FE exam calculator: 5 problems solved with the functions that save time

The functions that save the most time on the FE exam are a numeric equation solver, polynomial and linear-system solvers, one-variable statistics and a numeric integral. The TI-36X Pro and the Casio fx-115ES PLUS and fx-991EX all have them. Below are five FE-style problems solved with each function, with a check you can do by hand.

Last reviewed October 10, 2026 by Engineer Exam Lab. Exam facts link to NCEES. Problems are from our practice packs, written by a PhD environmental engineer, and every numeric answer was checked by a second, independent calculation.

What the calculator rules are

NCEES allows only certain models. The current list, the testing-room rules and how to choose a model are on our FE calculator policy guide. This page is about using the calculator: which built-in functions save minutes on real FE problem types.

Four functions worth learning before exam day

Function	Use it for	Where it is
Numeric equation solver	Any equation you can't rearrange: Manning depth, pipe friction, rate of return	TI-36X Pro: num-solv (2nd, sin key). Casio fx-115ES PLUS and fx-991EX: SOLVE (SHIFT, CALC)
Polynomial solver	Quadratics and cubics: alternate depths, roots	TI-36X Pro: poly-solv (2nd, cos key). Casio: equation mode
Linear system solver	2 or 3 unknowns: blending, truss joints, simultaneous mass balances	TI-36X Pro: sys-solv (2nd, tan key). Casio: equation mode
Statistics and numeric integral	Mean, standard deviation, regression; areas, volumes, flow rates	Both brands have a one-variable stats mode and a definite-integral function

Key locations are from the TI-36X Pro support pages and the Casio manuals. Menus differ between models, so check your own model's guidebook and practice on the exact calculator you will bring.

Five problems where the calculator does the heavy lifting

Work each one twice: once by hand or with the formula, once with the calculator function. If both agree, you know the function and your entry habits are right.

Problem 1 · Open-channel flow · numeric solver · Multiple choice

Water flows at 6.0 m³/s in a concrete rectangular channel 3.0 m wide, with Manning's n = 0.013 and a bed slope of 0.0010. The normal depth is most nearly:

A) 0.84 m

B) 0.89 m

C) 1.11 m

D) 1.80 m

Show the worked solution

Answer: C) 1.11 m

Manning (SI): Q = (1.0/n) A R^(2/3) S^(1/2), with A = 3.0y and R = 3.0y/(3.0 + 2y)
Depth appears inside A and R, so you can't isolate y. Enter 6.0 = (1/0.013)(3y)(3y/(3 + 2y))^(2/3)(0.0010)^(1/2) in the numeric solver and start from a guess of 1
Solver result: y = 1.11 m

Check: A = 3.0 × 1.11 = 3.33 m², V = 6.0/3.33 = 1.80 m/s, and putting y = 1.11 m back into Manning returns 6.0 m³/s.

Why the wrong choices are wrong:

A used the US-units constant 1.486 in an SI problem.
B used the wide-channel shortcut R = y, which a 3 m channel at 1 m depth is not.
D that is the velocity in m/s (Q/A), not the depth.

Handbook: in FE Reference Handbook 10.6, search for “Manning”.

Problem 2 · Open-channel flow · polynomial solver · Multiple choice

A rectangular channel carries 2.0 m³/s per meter of width with a specific energy of 1.5 m. The subcritical depth (the deeper of the two alternate depths) is most nearly:

A) 0.44 m

B) 0.74 m

C) 1.40 m

D) 1.50 m

Show the worked solution

Answer: C) 1.40 m

Specific energy: E = y + q²/(2gy²). Multiply through by y²: y³ − Ey² + q²/(2g) = 0
q²/(2g) = 2.0²/(2 × 9.81) = 0.2039 m³, so the cubic is y³ − 1.5y² + 0y + 0.2039 = 0 (enter the 0 for the y term)
Cubic solver roots: 1.395 m, 0.438 m and one negative root (not physical). The subcritical depth is the larger one, 1.40 m

Check: 1.395 + 2.0²/(2 × 9.81 × 1.395²) = 1.395 + 0.105 = 1.50 m. Critical depth 0.74 m sits between the two roots, as it must.

Why the wrong choices are wrong:

A is the supercritical alternate depth, the smaller positive root.
B is the critical depth (q²/g)^(1/3) = 0.74 m, where the two depths meet.
D ignores the velocity head and sets y = E.

Handbook: in FE Reference Handbook 10.6, search for “specific energy”.

