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FE · 5% of the exam

Mathematics practice questions

10 free Mathematics questions for the FE Environmental Exam (FE) exam, each with the correct answer and an explanation for every option.

Start the questionsFree 10-question FE quiz

What this domain covers

The mathematics an environmental engineer uses every day: geometry, calculus, differential equations and numerical methods applied to practical problems.

Exam weight
5%
On a real exam
≈ 6 questions
In our bank
30 questions

Topics that come up most

  • Analytic geometry, areas and volumes
  • Derivatives and integrals in applied problems
  • First-order differential equations such as exponential decay
  • Numerical methods and roots of equations
  • Linear algebra and simultaneous equations

How to study it

Practise solving first-order decay and growth equations by hand. The same form comes back in chemistry, reactor and groundwater questions.

Sample Mathematics questions

Try each one before opening the answer. Original questions written to the official exam objectives.

  1. Question 1

    A flow of 1 million gallons per day (MGD) is about:

    • A1,000 m³/day
    • B3,785 m³/day
    • C264 m³/day
    • D378,500 m³/day
    Show answer and explanation

    Answer: B) 3,785 m³/day

    • AIncorrect. A gallon is 3.785 L.
    • BCorrect. 1 gallon = 3.785 L, so 10⁶ gal = 3,785 m³.
    • CIncorrect. That divides by 3.785 instead of multiplying.
    • DIncorrect. Recheck the conversion.

  2. Question 2

    A circular clarifier is 20 m in diameter. Its surface area is about:

    • A314 m²
    • B1,257 m²
    • C62.8 m²
    • D400 m²
    Show answer and explanation

    Answer: A) 314 m²

    • ACorrect. A = πD²/4 = π(400)/4.
    • BIncorrect. That uses D as the radius.
    • CIncorrect. That's the circumference.
    • DIncorrect. Include π/4.

  3. Question 3

    A rectangular tank is 10 m long, 5 m wide and 4 m deep. Its volume is:

    • A19 m³
    • B200 m³
    • C50 m³
    • D2,000 m³
    Show answer and explanation

    Answer: B) 200 m³

    • AIncorrect. Multiply, don't add.
    • BCorrect. 10 × 5 × 4.
    • CIncorrect. Include the depth.
    • DIncorrect. Recheck the multiplication.

  4. Question 4

    A grade of 1 vertical in 200 horizontal, expressed as a percent, is:

    • A200%
    • B2%
    • C0.5%
    • D5%
    Show answer and explanation

    Answer: C) 0.5%

    • AIncorrect. That inverts it.
    • BIncorrect. That's 1 in 50.
    • CCorrect. 1/200 × 100.
    • DIncorrect. That's 1 in 20.

  5. Question 5

    A trapezoidal channel has a bottom width of 2 m, a top width of 6 m at the water surface, and a flow depth of 1 m. What is its flow area?

    • A6 m²
    • B8 m²
    • C12 m²
    • D4 m²
    Show answer and explanation

    Answer: D) 4 m²

    The area of a trapezoid is the average of the two parallel widths times the depth: (2 + 6)/2 × 1 = 4 m².

    • AIncorrect. This multiplies the top width by the depth, which treats the section as a rectangle as wide as the top.
    • BIncorrect. This adds the two widths and multiplies by the depth without dividing by 2.
    • CIncorrect. This multiplies the two widths together, which is not the area formula for a trapezoid.
    • DCorrect. A = (bottom + top)/2 × depth = (2 + 6)/2 × 1 = 4 m².

  6. Question 6

    A concentration follows \(C = 100e^{-0.1t}\) mg/L, with t in days. At t = 10 days, C is about:

    • A36.8 mg/L
    • B90.5 mg/L
    • C0 mg/L
    • D13.5 mg/L
    Show answer and explanation

    Answer: A) 36.8 mg/L

    • ACorrect. \(100e^{-1} = 36.8\).
    • BIncorrect. That's at t = 1 day.
    • CIncorrect. Exponential decay never reaches zero.
    • DIncorrect. That's \(100e^{-2}\), at t = 20 days.

