Worksheet Grade 11 Math

Rodmell's Digester Blend: Solving a Two-Stream Solids Balance

A grade 11 worksheet solving a system of two linear equations for the blend of thickened and primary sludge a digester needs, then checking solids in pounds per day.

Grade: Grade 11
Subject: Math
Material: Worksheet
Time: 45 min
Prep: No prep
Materials: Calculators

Standards

HSA-REI.C.6 Common Core Math

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

How this material aligns: Items 1 through 4 have students write and solve a system of two linear equations in two variables for the daily volume of each sludge stream.

All materials for HSA-REI.C.6 →

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: Items 5 through 9 carry percent solids, milligrams per liter, million gallons per day, and pounds per day through the loading formula with units labeled.

All materials for HSN-Q.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 2026-09-04.

Objectives

Students will write a system of two linear equations from a volume balance and a solids balance.
Students will solve that system and check the solution against a pounds-per-day calculation.
Students will decide whether a stated blend target is reachable from the available streams.

Vocabulary

Detention time: Detention time is how long water stays inside a tank or basin before flowing out. Water must stay long enough for each treatment step, like settling or disinfecting, to finish its job.

Also: percent solids, digester, thickened sludge, mass balance

Teacher overview

An anaerobic digester works best when its feed holds a steady share of solids, so operators blend a thick stream with a thin one to hit a target. That blending question is a system of two linear equations, one counting gallons and one counting the solids inside them. This worksheet uses the fictional Rodmell Water Reclamation Facility, which blends thickened waste activated sludge with primary sludge to feed its digester. The misconception it targets is that the blend percent is the average of the two stream percents, an answer that only holds when the two volumes happen to be equal, and the final items show a target that no blend of the two streams can reach at all.

Rodmell's Digester Blend: Solving a Two-Stream Solids Balance — Grade 11 Math

Name: ____________________________ Date: ______________

Student Worksheet

The fictional Rodmell Water Reclamation Facility blends two sludge streams for its digester.

Thickened waste activated sludge holds 5.0 percent solids. Call its daily volume x, in gallons per day.
Primary sludge holds 3.0 percent solids. Call its daily volume y, in gallons per day.
The digester takes 30,000 gallons per day of blended feed at 4.5 percent solids.

Write an equation stating that the two stream volumes add to the daily feed volume.
Write a second equation stating that the solids in the two streams add to the solids in the blend.
Solve the system for x, the gallons per day of thickened sludge.
Find y, the gallons per day of primary sludge.
Convert the 4.5 percent blend to milligrams per liter, using 1 percent equals 10,000 mg/L.
Convert 30,000 gallons per day to million gallons per day.
Find the pounds of dry solids fed to the digester each day. Use pounds per day equals concentration in mg/L, times flow in MGD, times 8.34.
Find the pounds of dry solids per day carried by the thickened stream alone.
Find the pounds of dry solids per day carried by the primary stream alone.
The digester holds 450,000 gallons. Find its detention time in days, using detention time equals volume divided by flow.
The thickener slips and delivers only 4.0 percent solids. The two volumes stay the same. Find the new blend percent, to two decimal places.
In one or two sentences, explain why the blend percent fell even though neither volume changed.
Rodmell wants the blend back at 4.5 percent while the thickener still delivers 4.0 percent. State whether any pair of volumes totaling 30,000 gallons per day can reach that target, and give a reason.

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: x + y = 30,000.
The two streams are the only sources of feed, so their daily volumes in gallons per day add to the digester's daily intake.
Answer: 0.05x + 0.03y = 0.045(30,000).
Each stream contributes its percent solids times its volume, and those contributions add to the solids in 30,000 gallons per day of 4.5 percent blend, which is 1,350 on the same scale.
Answer: 22,500 gal/day.
Substituting y = 30,000 - x gives 0.05x + 900 - 0.03x = 1,350, so 0.02x = 450 and x = 22,500 gallons per day.
Answer: 7,500 gal/day.
Subtracting gives y = 30,000 - 22,500 = 7,500 gallons per day.
Answer: 45,000 mg/L.
Multiplying gives 4.5 x 10,000 = 45,000 milligrams per liter.
Answer: 0.030 MGD.
Dividing gives 30,000 divided by 1,000,000 = 0.030 million gallons per day.
Answer: 11,259 lb/day.
Multiplying gives 45,000 x 0.030 x 8.34, which is 1,350 x 8.34 = 11,259 pounds per day.
Answer: 9,382.5 lb/day.
The stream holds 50,000 mg/L and flows at 0.0225 MGD, so 50,000 x 0.0225 x 8.34 = 1,125 x 8.34 = 9,382.5 pounds per day.
Answer: 1,876.5 lb/day.
The stream holds 30,000 mg/L and flows at 0.0075 MGD, so 30,000 x 0.0075 x 8.34 = 225 x 8.34 = 1,876.5 pounds per day. Adding items 8 and 9 returns the 11,259 pounds per day found in item 7.
Answer: 15 days.
Dividing gives 450,000 divided by 30,000 = 15 days.
Answer: 3.75%.
The solids now total 0.04(22,500) + 0.03(7,500) = 900 + 225 = 1,125, and 1,125 divided by 30,000 = 0.0375, or 3.75 percent.
Answer: The thicker stream lost solids while carrying the same volume.
The blend percent is a weighted average of the two stream percents, so lowering the larger stream's percent by one point pulls the weighted average down even though the weights are unchanged.
Answer: Not possible.
Setting 0.04x + 0.03(30,000 - x) = 1,350 gives 0.01x = 450 and x = 45,000 gallons per day, which is more than the whole 30,000 gallons per day of feed. A weighted average can never sit above the larger of the two values being averaged, and 4.5 percent is above both 4.0 and 3.0 percent.

Differentiation

Support

Give these students the two equations already written for the fictional Rodmell blend, so their work starts at the substitution step. Provide a filled-in units column beside items 5 through 9, and let them check item 7 against the sum of items 8 and 9 before going on.

Extension

Ask these students to write the fictional Rodmell blend percent as a function of x alone, with y replaced by 30,000 minus x, and to state the range of blend percents the two streams can produce. Then have them find the volume split that would hold the blend at 4.5 percent if the primary stream were thickened to 3.5 percent instead.

ELL supports

Pre-teach "thickened," "blend," and "percent solids" with two labeled jars, one crowded with particles and one nearly clear, poured into a third. Provide a two-column table headed "gallons" and "solids" to organize items 1 and 2, and sentence frames for item 12 such as "The blend percent fell because the ___ stream now holds ___ solids."

Extensions

Students can describe how an operator would confirm a percent solids reading in a laboratory and why a single bad reading would push the whole blend off target. A second task is to rewrite the fictional Rodmell system with three streams instead of two and explain why the volume and solids equations alone no longer determine a single answer.