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Worksheet Grade 12 Math

Thistledown GAC Contactors: PFAS Removal and Reporting Precision

A grade 12 worksheet on a fictional town's carbon filters, computing PFOA and PFOS removal efficiency and reporting results at the precision the lab method allows.

Objectives

  1. Students will calculate PFOA and PFOS removal efficiency across a carbon contactor.
  2. Students will report each result at a level of accuracy consistent with the laboratory method's precision.
  3. Students will compare treated-water readings against the current PFAS maximum contaminant level.

Vocabulary

Granular activated carbon
Small black grains of specially treated carbon that grab contaminants as water flows through them. Utilities use big tanks of it to remove PFAS, bad tastes, and odors.
Maximum contaminant level (MCL)
A maximum contaminant level, or MCL, is the highest amount of a certain substance that federal rules allow in drinking water. Water systems test regularly to prove their water stays under each limit.
Parts per trillion
A unit for measuring extremely small amounts of a chemical in water. PFAS limits are set in parts per trillion because these chemicals matter even in tiny doses.

Also: removal efficiency, significant figures

Teacher overview

The fictional town of Thistledown treats its groundwater with granular activated carbon to pull out PFOA and PFOS, two of the PFAS compounds the federal rule now regulates at 4.0 parts per trillion each. This worksheet has students compute removal efficiency across the carbon contactors and then decide how many digits an honest result should carry. A common error at this grade is reporting a calculator's full display, such as 93.5714286 percent, when the underlying readings are only good to the nearest tenth of a part per trillion; the last several digits are noise the instrument cannot support.

Thistledown GAC Contactors: PFAS Removal and Reporting Precision — Grade 12 Math
Name: ____________________________ Date: ______________

Student Worksheet

Thistledown's laboratory is a fictional operation, and all readings below come from its monthly monitoring. The carbon plant measures PFOA and PFOS at the inlet (influent) and outlet (effluent). The laboratory method reports each reading to the nearest 0.1 ppt. The federal maximum contaminant level (MCL) is 4.0 ppt for PFOA and 4.0 ppt for PFOS. Federal compliance is judged on a running annual average of samples; this worksheet compares single readings with the MCL as a screening check. Use the removal-efficiency formula: removal (%) = (influent − effluent) ÷ influent × 100.

This month's readings:

Compound Influent (ppt) Effluent (ppt)
PFOA 42.0 2.7
PFOS 28.0 3.6

  1. Calculate the carbon plant's removal efficiency for PFOA. Report your answer to the nearest tenth of a percent.

  2. Calculate the removal efficiency for PFOS. Report your answer to the nearest tenth of a percent.

  3. Does the effluent PFOA reading meet the 4.0 ppt MCL? State the margin, in ppt, by which it clears or misses the limit.

  4. Does the effluent PFOS reading meet the 4.0 ppt MCL? State the margin, in ppt, and explain why the operator should watch this compound closely.

  5. A student's calculator shows the PFOA removal as 93.5714286 percent. Because the lab reports only to the nearest 0.1 ppt, how many digits should the reported removal actually carry, and what is the correctly reported value?

  6. The state also tracks the simple sum of the two treated readings. Add the effluent PFOA and PFOS values to get that combined number.

  7. The federal rule requires water systems to comply with the 4.0 ppt limits by April 2029. In May 2026, EPA proposed letting eligible systems apply for an exemption lasting until April 2031. How many additional years would that exemption give an eligible system?

  8. Suppose the influent PFOA rises to 50.0 ppt while the carbon keeps the same removal efficiency you found in item 1. Predict the new effluent PFOA reading to the nearest tenth, and state whether it still meets the MCL.

  9. Explain, in one or two sentences, why reporting the effluent PFOS as 3.600000 ppt would misstate what the laboratory actually knows.

  10. The lab's readings are trustworthy to within about 0.05 ppt. Given the 3.6 ppt effluent PFOS reading, state the range the true value could fall within, and say whether any value in that range would exceed the MCL.

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

Answer Key

  1. Answer: 93.6 percent.
    (42.0 − 2.7) ÷ 42.0 × 100 = 39.3 ÷ 42.0 × 100 = 93.5714... percent, which rounds to 93.6 percent. Full credit for 93.6; accept 93.57 if a student keeps one extra digit, but flag it against item 5's precision point.

  2. Answer: 87.1 percent.
    (28.0 − 3.6) ÷ 28.0 × 100 = 24.4 ÷ 28.0 × 100 = 87.1428... percent, which rounds to 87.1 percent.

  3. Answer: Yes, by 1.3 ppt.
    The effluent PFOA is 2.7 ppt, which is below the 4.0 ppt MCL. 4.0 − 2.7 = 1.3 ppt of margin.

  4. Answer: Yes, by 0.4 ppt.
    The effluent PFOS is 3.6 ppt, below the 4.0 ppt MCL, but only by 4.0 − 3.6 = 0.4 ppt. A small rise in influent or a slight loss of carbon capacity could push it over, so this compound has almost no cushion.

  5. Answer: 93.6 percent.
    The readings carry three significant figures at most (42.0 and 2.7 are limited by the 0.1 ppt resolution), so the removal should be reported as 93.6 percent, roughly three significant figures. The trailing digits 5714286 imply a precision the instrument cannot deliver and should be dropped.

  6. Answer: 6.3 ppt.
    2.7 ppt + 3.6 ppt = 6.3 ppt. This is a simple sum of the two treated readings, not a comparison against the 4.0 ppt single-compound limit.

  7. Answer: 2 years.
    April 2031 − April 2029 = 2 years. Students should note that this is a proposal, not a final rule, as of September 2026, so the timeline could still change. It would not extend the deadline automatically for every system, since a system would have to be eligible and apply, and it would not change the 4.0 ppt limits.

  8. Answer: 3.2 ppt.
    Using the removal efficiency from item 1, effluent = 50.0 × (1 − 0.935714) = 50.0 × 0.064286 = 3.214, which rounds to 3.2 ppt. Since 3.2 ppt is below 4.0 ppt, it still meets the MCL, but the margin has shrunk to 0.8 ppt.

  9. Answer: It claims false precision.
    A reading of 3.600000 ppt states the value is known to the nearest millionth of a part per trillion, when the lab resolves only to 0.1 ppt. The honest report is 3.6 ppt.

  10. Answer: 3.55; the range is 3.55 to 3.65 ppt, and no value in it exceeds the MCL.
    A reading of 3.6 ppt with ±0.05 ppt of uncertainty spans 3.55 ppt to 3.65 ppt. Every value in that range is below the 4.0 ppt MCL, so the uncertainty alone does not put the sample above the MCL, though the thin margin still warrants attention.

Differentiation

Support

Provide the removal-efficiency formula worked through once with a neutral example (influent 20.0, effluent 5.0, giving 75 percent) before students begin item 1. Give a printed reminder that the readings stop at the tenths place, so answers should as well.

Extension

Ask students to determine the highest influent PFOA the carbon plant could receive while still keeping effluent PFOA at or below the maximum contaminant level, assuming the removal efficiency from item 1 holds. Have them explain why that ceiling matters for a plant whose source water is slowly worsening.

ELL supports

Pre-teach "influent," "effluent," "removal efficiency," and "margin" with a simple labeled flow diagram showing water entering and leaving a carbon tank. Provide sentence frames for items 4 and 9, such as "The reading meets the limit because ___" and "Reporting more digits is wrong because ___."