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Math problem set Grade 9 Math

Pinebluff Water Rates: Tiered Billing and Break-Even

A grade 9 problem set on the fictional Pinebluff utility, building a tiered water bill as a linear equation and solving for where two rate plans cost the same.

Objectives

  1. Students will compute a tiered water bill from a base charge and block rates.
  2. Students will write and use a linear equation for a household's monthly bill.
  3. Students will solve a system to find the break-even usage between two rate plans.

Vocabulary

Key words: tiered rate, base charge, break-even, system of equations

Teacher overview

The fictional Pinebluff utility bills each household a fixed monthly charge plus a price per thousand gallons that rises after a set amount. This set builds that bill into a linear equation, then compares it with a second rate plan to find the usage where the two cost the same. A common misconception is applying the higher tier price to every gallon; the higher price applies only to the gallons above the tier cutoff, and the base charge is added once no matter what.

Pinebluff Water Rates: Tiered Billing and Break-Even — Grade 9 Math
Name: ____________________________ Date: ______________

Problems

Pinebluff (a fictional utility) charges an $18 base charge each month. Water costs $4.00 per 1,000 gallons for the first 4,000 gallons, then $6.00 per 1,000 gallons for everything above 4,000. Let g stand for a household's use in thousands of gallons.

  1. The Rivera household uses 3,000 gallons. All of it is in the first tier. Find the total bill, including the base charge.
  2. The Okafor household uses 7,000 gallons. Find the total bill: base charge, the first 4,000 gallons at $4.00 per 1,000, and the rest at $6.00 per 1,000.
  3. Write an equation for the bill B, in dollars, for a household that uses g thousand gallons, where g is at least 4. Simplify it to the form B equals a number times g plus a number.
  4. Use your equation from problem 3 to find the bill for 10,000 gallons.
  5. A household's bill is $58. Using the equation from problem 3, find how many thousands of gallons it used.
  6. Next year Pinebluff raises the item-2 household's bill by 5 percent. Find the new bill for that household.
  7. Plan B is a different rate: a $30 base charge plus $3.00 per 1,000 gallons, so its bill is B equals 3g plus 30. Find Plan B's bill for 9,000 gallons.
  8. Using Plan A (B equals 6g plus 10) and Plan B (B equals 3g plus 30), solve the system to find the usage g where both plans cost the same.
  9. A household uses 4,000 gallons. Find each plan's bill and state which plan is cheaper, and by how much.
  10. Using your break-even answer, explain whether a low-use or a high-use household benefits more from Plan A.

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: $30. Volume charge = 3 times $4.00 = $12; plus the $18 base = $30.
  2. Answer: $52. First tier: 4 times $4.00 = $16. Above tier: (7 minus 4) times $6.00 = $18. Volume = $34; plus $18 base = $52.
  3. Answer: B equals 6g plus 10. Base 18, plus first tier 4 times 4 = 16, plus 6 times (g minus 4). That is 18 plus 16 plus 6g minus 24 = 6g plus 10, valid for g at least 4.
  4. Answer: $70. B = 6 times 10 plus 10 = 60 plus 10 = $70.
  5. Answer: 8 thousand gallons. 58 = 6g plus 10, so 6g = 48 and g = 8, which is 8,000 gallons.
  6. Answer: $54.60. The item-2 bill was $52; 52 times 1.05 = $54.60.
  7. Answer: $57. Plan B: 3 times 9 plus 30 = 27 plus 30 = $57.
  8. Answer: About 6.67 thousand gallons. Set 6g plus 10 = 3g plus 30, so 3g = 20 and g = 6.67, about 6,670 gallons. Each plan bills about $50 there.
  9. Answer: Plan A cheaper by $8. At g = 4, Plan A = 6 times 4 plus 10 = $34; Plan B = 3 times 4 plus 30 = $42; Plan A is $8 less.
  10. Answer: A low-use household benefits more from Plan A. Below the break-even near 6,670 gallons Plan A costs less, while above it Plan B costs less because Plan B's smaller per-gallon rate outweighs its bigger base charge.

Differentiation

Support

Give students a filled-in example bill that separates the base charge, the first tier, and the upper tier as three lines, using numbers not on this set. Provide the equation B equals 6g plus 10 already written for problems 4 and 5.

Extension

Ask students to design a third plan with a base charge and one flat per-thousand rate that costs the same as Plan A at exactly 5,000 gallons, then find where the new plan and Plan B break even.

ELL supports

Pre-teach "base charge," "tier," and "break-even" with a two-step bar picture: a fixed block for the base charge and a growing block for water used. Provide a sentence frame for problem 10: "Plan A is cheaper when a household uses ___ water."