Math problem set Grade 12 Math

Trenwick Regional Water: Rate Design and Revenue Requirement

A grade 12 problem set on utility finance: revenue requirement, fixed and volumetric charges, a break-even usage, debt-service coverage, and an affordability check.

Grade: Grade 12
Subject: Math
Material: Math problem set
Time: 50 min
Prep: No prep
Materials: Calculator

Standards

HSA-CED.A.1 Common Core Math

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

How this material aligns: Problems 3, 8, and 9 write and solve one-variable equations for the volumetric rate and the break-even usage that make revenue meet the utility's requirement.

All materials for HSA-CED.A.1 →

HSN-Q.A.1 Common Core Math

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How this material aligns: Problems 1 through 5 carry accounts, dollars, months, and thousand-gallon units through a multi-step bill calculation, keeping the money and volume units consistent.

All materials for HSN-Q.A.1 →

Download Word file

This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will split a utility's revenue requirement into fixed and volumetric charges and solve for the volumetric rate.
Students will calculate a customer bill, a break-even usage, a debt-service coverage ratio, and an affordability percentage.

Vocabulary

Revenue requirement: The total amount of money a utility must collect from customers each year to run the water system safely. Rates are set so that all the bills together add up to this amount.

Also: volumetric charge, break-even, debt-service coverage, affordability

Teacher overview

A water utility must collect exactly enough revenue to cover its yearly costs, a target called the revenue requirement, and it usually splits that target between a fixed monthly charge and a volumetric charge per unit of water. This set follows the fictional Trenwick Regional Water Authority as it sets a rate, tests a customer bill, and checks its bond covenant. A common misconception to correct is thinking the volumetric rate alone must cover the whole revenue requirement; the fixed charges collect part of it first, and only the remaining amount is spread across the water actually sold.

Trenwick Regional Water: Rate Design and Revenue Requirement — Grade 12 Math

Name: ____________________________ Date: ______________

Problems

Trenwick Regional Water Authority is a fictional utility. Use these figures for every problem. Water is billed per 1,000 gallons.

Annual revenue requirement: $6,500,000
Accounts served: 8,000
Fixed charge: $18 per account per month
Total water sold per year: 900 million gallons (900,000 thousand-gallon units)

Calculate the total annual revenue Trenwick collects from the fixed monthly charge across all accounts.
Subtract the fixed-charge revenue from the revenue requirement to find how much revenue must come from the volumetric charge.
Divide the volumetric revenue needed by the 900,000 thousand-gallon units sold to find the volumetric rate per 1,000 gallons. Round to the nearest cent.
A household uses 6,000 gallons in a month. Using the fixed charge plus your volumetric rate from problem 3, calculate the household's monthly bill.
Calculate that household's annual bill by multiplying its monthly bill by 12.
The household earns $28,000 per year. What percent of its income is the annual water bill? Round to the nearest tenth, and compare it to the 4.5 percent high-burden threshold, which applies to combined water and sewer costs.
Trenwick's bond covenant requires net revenue to be at least 1.25 times annual debt service. If net revenue is $2,300,000 and annual debt service is $1,700,000, calculate the coverage ratio and state whether the covenant is met. Round to the nearest hundredth.
A customer weighs a flat alternative plan costing $60 per month. Set the Trenwick bill (the fixed charge plus your rate from problem 3) equal to $60 and solve for the monthly usage, in gallons, where the two plans cost the same. Round to the nearest 5 gallons.
Next year the revenue requirement rises 8 percent to $7,020,000, while accounts, the fixed charge, and water sold stay the same. Calculate the new volumetric rate per 1,000 gallons. Round to the nearest cent.
Explain one tradeoff of collecting a larger share of the revenue requirement through the fixed charge instead of the volumetric rate.

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

Answer Key

Answer: $1,728,000. Fixed revenue equals 8,000 accounts times $18 times 12 months: 8,000 times 18 is 144,000 per month; times 12 is $1,728,000 per year.
Answer: $4,772,000. Volumetric revenue needed equals $6,500,000 minus $1,728,000, which is $4,772,000.
Answer: $5.30 per 1,000 gallons. The rate equals $4,772,000 divided by 900,000 units, which is 5.302, or about $5.30 per 1,000 gallons.
Answer: $49.80 per month. The bill equals the $18 fixed charge plus 6 units times $5.30, which is $18 plus $31.80, or $49.80.
Answer: $597.60 per year. The annual bill equals $49.80 times 12, which is $597.60.
Answer: 2.1 percent. $597.60 divided by $28,000 is 0.0213, or about 2.1 percent, which is below the 4.5 percent high-burden threshold. Because the threshold covers water and sewer together, the household's sewer bill would need to be added before judging its full burden.
Answer: 1.35. The coverage ratio equals $2,300,000 divided by $1,700,000, which is 1.353, or about 1.35; since 1.35 is at least 1.25, the covenant is met.
Answer: 7,925 gallons. Setting 18 plus 5.30 times x equal to 60 gives 5.30x equals 42, so x equals about 7.9245 thousand gallons, or 7,924.5 gallons, which rounds to 7,925 gallons to the nearest 5 gallons.
Answer: $5.88 per 1,000 gallons. The new volumetric revenue needed is $7,020,000 minus $1,728,000, which is $5,292,000; divided by 900,000 units is 5.88, or $5.88 per 1,000 gallons.
Answer: A larger fixed charge steadies the utility's revenue but weakens conservation and raises the burden on low users. Fixed revenue does not fall when customers save water, so it is predictable; however, it also means light users pay nearly as much as heavy users, and it gives customers less reason to conserve.

Differentiation

Support

Give students problem 1 fully worked and a labeled bill template for problem 4 that separates the fixed line from the volumetric line. Provide the revenue-requirement figure already broken into its fixed and volumetric parts for students who need problems 2 and 3 scaffolded.

Extension

Ask students to redesign the fictional Trenwick rate so the fixed charge covers only 15 percent of the revenue requirement, find the new volumetric rate, and describe how the typical household's bill would change.

ELL supports

Pre-teach "revenue," "fixed charge," "volumetric," and "break-even" with a simple two-part bill diagram. Provide the sentence frame "The volumetric rate is ___ per 1,000 gallons because ___" for problem 3.

Extensions

Have students find their own community's water rate structure and identify its fixed and volumetric parts, then compare the fixed-charge share with the fictional Trenwick example and discuss what that share signals about the utility's priorities.