MATHEMATICS MANUAL FOR WATER AND WASTEWATER TREATMENT PLANT OPERATORS - PART 1 potx

for
Mathematics Manual
WATER AND WASTEWATER
TREATMENT PLANT OPERATORS
© 2004 by CRC Press LLC
CRC PRESS
Boca Raton London New York Washington, D.C.
Frank R. Spellman
for
Mathematics Manual
WATER AND WASTEWATER
TREATMENT PLANT OPERATORS
© 2004 by CRC Press LLC

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with
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International Standard Book Number 1-56670-675-0
Library of Congress Card Number 2003065830
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Spellman, Frank R.
Mathematics manual for water and wastewater treatment plant operators / Frank R. Spellman.
p. cm.
Includes index.
ISBN 1-56670-675-0 (alk. paper)
1. Water—PuriÞcation—Mathematics. 2. Water quality management—Mathematics. 3.
Water—PuriÞcation—Problems, exercises, etc. 4. Water quality management—Problems,
exercises, etc. 5. Sewage—PuriÞcation—Mathematics. 6. Sewage disposal—Mathematics. 7.
Sewage—PuriÞcation—Problems, exercises, etc. 8. Sewage disposal—Problems, exercises,
etc. I. Title.
TD430.S64 2004
628.1

¢

01

¢

51—dc22 2003065830

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© 2004 by CRC Press LLC

PREFACE

To properly operate a waterworks or wastewater treatment plant and to pass the examination for a
waterworks/wastewater operator’s license, it is necessary to know how to perform certain calcula-
tions. In reality, most of the calculations that operators at the lower level of licensure need to know
how to perform are not difÞcult, but all operators need a basic understanding of arithmetic and
problem-solving techniques to be able to solve the problems they typically encounter.
How about waterworks/wastewater treatment plant operators at higher levels of licensure —
do they also need to be well versed in mathematical operations? The short answer is absolutely.
The long answer is that anyone who works in water or wastewater treatment and who expects to
have a successful career that includes advancement to the highest levels of licensure or certiÞcation
(usually prerequisites for advancement to higher management levels) must have knowledge of math
at both the basic or fundamental level and at the advanced practical level. It is simply not possible
to succeed in this Þeld without the ability to perform mathematical operations.
Keep in mind that mathematics is a universal language. Mathematical symbols have the same
meaning to people speaking many different languages throughout the world. The key to using
mathematics is learning the language, symbols, deÞnitions, and terms of mathematics that allow
us to grasp the concepts necessary to solve equations.
In

Mathematics Manual for Water/Wastewater Treatment Plant Operators

, we begin by intro-
ducing and reviewing concepts critical to the qualiÞed operators at the fundamental or entry level;
however, this does not mean that these are the only math concepts that a competent operator must
know to solve routine operation and maintenance problems.

After covering the basics, therefore,
the text progressively advances, step-by-step, to higher more practical applications of mathematical
calculations — that is, the math operations that operators at the highest level of licensure would
be expected to know how to perform.
The basic level reviews fractions and decimals, rounding numbers, determining the correct
number of signiÞcant digits, raising numbers to powers, averages, proportions, conversion factors,
calculating ßow and detention times, and determining the areas and volumes of different shapes.
This review also explains how to keep track of units of measurement (inches, feet, gallons, etc.)
during calculations and demonstrates how to solve real-life problems that require calculations.
After building a strong foundation based on theoretical math concepts (the basic tools of
mathematics, such as fractions, decimals, percents, areas, volumes), we move on to applied math
— basic math concepts applied when solving practical water/wastewater operational problems.
Even though considerable crossover of basic math operations used by both waterworks and waste-
water operators occurs, this book separates applied math problems for wastewater and water to aid
operators dealing with speciÞc unit processes unique to either waterworks or wastewater operations.
The text is divided into Þve parts. Part I covers basic math concepts used in both water and
wastewater treatment. Part II covers advanced math concepts for waterworks operators. Part III
covers advanced math concepts for wastewater operators. Part IV covers fundamental laboratory
calculations used in both water and wastewater treatment operations. Part V presents a comprehen-
sive workbook of more than 1400 practical math problems that highlight the type of math exam
questions operators can expect to see on state licensure examinations.
What makes

Mathematics Manual for Water/Wastewater Treatment Plant Operators

different
from other math books available? Consider the following:

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© 2004 by CRC Press LLC

• The author has worked in and around water/wastewater treatment and taught water/waste-
water math for several years.
• The author has sat at the table of licensure examination preparation boards to review,
edit, and write state licensure exams.
• This step-by-step training manual provides concise, practical instruction in the math
skills that operators must have to pass certiÞcation tests.
• The text is completely self-contained in one complete volume. The advantage should be
obvious — one text that combines math basics and advanced operator math concepts
eliminates shufßing from one volume to another to Þnd the solution to a simple or more
complex problem.
• The text is user friendly; no matter the difÞculty of the problem to be solved, each
operation is explained in straightforward, plain English. Moreover, numerous example
problems (several hundred) are presented to enhance the learning process.
To assure correlation to modern practice and design, the text provides illustrative problems
dealing with commonly encountered waterworks/wastewater treatment operations and associated
parameters and covers typical math concepts for waterworks/wastewater treatment unit process
operations found in today’s waterworks/wastewater treatment facilities.

