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Math problem set Grade 11 Engineering & Design

Calderwick Water Rates: A Two-Part Bill and Rate-Increase Problem Set

A grade 11 engineering problem set on a utility's two-part water rate: writing bill equations, comparing a rate increase, and testing an affordability threshold.

Objectives

  1. Students will write and interpret linear bill equations for a two-part water rate.
  2. Students will calculate a rate increase and test a household bill against an affordability threshold.

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.
Volumetric rate
The price a customer pays for each unit of water actually used. Using more water makes this part of the bill bigger, so it rewards saving water.

Also: affordability threshold, slope

Teacher overview

The fictional town of Calderwick charges a two-part water rate, a flat monthly charge plus a price for each 1,000 gallons used, and it has proposed raising both parts to help pay for replacing aging water mains. This set has students model each rate as a linear equation, find the increase a typical household would face, and check that household bill against the affordability threshold utilities watch. A common misconception in this math is reporting the percent increase as the difference between the two per-gallon prices rather than dividing the dollar increase by the original bill; the percent increase always compares the change to the starting amount.

Calderwick Water Rates: A Two-Part Bill and Rate-Increase Problem Set — Grade 11 Engineering & Design
Name: ____________________________ Date: ______________

Problems

Use these values for the fictional Calderwick water utility. Let g stand for the number of 1,000-gallon units a household uses in a month.

  • Current rate: a fixed charge of $18.00 per month plus $4.50 per 1,000 gallons.
  • Proposed rate: a fixed charge of $22.00 per month plus $5.25 per 1,000 gallons.
  • A typical household uses 6,000 gallons per month, so g = 6.
  • The affordability threshold treats combined water and sewer costs above 4.5 percent of household income as a high burden.
  1. Write an equation for the monthly bill B, in dollars, under the current rate, in terms of g.
  2. Find the current monthly bill for a household that uses 6,000 gallons, so g = 6.
  3. Write the equation for the proposed rate, then find the same household's proposed monthly bill.
  4. Find the dollar increase in that household's monthly bill, then find what percent increase that is over the current bill. Round the percent to the nearest whole number.
  5. Over a full year of 12 months, how much more will the household pay under the proposed rate?
  6. In each bill equation, identify the slope and state what it represents in this situation.
  7. Under the proposed rate, one household's combined water and sewer bill for the year is $1,620, and the household earns $30,000 a year. What percent of income goes to water and sewer, and does it cross the 4.5 percent high-burden threshold?
  8. The increase helps fund replacing about 1 mile of old water main at a cost of $1,200,000. If the increase raises an extra $102 per year from each of 5,000 households, how many years of the increase are needed to fund the replacement? Round to the nearest tenth.

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 + 4.50g dollars. The fixed charge of $18.00 is the constant, and $4.50 is added for each 1,000-gallon unit g.
  2. Answer: $45.00. Substituting g = 6 gives B = 18 + 4.50 times 6, which is 18 + 27, equal to $45.00.
  3. Answer: $53.50. The proposed equation is B = 22 + 5.25g; with g = 6, B = 22 + 5.25 times 6, which is 22 + 31.50, equal to $53.50.
  4. Answer: 19 percent. The dollar increase is $53.50 minus $45.00, which is $8.50; dividing 8.50 by 45.00 gives 0.189, or about 19 percent.
  5. Answer: $102.00. The monthly increase of $8.50 over 12 months is 8.50 times 12, which equals $102.00.
  6. Answer: $4.50 and $5.25 per 1,000 gallons. The slope of the current equation is 4.50 and the slope of the proposed equation is 5.25; each slope is the added cost for every extra 1,000 gallons used.
  7. Answer: 5.4 percent, yes. Dividing $1,620 by $30,000 gives 0.054, or 5.4 percent, which is above the 4.5 percent threshold, so the bill crosses into high burden.
  8. Answer: 2.4 years. The extra revenue is $102 times 5,000 households, which is $510,000 per year; dividing $1,200,000 by $510,000 gives 2.35, or about 2.4 years.

Differentiation

Support

Provide the two bill equations already written for students so they can focus on substituting g = 6 and on the percent and affordability steps. Give a worked model of a percent increase, using a non-water example, right before problem 4.

Extension

Ask students to find the monthly water use, in gallons, at which the proposed bill would reach $60.00, by setting 22 + 5.25g equal to 60 and solving for g. Have them explain what that usage level says about a heavy-use household in the fictional town.

ELL supports

Pre-teach "fixed charge," "per unit," and "increase" with a simple two-column example bill. Provide a sentence frame for problem 6: "The slope is ___, which means each extra 1,000 gallons adds ___ dollars to the bill."