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10 FE practice problems: rational method peak runoff, with solutions

By Dr. Mohamed Elansary, PhD Environmental EngineeringUpdated

The rational method estimates peak runoff from a small drainage area from three inputs: a runoff coefficient, a rainfall intensity, and the area. It appears in hydrology topics in the FE Civil and FE Environmental specifications. These ten fill-in problems include composite coefficients for mixed land use. Watch the units: the handbook states the U.S. form with area in acres and intensity in inches per hour.

How to use these problems

Give each problem an honest attempt before you open the solution: write the given values with units, find the equation in the FE Reference Handbook PDF, and commit to an answer. Then compare line by line. If you picked a wrong option, read the note for that option; each one names the mistake that produces it.

The problems

Problem 1 · FE, Rational method peak runoff

A 15.5 acre drainage area is made up of 8.0 acres of paved parking (C = 0.72) and 7.5 acres of wooded park land (C = 0.22). The design rainfall intensity for the time of concentration is 3.7 in./hr. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
27.4 cfs
Given
C = 0.72, 0.22, A = 8.0 acres, 7.5 acres, I = 3.7 in./hr, ΣA = 15.5 acres
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
Q = CIA, with the area-weighted C = Σ(C·A)/ΣA
Substitute
  1. Weighted C = Σ(C·A)/ΣA = [(0.72)(8.0) + (0.22)(7.5)]/15.5 = 7.410/15.5 = 0.4781
  2. Q = CIA = (0.4781)(3.7 in./hr)(15.5 acres) = 27.4 cfs
Result
27.4 cfs, 3 significant figures
Check
the weighted C (0.4781) lies between the smallest and largest C (0.22 and 0.72); with I in in./hr and A in acres, CIA is read directly in cfs.
Common wrong answers
  • 57.4 cfs: left out the runoff coefficient (C = 1)
  • 41.3 cfs: applied the largest C to the whole drainage area

Problem 2 · FE, Rational method peak runoff

A 60.0 acre watershed is mostly gravel surfaces, with a runoff coefficient of 0.49. The local intensity-duration relation is I = 80/(t + 12), with I in in./hr and t in minutes. The time of concentration is 13 min. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
94.1 cfs
Given
C = 0.49, A = 60.0 acres, a = 80, b = 12 min, tc = 13 min
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
I = a/(tc + b); Q = CIA
Substitute
  1. I = 80/(tc + 12) = 80/(13 + 12) = 3.200 in./hr (the storm lasting tc gives the peak)
  2. Q = CIA = (0.49)(3.200 in./hr)(60.0 acres) = 94.1 cfs
Result
94.1 cfs, 3 significant figures
Check
the design storm duration equals tc, the time for the whole area to contribute; CIA is read in cfs.
Common wrong answers
  • 181 cfs: left the constant b out of the intensity relation
  • 193 cfs: put the time of concentration in hours into a relation written for minutes

Problem 3 · FE, Rational method peak runoff

A 50.5 acre watershed is mostly lawns on sandy soil, with a runoff coefficient of 0.16. The local intensity-duration relation is I = 115/(t + 19), with I in in./hr and t in minutes. The time of concentration is 29 min. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
19.4 cfs
Given
C = 0.16, A = 50.5 acres, a = 115, b = 19 min, tc = 29 min
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
I = a/(tc + b); Q = CIA
Substitute
  1. I = 115/(tc + 19) = 115/(29 + 19) = 2.396 in./hr (the storm lasting tc gives the peak)
  2. Q = CIA = (0.16)(2.396 in./hr)(50.5 acres) = 19.4 cfs
Result
19.4 cfs, 3 significant figures
Check
the design storm duration equals tc, the time for the whole area to contribute; CIA is read in cfs.
Common wrong answers
  • 47.7 cfs: put the time of concentration in hours into a relation written for minutes
  • 121 cfs: left out the runoff coefficient (C = 1)
  • 32.0 cfs: left the constant b out of the intensity relation

