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Math problem set Grade 9 Math

Hartwell Water Rates: Funding a Pipe Replacement Plan

A grade 9 problem set where students model a fictional town's water bill, calculate a rate increase and affordability burden, and graph a pipe-replacement schedule.

Objectives

  1. Students will write and use a two-variable equation for a tiered water bill.
  2. Students will calculate a rate increase as a dollar amount and as a percent.
  3. Students will graph a linear pipe-replacement schedule and read values from it.

Vocabulary

Fixed charge
The part of a water bill that stays the same every month no matter how much water a family uses. It helps pay for the pipes, meters, and workers the utility needs even when no water is flowing.

Also: volume rate, affordability, rate increase

Teacher overview

Most water bills have two parts: a fixed monthly charge that every customer pays, plus a volume rate that depends on how much water the household uses. When a town needs money to replace old pipes, it usually raises the volume rate, which changes every bill. This set follows the fictional town of Hartwell as it plans a rate increase to fund a pipe-replacement project, then checks whether the new bill crosses a common affordability threshold of 4.5 percent of household income.

A common misconception is to figure a percent increase by dividing the size of the increase by the new bill. The percent increase is always the increase divided by the original bill, so students should keep the old amount as the base.

Hartwell Water Rates: Funding a Pipe Replacement Plan — Grade 9 Math
Name: ____________________________ Date: ______________

Problems

The town of Hartwell (a fictional town) charges an $18 fixed monthly charge plus a volume rate of $4.20 per hundred cubic feet (CCF) of water used. A typical household uses 7 CCF per month. Use these numbers throughout the set.

  1. Write an equation for the monthly bill B, in dollars, for a household that uses c hundred cubic feet, using the $18 fixed charge and the $4.20 volume rate.
  2. Use your equation to find the monthly bill for a household that uses 7 CCF.
  3. Hartwell proposes raising the volume rate from $4.20 to $5.10 per CCF, keeping the $18 fixed charge. Find the new bill for the same 7 CCF household.
  4. How much more per month is the new bill (item 3) than the old bill (item 2)?
  5. What percent increase is that, compared with the old bill from item 2? Round to the nearest tenth of a percent.
  6. Hartwell says a household earning $2,400 per month faces a high burden if its combined water and sewer bill tops 4.5 percent of monthly income. Find 4.5 percent of $2,400.
  7. This household's sewer bill is $60. Add it to the new water bill from item 3, then state whether the combined bill is above or below the threshold from item 6, and by how much.
  8. The replacement project will renew 12 miles of pipe over 4 years at a steady pace of 3 miles per year. Write an equation for the miles remaining M after t years, starting from 12 miles.
  9. Graph the equation from item 8 for t = 0 to t = 4 on labeled, scaled axes, and state how many miles remain after 3 years.
  10. Hartwell could keep rates flat and delay the pipe project. Give one reason that delaying could cost more later, using the idea that old pipes break more often as they age.

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: B = 18 + 4.20c. The $18 fixed charge is added to $4.20 times the number of hundred-cubic-foot units c.
  2. Answer: $47.40. B = 18 + 4.20 × 7 = 18 + 29.40 = 47.40 dollars.
  3. Answer: $53.70. B = 18 + 5.10 × 7 = 18 + 35.70 = 53.70 dollars.
  4. Answer: $6.30. 53.70 − 47.40 = 6.30 dollars more per month.
  5. Answer: 13.3 percent. 6.30 ÷ 47.40 = 0.1329, which rounds to 13.3 percent, using the old bill as the base.
  6. Answer: $108. 0.045 × 2,400 = 108 dollars.
  7. Answer: Above the threshold, by $5.70. 60 + 53.70 = 113.70 dollars, and 113.70 − 108 = 5.70 dollars above the 108 threshold.
  8. Answer: M = 12 − 3t. Starting from 12 miles, the town renews 3 miles each year, so 3t miles are gone after t years.
  9. Answer: 3 miles remain after 3 years. The line falls from (0, 12) to (4, 0); at t = 3, M = 12 − 3 × 3 = 3 miles.
  10. Answer: Accept any reasoned response noting that older pipes break more often, and emergency repairs, water loss, and damage from a break can cost more than replacing the pipe on a planned schedule.

Differentiation

Support

Provide the bill equation from item 1 already written so students can focus on substitution, and give a partly labeled coordinate grid for item 9 with the axes and scale marked.

Extension

Have students find the volume rate Hartwell would need to charge so that the 7 CCF household's combined water-and-sewer bill lands exactly at the affordability threshold found in item 6, working backward from the bill equation.

ELL supports

Pre-teach "fixed charge," "volume rate," and "affordability" with a simple labeled sample bill showing the two parts added together. Provide a sentence frame for item 10: "Waiting could cost more because old pipes ___."