridge (a fictional town used for this problem set) is served by one water utility that offers residential customers a choice between two monthly billing plans. Let x stand for the number of gallons a household uses in a month, and let y stand for the total monthly bill in dollars.

  • Plan A (Base Plan): a flat charge of $15.00 per month, plus $0.006 for every gallon used.
  • Plan B (Pay-as-you-go Plan): no flat charge, just $0.009 for every gallon used.

  1. Write an equation for Plan A's total monthly cost, y, in terms of gallons used, x.

  2. Write an equation for Plan B's total monthly cost, y, in terms of gallons used, x.

  3. Complete the table below by calculating the cost of each plan at each usage level.

    Gallons used (x) Plan A cost Plan B cost
    1,000
    3,000
    5,000
    7,000

  4. Based on your completed table, at what usage level do the two plans cost the same, and what is that shared monthly cost?

  5. Solve the system of equations from problems 1 and 2 algebraically to confirm the breakeven usage level you found in problem 4. Show every step.

  6. For each plan's equation, name the rate of change (the per-gallon charge) and the initial value (the flat fee). Which plan has the higher rate of change, and which has the higher initial value?

  7. For usage levels below the breakeven point you found, which plan costs less? Use the two rates of change to explain why, even though that plan charges more per gallon than the other.

  8. The average American household uses roughly 9,000 gallons of water in a month (about 300 gallons per day, the EPA's national estimate, times a 30-day month). Which plan should a typical Denton Ridge household choose, and how much would that household save per month compared with the other plan?

  9. Challenge. Suppose the utility raises Plan A's flat fee to $24.00 per month to help cover new treatment costs, while keeping both per-gallon rates the same. Find the new breakeven usage level.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key

  1. Answer: y = 15 + 0.006x.
    The $15.00 flat charge is the fixed part of the bill regardless of usage, and $0.006 is charged for every one of the x gallons used.

  2. Answer: y = 0.009x.
    Plan B has no flat fee, so the entire bill is the per-gallon rate, 0.009, multiplied by gallons used, x.

  3. Answer: table values.
    At x = 1,000: Plan A = 15 + 0.006(1,000) = 15 + 6 = $21.00; Plan B = 0.009(1,000) = $9.00.
    At x = 3,000: Plan A = 15 + 0.006(3,000) = 15 + 18 = $33.00; Plan B = 0.009(3,000) = $27.00.
    At x = 5,000: Plan A = 15 + 0.006(5,000) = 15 + 30 = $45.00; Plan B = 0.009(5,000) = $45.00.
    At x = 7,000: Plan A = 15 + 0.006(7,000) = 15 + 42 = $57.00; Plan B = 0.009(7,000) = $63.00.

  4. Answer: 5,000 gallons, at a shared cost of $45.00.
    The table shows both plans reaching $45.00 when x = 5,000, the only row where the two costs match.

  5. Answer: 5,000 gallons (x = 5,000).
    Set the equations equal: 15 + 0.006x = 0.009x. Subtract 0.006x from both sides: 15 = 0.003x. Divide both sides by 0.003: x = 5,000. Substituting back, y = 0.009(5,000) = $45.00, which matches problem 4.

  6. Answer: Plan A's rate of change is 0.006 and its initial value is 15; Plan B's rate of change is 0.009 and its initial value is 0.
    Plan B has the higher rate of change (it charges more per gallon), while Plan A has the higher initial value (its flat fee).

  7. Answer: Plan B costs less below 5,000 gallons.
    Plan B has no flat fee to overcome, so at low usage its lack of a starting cost outweighs its higher per-gallon rate; only once enough gallons accumulate does Plan A's lower rate of change let it catch up and then cost less.

  8. Answer: $12.00 per month savings (Plan A is cheaper).
    At x = 9,000 gallons: Plan A = 15 + 0.006(9,000) = 15 + 54 = $69.00; Plan B = 0.009(9,000) = $81.00. Since 9,000 gallons is above the 5,000-gallon breakeven point, Plan A is cheaper, saving 81 − 69 = $12.00 per month.

  9. Answer: 8,000 gallons.
    Set the new equations equal: 24 + 0.006x = 0.009x. Subtract 0.006x: 24 = 0.003x. Divide by 0.003: x = 8,000. Checking: Plan A = 24 + 0.006(8,000) = 24 + 48 = $72.00, and Plan B = 0.009(8,000) = $72.00, confirming the new breakeven point.

Differentiation

Support (below grade level)

Give students the two equations already written (y = 15 + 0.006x and y = 0.009x) and start with problem 3's table before problems 1 and 2, so the linear pattern is visible before they are asked to write equations from scratch. Allow calculators for all multiplication.

Extension (above grade level)

Ask students to find what Plan B's per-gallon rate would need to become, keeping Plan A unchanged, for the breakeven point to fall at exactly 4,000 gallons instead of 5,000, and to show the algebra that supports their answer.

ELL supports

Provide a labeled diagram showing a water bill with the flat fee and the per-gallon charge marked as two separate parts before problem 1. Allow students to complete problem 7 by circling the cheaper cost in each row of the table from problem 3, rather than writing a full explanation, before attempting the written response.