Problem 4 · FE, Rational method peak runoff

A 28.5 acre watershed is mostly lawns on clay soil, with a runoff coefficient of 0.30. The local intensity-duration relation is I = 65/(t + 9), with I in in./hr and t in minutes. The time of concentration is 31 min. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
13.9 cfs
Given
C = 0.30, A = 28.5 acres, a = 65, b = 9 min, tc = 31 min
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
I = a/(tc + b); Q = CIA
Substitute
  1. I = 65/(tc + 9) = 65/(31 + 9) = 1.625 in./hr (the storm lasting tc gives the peak)
  2. Q = CIA = (0.30)(1.625 in./hr)(28.5 acres) = 13.9 cfs
Result
13.9 cfs, 3 significant figures
Check
the design storm duration equals tc, the time for the whole area to contribute; CIA is read in cfs.
Common wrong answers
  • 58.4 cfs: put the time of concentration in hours into a relation written for minutes
  • 17.9 cfs: left the constant b out of the intensity relation
  • 46.3 cfs: left out the runoff coefficient (C = 1)

Problem 5 · FE, Rational method peak runoff

A 41,000 ft² site is covered by single-family residential lots, with a runoff coefficient of 0.42. The design rainfall intensity is 5.5 in./hr. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
2.17 cfs
Given
C = 0.42, A = 41,000 ft², I = 5.5 in./hr, 43,560 ft²/acre (handbook)
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula; Units and Conversion Factors (acre), pages 3 and 296
Equation
Q = CIA, with A in acres
Substitute
  1. A = 41,000 ft²/(43,560 ft²/acre) = 0.9412 acres
  2. Q = CIA = (0.42)(5.5 in./hr)(0.9412 acres) = 2.17 cfs
Result
2.17 cfs, 3 significant figures
Check
with I in in./hr and A in acres, CIA is read directly in cfs (the handbook's form of the formula).
Common wrong answers
  • 0.395 cfs: multiplied C by A and left out the intensity
  • 5.18 cfs: left out the runoff coefficient (C = 1)
  • 23.4 cfs: divided by 4,047 (square meters per acre) instead of 43,560 ft² per acre

Problem 6 · FE, Rational method peak runoff

A 32.0 acre watershed is mostly single-family residential lots, with a runoff coefficient of 0.36. The local intensity-duration relation is I = 80/(t + 15), with I in in./hr and t in minutes. The time of concentration is 32 min. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
19.6 cfs
Given
C = 0.36, A = 32.0 acres, a = 80, b = 15 min, tc = 32 min
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
I = a/(tc + b); Q = CIA
Substitute
  1. I = 80/(tc + 15) = 80/(32 + 15) = 1.702 in./hr (the storm lasting tc gives the peak)
  2. Q = CIA = (0.36)(1.702 in./hr)(32.0 acres) = 19.6 cfs
Result
19.6 cfs, 3 significant figures
Check
the design storm duration equals tc, the time for the whole area to contribute; CIA is read in cfs.
Common wrong answers
  • 28.8 cfs: left the constant b out of the intensity relation
  • 59.3 cfs: put the time of concentration in hours into a relation written for minutes
  • 54.5 cfs: left out the runoff coefficient (C = 1)

Problem 7 · FE, Rational method peak runoff

A 62.5 acre drainage area is made up of 34.0 acres of lawns on clay soil (C = 0.20) and 28.5 acres of gravel surfaces (C = 0.42). The design rainfall intensity for the time of concentration is 3.0 in./hr. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
56.3 cfs
Given
C = 0.20, 0.42, A = 34.0 acres, 28.5 acres, I = 3.0 in./hr, ΣA = 62.5 acres
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
Q = CIA, with the area-weighted C = Σ(C·A)/ΣA
Substitute
  1. Weighted C = Σ(C·A)/ΣA = [(0.20)(34.0) + (0.42)(28.5)]/62.5 = 18.77/62.5 = 0.3003
  2. Q = CIA = (0.3003)(3.0 in./hr)(62.5 acres) = 56.3 cfs
Result
56.3 cfs, 3 significant figures
Check
the weighted C (0.3003) lies between the smallest and largest C (0.20 and 0.42); with I in in./hr and A in acres, CIA is read directly in cfs.
Common wrong answers
  • 58.1 cfs: averaged the C values without weighting them by area
  • 78.8 cfs: applied the largest C to the whole drainage area
  • 188 cfs: left out the runoff coefficient (C = 1)