Problem 3 · Water supply · linear system solver · Multiple choice

Three wells are blended to supply 10.0 ML/d. Nitrate is 2, 8 and 14 mg/L and hardness is 300, 120 and 200 mg/L as CaCO₃ in wells A, B and C. The blend must have 6.0 mg/L nitrate and 200 mg/L hardness. The flow from well A is most nearly:

A) 0.77 ML/d

B) 3.33 ML/d

C) 4.10 ML/d

D) 5.13 ML/d

Show the worked solution

Answer: C) 4.10 ML/d

Flow: A + B + C = 10.0
Nitrate mass: 2A + 8B + 14C = 6.0 × 10.0 = 60
Hardness mass: 300A + 120B + 200C = 200 × 10.0 = 2,000
Enter the 3×3 system row by row in the same variable order: A = 4.10, B = 5.13, C = 0.77 ML/d

Check: 4.10 + 5.13 + 0.77 = 10.0; nitrate (8.2 + 41.0 + 10.8)/10 = 6.0 mg/L; hardness (1,231 + 615 + 154)/10 = 200 mg/L.

Why the wrong choices are wrong:

A is the flow from well C.
B splits the flow equally, which ignores both quality limits.
D is the flow from well B.

Handbook: in FE Reference Handbook 10.6, search for “mass balance”.

Problem 4 · Statistics · one-variable stats mode · Multiple choice

Six concrete cylinders from one pour break at 4,150, 3,980, 4,320, 4,060, 4,210 and 3,890 psi. The sample standard deviation is most nearly:

A) 143 psi

B) 157 psi

C) 4,100 psi

D) 24,600 psi

Show the worked solution

Answer: B) 157 psi

Enter the six values in the stats list and run one-variable statistics
Mean = 24,610/6 = 4,101.7 psi
Sample standard deviation s = √[Σ(x − mean)²/(n − 1)] = √(123,083/5) = 157 psi. On the calculator that is Sx (TI) or sx (Casio)

Check: 157² = 24,600, and dividing by 6 instead of 5 gives 143, matching choice A, so the two calculator outputs are easy to confuse.

Why the wrong choices are wrong:

A is the population value σx, dividing by n = 6 instead of n − 1 = 5.
C is the mean.
D is the sample variance s², before the square root.

Handbook: in FE Reference Handbook 10.6, search for “standard deviation”.

Problem 5 · Fluid mechanics · numeric integral · Multiple choice

Laminar flow in a pipe of radius 0.05 m has the velocity profile u(r) = 0.40[1 − (r/0.05)²] m/s. The flow rate, found by integrating u(r) over the cross section, is most nearly:

A) 1.57 L/s

B) 2.09 L/s

C) 3.14 L/s

D) 6.28 L/s

Show the worked solution

Answer: A) 1.57 L/s

Q = ∫ u dA, with dA = 2πr dr for a ring of radius r: Q = ∫ from 0 to 0.05 of 0.40[1 − (r/0.05)²] 2πr dr
Numeric integral on the calculator (use x for r): 0.001571 m³/s
Q = 0.001571 m³/s × 1,000 L/m³ = 1.57 L/s

Check: For laminar pipe flow the average velocity is half the maximum: 0.20 m/s × π(0.05)² = 0.001571 m³/s, the same as the integral.

Why the wrong choices are wrong:

B used an average velocity of 2/3 of the maximum (the parallel-plate result) instead of 1/2.
C multiplied the maximum velocity by the whole area.
D used the 0.10 m diameter as the radius.

Handbook: in FE Reference Handbook 10.6, search for “laminar”.

Habits that prevent calculator errors

Check the angle mode (DEG or RAD) at the start of every practice session and on exam day.
Give the solver a sensible starting guess. A numeric solver returns one root near the guess; a depth guess of 1 m beats 0.
Put the answer back in. Substituting the result into the original equation takes 15 seconds and catches entry errors.
Know which output you need. Sample (Sx) or population (σx) standard deviation, and which root of a cubic is physical.

Want 100 more like these, plus a timed 110-question mock? Each pack is $39, one time, with every solution tied to the handbook page.

FE Environmental pack · $39 FE Civil pack · $39

What this page doesn't cover

Which models are approved, and the testing-room rules, are on the calculator policy guide. Complex numbers, matrices, vectors, unit conversions and base-n functions also exist on these calculators but are not shown here.

Questions

What is the best calculator for the FE exam?

One from the NCEES approved list that you already know well. For speed, pick a model with a numeric equation solver, polynomial and system solvers, statistics and numeric integrals, such as the TI-36X Pro or the Casio fx-991EX or fx-115ES PLUS. Check the current list before you buy.

Is the TI-36X Pro allowed on the FE exam?

Texas Instruments models with TI-36X in the name are on the NCEES list for 2026. NCEES can change the list, so confirm it before your exam.

Is there a calculator on screen during the exam?

Yes. A TI-30XS calculator is available on screen, and you may also bring one approved physical calculator.

Should I rely on the solver for every problem?

No. Use it when an equation can't be rearranged easily. For simple formulas, direct entry is faster, and you should always put the answer back into the equation as a check.

Related guides

Engineer Exam Lab is not affiliated with or endorsed by NCEES. The problems here are original practice problems, not questions from a real exam.