Rational Formula

Hydrology and Water Resources · FE Reference Handbook section

Hydrology and Water Resources

5 formulas

10 exam-style examples

~55 min

All Hydrology and Water Resources lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Rational Formula within Hydrology and Water Resources. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what rational formula describes physically and when it applies.
State every one of the 5 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: acre-in/hr ≈ cfs makes the rational formula work in US units.

Lecture

Why this section exists. Rational Formula is the part of Hydrology and Water Resources that lets you connect a watershed, aquifer or detention facility to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a rainfall-runoff or well-drawdown calculation with one lookup. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. acre-in/hr ≈ cfs makes the rational formula work in US units. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Row of centrifugal pumps and valved steel piping inside a water pumping station. — Photo 1. Where this shows up in practice: rational formula.
Capstone Studio instructional photograph

Hydrology and Water Resources — Rational Formula: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a watershed, aquifer or detention facility. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 5 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

Q	Quantity produced by "Q = CIA" — read its definition and unit from the handbook line directly above the equation.
A	Quantity produced by "A = watershed area (acres)" — read its definition and unit from the handbook line directly above the equation.
C	Quantity produced by "C = runoff coefficient" — read its definition and unit from the handbook line directly above the equation.
I	Quantity produced by "I = rainfall intensity (in./hr)" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

where

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula

A 22.5 acre catchment is 5.5 acres of pavement (C = 0.95) and 17.0 acres of lawn (C = 0.35). For a design intensity of 2.9 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 5.5 ac, C₁ = 0.95
A₂ = 17.0 ac, C₂ = 0.35
i = 2.9 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.95 (5.5) + 0.35 (17.0)] /22.5 = 0.497$
Formula
$Q = C i A$
Substituting
$Q = 0.497 (2.9) (22.5) = 32.41 c f s$
Interpretation — the composite C of 0.50 means about 50% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.497, Q_peak = 32.4 cfs

Why the other options are there

C = 0.650 (simple average of the two coefficients)
Q = 65.3 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 2

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (2)

A 12.5 acre catchment is 8.0 acres of pavement (C = 0.85) and 4.5 acres of lawn (C = 0.30). For a design intensity of 3.2 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 8.0 ac, C₁ = 0.85
A₂ = 4.5 ac, C₂ = 0.30
i = 3.2 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.85 (8.0) + 0.30 (4.5)] /12.5 = 0.652$
Formula
$Q = C i A$
Substituting
$Q = 0.652 (3.2) (12.5) = 26.08 c f s$
Interpretation — the composite C of 0.65 means about 65% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.652, Q_peak = 26.1 cfs

Why the other options are there

C = 0.575 (simple average of the two coefficients)
Q = 40.0 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 3

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (3)

A 8.5 acre catchment is 3.0 acres of pavement (C = 0.85) and 5.5 acres of lawn (C = 0.35). For a design intensity of 3.9 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 3.0 ac, C₁ = 0.85
A₂ = 5.5 ac, C₂ = 0.35
i = 3.9 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.85 (3.0) + 0.35 (5.5)] /8.5 = 0.526$
Formula
$Q = C i A$
Substituting
$Q = 0.526 (3.9) (8.5) = 17.45 c f s$
Interpretation — the composite C of 0.53 means about 53% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.526, Q_peak = 17.5 cfs

Why the other options are there

C = 0.600 (simple average of the two coefficients)
Q = 33.2 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 4

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (4)

A 17.5 acre catchment is 7.0 acres of pavement (C = 0.90) and 10.5 acres of lawn (C = 0.20). For a design intensity of 4.1 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 7.0 ac, C₁ = 0.90
A₂ = 10.5 ac, C₂ = 0.20
i = 4.1 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.90 (7.0) + 0.20 (10.5)] /17.5 = 0.480$
Formula
$Q = C i A$
Substituting
$Q = 0.480 (4.1) (17.5) = 34.44 c f s$
Interpretation — the composite C of 0.48 means about 48% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.480, Q_peak = 34.4 cfs

Why the other options are there

C = 0.550 (simple average of the two coefficients)
Q = 71.8 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 5

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (5)

A 24.5 acre catchment is 9.0 acres of pavement (C = 0.95) and 15.5 acres of lawn (C = 0.20). For a design intensity of 5.2 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 9.0 ac, C₁ = 0.95
A₂ = 15.5 ac, C₂ = 0.20
i = 5.2 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.95 (9.0) + 0.20 (15.5)] /24.5 = 0.476$
Formula
$Q = C i A$
Substituting
$Q = 0.476 (5.2) (24.5) = 60.58 c f s$
Interpretation — the composite C of 0.48 means about 48% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.476, Q_peak = 60.6 cfs

