Worksheet Grade 10 Math

Quillfeather RO Plant: Percent Recovery and Flow Conversions

A grade 10 worksheet in which students calculate reverse osmosis percent recovery, reject flow, and gallon-to-MGD conversions for a fictional groundwater plant.

Grade: Grade 10
Subject: Math
Material: Worksheet
Time: 35 min
Prep: No prep
Materials: Calculator

Standards

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: Students carry gallons per minute through a conversion to million gallons per day in item 3, choosing and interpreting units consistently across every step of the worksheet.

All materials for HSN-Q.A.1 →

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: Students set up and solve a one-variable equation in item 5 to find the permeate flow rate that would raise the plant's recovery to a stated target percentage.

All materials for HSA-CED.A.1 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-16.

Objectives

Students will calculate percent recovery for a reverse osmosis system from feed and permeate flow rates.
Students will convert flow rates between gallons per minute and million gallons per day.
Students will solve a one-variable equation to find a flow rate that meets a target percent recovery.

Vocabulary

Reverse osmosis: Reverse osmosis cleans water by pushing it through a membrane with holes so tiny that salt and most other dissolved things cannot fit through. It can even turn seawater into fresh water.

Also: permeate, reject stream, percent recovery, feed flow

Teacher overview

Quillfeather Regional Water Authority (a fictional utility used for this worksheet) treats brackish groundwater with a reverse osmosis system before sending it into the distribution system. Every RO system splits its incoming feed water into two streams: clean permeate that becomes drinking water, and a concentrated reject stream that carries away the salts the membrane filters out. This worksheet uses one plant's feed and permeate readings to practice percent recovery, unit conversion, and solving for an unknown flow rate.

Quillfeather RO Plant: Percent Recovery and Flow Conversions — Grade 10 Math

Name: ____________________________ Date: ______________

Student Worksheet

Quillfeather's RO train (a fictional treatment train used for this worksheet) currently takes in a feed flow of 900 gallons per minute (gpm) and produces a permeate flow of 630 gpm. Use these numbers, along with the conversion that 1 million gallons per day (MGD) equals 694 gpm, to answer items 1 through 10.

Calculate the system's percent recovery, defined as permeate flow divided by feed flow, times 100.
Calculate the reject (concentrate) flow rate in gpm.
Convert the feed flow (900 gpm) to million gallons per day (MGD). Round to the nearest hundredth.
Convert the permeate flow (630 gpm) to gallons per day.
Plant engineers want to raise recovery to 75 percent while keeping the feed flow at 900 gpm. Write and solve a one-variable equation to find the permeate flow rate this would require.
Using your answer to item 5, calculate the new reject flow rate at 75 percent recovery. How many fewer gallons per minute does the plant send to reject compared to its original 70 percent recovery?
In one or two sentences, explain why RO recovery can never reach 100 percent, referencing the reject stream in your answer.
The reject stream is trucked off-site for disposal at a hypothetical rate of $0.03 per gallon. At the original reject flow rate, running 20 hours per day, what is the daily disposal cost?
The plant expands and feed flow increases by 20 percent, but recovery stays at 70 percent. What is the new permeate flow rate in gpm?
Using your answers above, explain whether the reject flow rate increases or decreases as percent recovery increases, assuming feed flow stays constant.

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: 70 percent.
630 ÷ 900 × 100 = 70%.
Answer: 270 gpm.
900 gpm − 630 gpm = 270 gpm.
Answer: 1.30 MGD.
900 gpm ÷ 694 gpm/MGD = 1.2968, which rounds to 1.30 MGD.
Answer: 907,200 gal.
630 gpm × 1,440 minutes/day = 907,200 gallons per day.
Answer: 675 gpm.
Let p = the new permeate flow. 0.75 = p ÷ 900, so p = 0.75 × 900 = 675 gpm.
Answer: 45 gpm.
New reject = 900 − 675 = 225 gpm. Original reject (item 2) was 270 gpm, so the plant sends 270 − 225 = 45 fewer gpm to reject.
Answer: The salts the membrane holds back must leave the system in some of the water.
Some water must always carry the rejected salts away as concentrate, so a fraction of the feed flow is permanently unavailable as permeate, which is why recovery stays below 100 percent.
Answer: $9,720.
Reject flow of 270 gpm × 60 min/hr × 20 hr/day = 324,000 gallons per day. 324,000 × $0.03 = $9,720 per day.
Answer: 756 gpm.
New feed = 900 × 1.20 = 1,080 gpm. New permeate = 1,080 × 0.70 = 756 gpm.
Answer: Reject flow decreases as recovery increases.
Comparing items 2 and 6 shows that raising recovery from 70% to 75% at the same feed flow lowered the reject flow from 270 gpm to 225 gpm, so a higher percent recovery leaves less water in the reject stream.

Differentiation

Support (below grade level)

Provide the percent recovery formula and the MGD conversion factor (694 gpm per MGD) as a printed reference card for this fictional scenario, and complete item 1 together as a class before students work items 2 through 10 independently.

Extension (above grade level)

Ask students to find the feed flow rate the fictional Quillfeather plant would need, at 70 percent recovery, to produce a permeate flow of 900 gpm, and to explain what that result implies about scaling up the plant.

ELL supports

Provide a labeled diagram of an RO train showing the feed stream splitting into permeate and reject arrows, with the home-language terms for "feed," "permeate," and "reject" alongside the English vocabulary list.

Extensions

Have students use the plant's given flow numbers to show how the daily disposal cost in item 8 would change if the fictional utility negotiated a lower disposal rate, recomputing the cost at a rate of $0.02 per gallon instead of $0.03.