Worksheet Grade 12 Math

Marchwood Softening Plant: Hardness, Alkalinity, and Lime Dose

A grade 12 operator-math worksheet on lime softening: hardness in grains and mg/L as CaCO3, removal efficiency, lime feed in pounds per day, and detention time.

Grade: Grade 12
Subject: Math
Material: Worksheet
Time: 40 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: Items 1, 4, and 10 carry units through the grains-per-gallon conversion and the pounds formula, choosing and interpreting units consistently across a multi-step softening calculation.

All materials for HSN-Q.A.1 →

HSN-Q.A.3 Common Core Math

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

How this material aligns: Item 3 asks students to report removal efficiency to a level of accuracy matched to the stated hardness readings rather than carrying false precision from the calculator.

All materials for HSN-Q.A.3 →

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

Objectives

Students will convert water hardness between grains per gallon and mg/L as calcium carbonate.
Students will calculate lime feed in pounds per day using the pounds formula and interpret carbonate versus noncarbonate hardness.

Vocabulary

Alkalinity: Alkalinity is water's ability to resist becoming acidic, acting like a buffer that keeps pH from swinging wildly when acids are added. Water with good alkalinity helps protect fish and other aquatic life from sudden pH changes.
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: hardness, as CaCO3, lime softening

Teacher overview

Lime softening removes hardness by raising pH so that calcium and magnesium precipitate out of the water. This worksheet uses the fictional Marchwood Softening Plant to practice the exact calculations a licensed operator runs on a softening shift: converting hardness units, sizing the lime feed with the pounds formula, and separating carbonate from noncarbonate hardness. A common misconception to correct is treating "mg/L as CaCO3" as a measure of actual calcium carbonate in the water; it is a common reference scale that lets operators add hardness, alkalinity, and lime doses on the same basis.

Marchwood Softening Plant: Hardness, Alkalinity, and Lime Dose — Grade 12 Math

Name: ____________________________ Date: ______________

Student Worksheet

Marchwood Softening Plant is a fictional plant that treats 3.2 MGD. Use these plant readings for every item, and use the pounds formula (pounds per day = concentration in mg/L times flow in MGD times 8.34) where a feed rate is asked for.

Raw (incoming) total hardness: 280 mg/L as CaCO3
Total alkalinity: 240 mg/L as CaCO3
Finished-water hardness goal: 85 mg/L as CaCO3
Solids-contact basin volume: 0.45 MG
Lime feed dose: 190 mg/L
Lab conversion: 1 grain per gallon equals 17.1 mg/L as CaCO3
Finished-water pH: 8.4

Convert the raw total hardness of 280 mg/L as CaCO3 to grains per gallon. Round to the nearest tenth.
How much hardness, in mg/L as CaCO3, must the plant remove to reach its finished-water goal?
Calculate the plant's hardness removal efficiency as a percent. Report it to the nearest tenth of a percent, which matches the precision of the hardness readings.
Using the pounds formula, calculate the lime feed in pounds per day at the 190 mg/L dose and the 3.2 MGD flow. Round to the nearest whole pound.
Lime costs $0.14 per pound at Marchwood. Using your answer from item 4, find the plant's daily lime cost. Round to the nearest dollar.
Carbonate hardness equals the smaller of the total hardness and the total alkalinity. How much of the raw hardness is noncarbonate hardness, in mg/L as CaCO3?
Calculate the detention time of the solids-contact basin in hours, using detention time equals volume divided by flow. Round to the nearest tenth of an hour.
Explain why the plant must add soda ash, not lime alone, to remove the noncarbonate hardness you found in item 6.
The finished-water pH is 8.4, and the secondary standard for pH is 6.5 to 8.5. Does the finished water meet the secondary pH standard?
Suppose demand rises and the plant treats 4.0 MGD while holding the same 190 mg/L lime dose. Calculate the new lime feed in pounds per day, and state whether it is higher or lower than your answer in item 4.

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: 16.4 grains per gallon. 280 mg/L divided by 17.1 mg/L per grain equals 16.37, which rounds to 16.4 gpg.
Answer: 195 mg/L as CaCO3. The plant removes 280 minus 85, which is 195 mg/L as CaCO3.
Answer: 69.6 percent. Removal efficiency equals (280 minus 85) divided by 280, times 100: 195 divided by 280 equals 0.6964, which is 69.6 percent.
Answer: 5,071 pounds per day. Pounds per day equals 190 times 3.2 times 8.34. First 190 times 3.2 equals 608; then 608 times 8.34 equals 5,070.72, which rounds to 5,071 pounds per day.
Answer: $710 per day. 5,070.72 pounds per day times $0.14 per pound equals $709.90, which rounds to about $710 per day.
Answer: 40 mg/L as CaCO3. Alkalinity (240) is smaller than total hardness (280), so carbonate hardness is 240 mg/L as CaCO3. Noncarbonate hardness is 280 minus 240, which is 40 mg/L as CaCO3.
Answer: 3.4 hours. Detention time equals 0.45 MG divided by 3.2 MGD, which is 0.1406 days; times 24 hours per day equals 3.375 hours, or about 3.4 hours.
Answer: Soda ash supplies the carbonate the water lacks. Noncarbonate hardness has no matching alkalinity, so adding lime alone raises pH without providing the carbonate ions needed to precipitate that calcium (lime turns noncarbonate magnesium into magnesium hydroxide, but it leaves the same amount of calcium noncarbonate hardness behind); soda ash adds those carbonate ions.
Answer: Yes. A pH of 8.4 falls inside the secondary range of 6.5 to 8.5, so the finished water meets the secondary pH standard.
Answer: 6,338 pounds per day. At 4.0 MGD the feed is 190 times 4.0 times 8.34, which equals 6,338.4, or about 6,338 pounds per day. It is higher than the 5,071 pounds per day in item 4 because the same dose treats more flow.

Differentiation

Support

Provide a labeled reference strip for the fictional plant with the two directions of the pounds formula and the grains conversion written as arrow rules. Work item 4 as a modeled example, then let students mirror the steps in item 10 with the flow changed.

Extension

Ask students to find the lime cost per million gallons treated at the original 3.2 MGD flow, then explain how that unit cost would change if the raw hardness dropped while the finished-water goal stayed the same.

ELL supports

Pre-teach "hardness," "alkalinity," "carbonate," and "dose" with a simple diagram of ions entering and precipitate settling out. Provide the sentence frames "The plant must remove ___ mg/L because ___" for item 2 and "Soda ash is needed because ___" for item 8.

Extensions

Invite a softening-plant or laboratory operator to describe how they log lime feed and hardness across a shift. Students can compare the fictional Marchwood readings in this worksheet with the numbers on a real plant's daily operating report, if a local utility will share a redacted one.