Why the other options are there

C = 0.575 (simple average of the two coefficients)
Q = 127.4 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 6

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (6)

A 19.5 acre catchment is 9.0 acres of pavement (C = 0.85) and 10.5 acres of lawn (C = 0.35). For a design intensity of 5.0 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 9.0 ac, C₁ = 0.85
A₂ = 10.5 ac, C₂ = 0.35
i = 5.0 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.85 (9.0) + 0.35 (10.5)] /19.5 = 0.581$
Formula
$Q = C i A$
Substituting
$Q = 0.581 (5.0) (19.5) = 56.62 c f s$
Interpretation — the composite C of 0.58 means about 58% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.581, Q_peak = 56.6 cfs

Why the other options are there

C = 0.600 (simple average of the two coefficients)
Q = 97.5 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 7

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (7)

A 12.5 acre catchment is 2.5 acres of pavement (C = 0.95) and 10.0 acres of lawn (C = 0.35). For a design intensity of 5.9 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 2.5 ac, C₁ = 0.95
A₂ = 10.0 ac, C₂ = 0.35
i = 5.9 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.95 (2.5) + 0.35 (10.0)] /12.5 = 0.470$
Formula
$Q = C i A$
Substituting
$Q = 0.470 (5.9) (12.5) = 34.66 c f s$
Interpretation — the composite C of 0.47 means about 47% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.470, Q_peak = 34.7 cfs

Why the other options are there

C = 0.650 (simple average of the two coefficients)
Q = 73.8 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 8

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (8)

A 16.5 acre catchment is 4.5 acres of pavement (C = 0.95) and 12.0 acres of lawn (C = 0.30). For a design intensity of 3.9 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 4.5 ac, C₁ = 0.95
A₂ = 12.0 ac, C₂ = 0.30
i = 3.9 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.95 (4.5) + 0.30 (12.0)] /16.5 = 0.477$
Formula
$Q = C i A$
Substituting
$Q = 0.477 (3.9) (16.5) = 30.71 c f s$
Interpretation — the composite C of 0.48 means about 48% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.477, Q_peak = 30.7 cfs

Why the other options are there

C = 0.625 (simple average of the two coefficients)
Q = 64.4 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 9

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (9)

A 13.0 acre catchment is 10.0 acres of pavement (C = 0.95) and 3.0 acres of lawn (C = 0.30). For a design intensity of 5.3 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 10.0 ac, C₁ = 0.95
A₂ = 3.0 ac, C₂ = 0.30
i = 5.3 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.95 (10.0) + 0.30 (3.0)] /13.0 = 0.800$
Formula
$Q = C i A$
Substituting
$Q = 0.800 (5.3) (13.0) = 55.12 c f s$
Interpretation — the composite C of 0.80 means about 80% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.800, Q_peak = 55.1 cfs

Why the other options are there

C = 0.625 (simple average of the two coefficients)
Q = 68.9 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Example 10

Peak discharge by the rational formula with a composite runoff coefficient — Rational Formula (10)

A 16.0 acre catchment is 9.0 acres of pavement (C = 0.90) and 7.0 acres of lawn (C = 0.35). For a design intensity of 5.7 in/hr, compute the composite runoff coefficient and the peak discharge.

Given

A₁ = 9.0 ac, C₁ = 0.90
A₂ = 7.0 ac, C₂ = 0.35
i = 5.7 in/hr

Find

Composite C and Q_peak

Start with the thinking

Q = CiA is numerically convenient in U.S. units: acres × in/hr ≈ cfs because the conversion factor is 1.008.
The composite C is an area-weighted average, never a simple average.

Step-by-step solution

Formula — C_w = \dfrac{\sum C_iA_i}{\sum A_i}
Substituting
$C_{w} = [0.90 (9.0) + 0.35 (7.0)] /16.0 = 0.659$
Formula
$Q = C i A$
Substituting
$Q = 0.659 (5.7) (16.0) = 60.14 c f s$
Interpretation — the composite C of 0.66 means about 66% of the rainfall appears as direct runoff at the peak.

Answer: C_w = 0.659, Q_peak = 60.1 cfs

Why the other options are there

C = 0.625 (simple average of the two coefficients)
Q = 91.2 cfs (runoff coefficient omitted)

Reference: FE Reference Handbook — Hydrology and Water Resources → Rational Formula

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a watershed, aquifer or detention facility, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Rational Formula contains 5 relations; you must be able to find this page in under 15 seconds.
Exam style: a rainfall-runoff or well-drawdown calculation with one lookup.
Unit rule: acre-in/hr ≈ cfs makes the rational formula work in US units.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

acre-in/hr ≈ cfs makes the rational formula work in US units
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.