Worksheet Grade 6 Math
Quarry Ridge's Water Bill: Expressions and a Percent Rate Increase
A grade 6 worksheet where students write and evaluate an expression for a fictional town's two-part water bill, then compare percent increases in a rate change.
Objectives
- Students will write and evaluate an algebraic expression for a two-part water bill.
- Students will calculate percent increase for both a unit rate and a total cost.
- Students will complete a table to find total cost at several usage levels.
Vocabulary
Key words: expression, percent increase, unit rate, service fee
Teacher overview
The town of Quarry Ridge (a fictional town used for this worksheet) bills water customers a flat monthly service fee plus a per-unit charge for water used, where one unit equals 1,000 gallons. This worksheet has students write that bill as an algebraic expression, evaluate it for different amounts of water use, and compare two ways a rate increase can be described: as a percent change in the per-unit rate and as a percent change in a customer's total bill.
A common misconception here is assuming the two percent increases must be equal. Because the fixed service fee does not change, the total bill always rises by a smaller percent than the per-unit rate does, and the worksheet asks students to explain why directly.
Student Worksheet
Quarry Ridge (fictional) charges every home a $15 fee each month. It also charges $4.25 for each unit of water used. One unit equals 1,000 gallons. Next month the rate per unit will rise to $4.50. The $15 fee stays the same. Use these facts for all problems.
Write an expression for a home's total bill, C, at the current rate. Use u for the number of units used.
Use your expression from problem 1 to find the bill for a home that uses 8 units this month.
Use your expression from problem 1 to find the bill for a home that uses 12 units this month.
Write a new expression for a home's total bill, C, at next month's rate. Use u for units used.
Use your expression from problem 4 to find the bill for a home that uses 8 units next month.
Find the percent increase in the rate per unit, from $4.25 to $4.50. Round to the nearest tenth of a percent.
Look at your answers to problems 2 and 5. Find the percent increase in the total bill for a home that uses 8 units both months. Round to the nearest tenth of a percent.
Compare your answers to problems 6 and 7. The rate went up by the same dollar amount for every home. Why is the percent increase in the total bill smaller than the percent increase in the rate? Explain.
Complete the table below, showing the total bill at next month's rate for several amounts of water used.
Units used Total bill ($) 5 10 15 20 Challenge. The utility wants a home that uses 10 units to pay exactly 10 percent more. The $15 fee will not change. What rate per unit, to the nearest cent, would do this? Show the current bill for 10 units, the target bill after a 10 percent increase, and the rate that produces it.
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: C = 4.25u + 15.
The bill equals the per-unit rate (4.25) times the number of units (u), plus the fixed service fee (15).Answer: $49.
C = 4.25(8) + 15 = 34 + 15 = 49.Answer: $66.
C = 4.25(12) + 15 = 51 + 15 = 66.Answer: C = 4.50u + 15.
Same structure as problem 1, using next month's per-unit rate of 4.50.Answer: $51.
C = 4.50(8) + 15 = 36 + 15 = 51.Answer: 5.9 percent.
Increase = 4.50 − 4.25 = 0.25. Percent increase = 0.25 ÷ 4.25 = 0.0588..., which rounds to 5.9 percent.Answer: 4.1 percent.
Increase = 51 − 49 = 2. Percent increase = 2 ÷ 49 = 0.0408..., which rounds to 4.1 percent.Answer: The $15 service fee stays fixed, so it dilutes the effect of the rate increase.
The rate increase applies only to the per-unit part of the bill, which goes from 4.25u to 4.50u. Because the $15 fee does not change, it makes up a smaller share of the new, larger total, which pulls the overall percent increase in the total bill below the percent increase in the per-unit rate alone.Answer: $37.50, $60.00, $82.50, $105.00 for 5, 10, 15, and 20 units.
5 units: 4.50(5) + 15 = 22.50 + 15 = 37.50. 10 units: 4.50(10) + 15 = 45 + 15 = 60.00. 15 units: 4.50(15) + 15 = 67.50 + 15 = 82.50. 20 units: 4.50(20) + 15 = 90 + 15 = 105.00. This is not a ratio table of equivalent ratios: 10 units costs $60.00, not double the $37.50 for 5 units, because the $15 fee stays the same.Answer: $4.83 per unit.
Current total for 10 units: 4.25(10) + 15 = 42.50 + 15 = 57.50. A 10 percent increase brings the target total to 57.50 × 1.10 = 63.25. Subtracting the unchanged $15 fee leaves 63.25 − 15 = 48.25 for the variable portion. Dividing by 10 units gives a rate of 48.25 ÷ 10 = 4.825, which rounds to $4.83 per unit.
Differentiation
Support (below grade level)
Provide the expression C = 4.25u + 15 already written on the page, along with a labeled order-of-operations checklist ("multiply the rate by the units first, then add the fee"), so students can focus on evaluating rather than deriving the expression. Reduce the table in problem 9 to two rows (5 and 15 units) instead of four.
Extension (above grade level)
Ask students to write a single combined expression that gives the percent increase in the total bill directly in terms of u, the number of units used, and to test it against their answer to problem 7.
ELL supports
Pre-teach "service fee," "per-unit rate," and "percent increase" using a simple bill mock-up with each part labeled and, where possible, translated into students' home languages. Allow the explanation in problem 8 to be given orally or supported with an arrow diagram instead of a full written paragraph.