  7. Question 7

    A gas-law calculation needs the absolute temperature of water at 20 °C in SI units. The value to use is about:

    • A68 K
    • B293 K
    • C253 K
    • D528 K
    Show answer and explanation

    Answer: B) 293 K

    To convert Celsius to kelvins, add 273.15: 20 °C ≈ 293 K. Fahrenheit and Rankine values are common mix-ups.

    • AIncorrect. 68 is the Fahrenheit equivalent of 20 °C (68 °F), not a kelvin value.
    • BCorrect. T(K) = T(°C) + 273.15 = 20 + 273.15 ≈ 293 K.
    • CIncorrect. This subtracts 20 from 273 instead of adding the offset to the Celsius value.
    • DIncorrect. About 528 is the Rankine value (68 °F + 459.67 ≈ 528 °R). It is the U.S. Customary absolute scale, not kelvins.

  8. Question 8

    The concentration of a constituent in a stream rises from 40 mg/L to 50 mg/L. The percent increase is:

    • A10%
    • B20%
    • C25%
    • D125%
    Show answer and explanation

    Answer: C) 25%

    Percent change = (new − original)/original × 100 = (50 − 40)/40 × 100 = 25%.

    • AIncorrect. 10 is the absolute change in mg/L (50 − 40). It is not a percentage.
    • BIncorrect. This divides the change by the final value (10/50). Percent change is based on the original value.
    • CCorrect. Percent increase = (50 − 40)/40 × 100 = 25%.
    • DIncorrect. 125% is the ratio of the new value to the old one (50/40). It is not the increase.

  9. Question 9

    A treatment process achieves a 3-log removal of Giardia cysts. This equals a percent removal of:

    • A90%
    • B99%
    • C99.9%
    • D99.99%
    Show answer and explanation

    Answer: C) 99.9%

    • AIncorrect. That's 1-log.
    • BIncorrect. That's 2-log.
    • CCorrect. 3-log means only 10⁻³ of the cysts remain.
    • DIncorrect. That's 4-log.

  10. Question 10

    A circular pipe with an inside diameter of 2 m flows full. The cross-sectional flow area is about:

    • A6.28 m²
    • B12.6 m²
    • C3.14 m²
    • D4.00 m²
    Show answer and explanation

    Answer: C) 3.14 m²

    For a circular section, A = πD²/4 = πr². With D = 2 m, A = π ≈ 3.14 m².

    • AIncorrect. 6.28 is πD, the circumference of the pipe in meters. That is a length, not an area.
    • BIncorrect. This uses the 2-m diameter as the radius: π(2)² ≈ 12.6. The radius is 1 m.
    • CCorrect. A = πD²/4 = π(2)²/4 = π(1)² ≈ 3.14 m².
    • DIncorrect. 4.00 is D², the area of the square that encloses the circle. It leaves out the π/4 factor.

Mathematics: FAQs

How much of the FE exam is Mathematics?

Mathematics is 5% of the FE exam by the official exam guide's weighting, which on a 110-question form works out to roughly 6 questions. The real exam's exact split varies by form.

What does the Mathematics domain cover?

The mathematics an environmental engineer uses every day: geometry, calculus, differential equations and numerical methods applied to practical problems.

How should I study Mathematics for FE?

Practise solving first-order decay and growth equations by hand. The same form comes back in chemistry, reactor and groundwater questions.

Where can I practise more Mathematics questions?

This page has 10 free Mathematics questions with answers. The ExamWize FE question bank has 30 questions in this domain. Every practice exam result breaks your score down by domain, and Review Mode re-drills the questions you missed.

Other FE topics

Also: 15 free questions across all topics, the FE cheat sheet and the study guide.

Get Mathematics exam-ready

The full FE bank has 30 Mathematics questions. Timed practice exams match the real exam's length and domain weights, every result shows your score per domain, and Review Mode re-drills the questions you missed.

See the full FE practice examsTake the free diagnostic

Not affiliated with NCEES; questions are original, written from the official exam outline, and never copied from the real exam.