Problem 8 · FE, Rational method peak runoff

A 55.0 acre watershed is mostly lawns on clay soil, with a runoff coefficient of 0.30. The local intensity-duration relation is I = 70/(t + 14), with I in in./hr and t in minutes. The time of concentration is 15 min. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
39.8 cfs
Given
C = 0.30, A = 55.0 acres, a = 70, b = 14 min, tc = 15 min
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
I = a/(tc + b); Q = CIA
Substitute
  1. I = 70/(tc + 14) = 70/(15 + 14) = 2.414 in./hr (the storm lasting tc gives the peak)
  2. Q = CIA = (0.30)(2.414 in./hr)(55.0 acres) = 39.8 cfs
Result
39.8 cfs, 3 significant figures
Check
the design storm duration equals tc, the time for the whole area to contribute; CIA is read in cfs.
Common wrong answers
  • 133 cfs: left out the runoff coefficient (C = 1)
  • 77.0 cfs: left the constant b out of the intensity relation
  • 81.1 cfs: put the time of concentration in hours into a relation written for minutes

Problem 9 · FE, Rational method peak runoff

A 316,000 ft² site is covered by wooded park land, with a runoff coefficient of 0.18. The design rainfall intensity is 2.8 in./hr. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
3.66 cfs
Given
C = 0.18, A = 316,000 ft², I = 2.8 in./hr, 43,560 ft²/acre (handbook)
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula; Units and Conversion Factors (acre), pages 3 and 296
Equation
Q = CIA, with A in acres
Substitute
  1. A = 316,000 ft²/(43,560 ft²/acre) = 7.254 acres
  2. Q = CIA = (0.18)(2.8 in./hr)(7.254 acres) = 3.66 cfs
Result
3.66 cfs, 3 significant figures
Check
with I in in./hr and A in acres, CIA is read directly in cfs (the handbook's form of the formula).
Common wrong answers
  • 20.3 cfs: left out the runoff coefficient (C = 1)
  • 39.4 cfs: divided by 4,047 (square meters per acre) instead of 43,560 ft² per acre
  • 1.31 cfs: multiplied C by A and left out the intensity

Problem 10 · FE, Rational method peak runoff

A 4.5 acre watershed is mostly lawns on clay soil, with a runoff coefficient of 0.25. The local intensity-duration relation is I = 75/(t + 11), with I in in./hr and t in minutes. The time of concentration is 43 min. Using the rational method, find the peak runoff in cfs.

Enter your answer in cfs, 3 significant figures.

Handbook: Civil Engineering, Hydrology/Water Resources, Rational Formula, FE Reference Handbook 10.6

Show the worked solution
Answer
1.56 cfs
Given
C = 0.25, A = 4.5 acres, a = 75, b = 11 min, tc = 43 min
Find
peak runoff Q (cfs)
Handbook
Civil Engineering, Hydrology/Water Resources, Rational Formula, page 296
Equation
I = a/(tc + b); Q = CIA
Substitute
  1. I = 75/(tc + 11) = 75/(43 + 11) = 1.389 in./hr (the storm lasting tc gives the peak)
  2. Q = CIA = (0.25)(1.389 in./hr)(4.5 acres) = 1.56 cfs
Result
1.56 cfs, 3 significant figures
Check
the design storm duration equals tc, the time for the whole area to contribute; CIA is read in cfs.
Common wrong answers
  • 6.25 cfs: left out the runoff coefficient (C = 1)
  • 1.96 cfs: left the constant b out of the intensity relation
  • 7.20 cfs: put the time of concentration in hours into a relation written for minutes

A new problem posts every day inside the lab, with the full worked solution the same evening.

Frequently asked questions

What units does the rational method use on the FE?

The handbook gives the U.S. customary form: area in acres, intensity in inches per hour, and flow in cubic feet per second. Our problems use that form.

How do I find a composite runoff coefficient?

Weight each land use’s coefficient by its share of the area, then add. The solution shows the weighted sum line by line.

Why are these fill-in questions?

The FE uses fill-in-the-blank items, and they punish rounding and unit slips. Practicing without options builds the habit of checking units.

Sources

  1. NCEES FE Civil CBT exam specifications (PDF). Retrieved October 3, 2026.
  2. NCEES FE Environmental CBT exam specifications (PDF). Retrieved October 3, 2026.
  3. NCEES FE Reference Handbook 10.6 (free PDF in MyNCEES). Retrieved October 3, 2026.