ߜ

Note:

The symbol

ߜ

displayed in various locations throughout this manual indicates or emphasizes
an important point or points to study carefully.

This text is accessible to those who have little or no experience in treatment plant math
operations. Readers who work through the text systematically will be surprised at how easily they
can acquire an understanding of water/wastewater math concepts, thus adding another critical
component to their professional knowledge.
A Þnal point before beginning our discussion of math concepts: It can be said with some
accuracy and certainty that without the ability to work basic math problems (i.e., those typical to
water/wastewater treatment) candidates for licensure will Þnd any attempts to successfully pass
licensure exams a much more difÞcult proposition.

Frank R. Spellman

Norfolk, Virginia

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© 2004 by CRC Press LLC

Table of Contents

PART I

Basic Math Concepts

Chapter 1

Introduction
Math Terminology and DeÞnitions
Calculation Steps

Key Words
Calculators

Chapter 2

Sequence of Operations
Sequence of Operations — Rules
Sequence of Operations — Examples

Chapter 3

Fractions, Decimals, and Percent
Fractions
Decimals
Percent

Chapter 4

Rounding and SigniÞcant Digits
Rounding Numbers
Determining SigniÞcant Figures

Chapter 5

Powers of Ten and Exponents
Rules
Examples

Chapter 6

Averages (Arithmetic Mean) and Median
Averages
Median

Chapter 7

Solving for the Unknown
Equations
Axioms
Solving Equations
Setting Up Equations

Chapter 8

Ratio and Proportion
Ratio
Proportion
Working with Ratio and Proportion

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Chapter 9

Dimensional Analysis
Dimensional Analysis in Problem Solving
Basic Operation 1
Basic Operation 2
Basic Operation 3

Chapter 10

Units of Measurement, Conversions, and Electrical Calculations
Units of Measurement
Conversion Factors
Weight, Concentration, and Flow
Conversions
Typical Water/Wastewater Conversion Examples
Temperature Conversions
Population Equivalent (PE) or Unit Loading Factor
SpeciÞc Gravity and Density
Flow
Detention Time
Chemical Addition Conversions
Horsepower and Energy Costs
Electrical Power
Electrical Calculations
Ohm’s Law
Electric Power
Electric Energy
Series D-C Circuit Characteristics
Parallel D-C Circuits

Chapter 11

Measurements: Circumference, Area, and Volume
Perimeter and Circumference
Perimeter
Circumference
Area

Area of a Rectangle
Area of a Circle
Surface Area of a Circular or Cylindrical Tank
Volume
Volume of a Rectangular Basin
Volume of Round Pipe and Round Surface Areas
Volume of a Cone and Sphere
Volume of a Circular or Cylindrical Tank

Chapter 12

Force, Pressure, and Head Calculations
Force and Pressure
Head
Static Head
Friction Head
Velocity Head
Total Dynamic Head (Total System Head)

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Pressure/Head
Head/Pressure
Force, Pressure, and Head Example Problems

Chapter 13

Mass Balance and Measuring Plant Performance
Mass Balance for Settling Tanks

Mass Balance Using BOD Removal
Measuring Plant Performance
Plant Performance/EfÞciency
Unit Process Performance/EfÞciency
Percent Volatile Matter Reduction in Sludge

PART II

Applied Math Concepts: Water Treatment

Chapter 14

Pumping Calculations.
Pumping
Basic Water Hydraulics Calculations
Weight of Water
Weight of Water Related to the Weight of Air
Gauge Pressure
Water in Motion
Pipe Friction
Basic Pumping Calculations
Pumping Rates
Calculating Head Loss
Calculating Horsepower and EfÞciency
SpeciÞc Speed
Positive Displacement Pumps
Volume of Biosolids Pumped (Capacity)

Chapter 15

Water Source and Storage Calculations
Water Sources
Water Source Calculations
Well Drawdown
Well Yield
SpeciÞc Yield
Well-Casing Disinfection
Deep-Well Turbine Pump Calculations
Vertical Turbine Pump Calculations
Water Storage
Water Storage Calculations
Copper Sulfate Dosing

Chapter 16

Coagulation and Flocculation Calculations
Coagulation
Flocculation

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Coagulation and Flocculation Calculations.
Chamber and Basin Volume Calculations
Detention Time
Determining Dry Chemical Feeder Setting (lb/d)
Determining Chemical Solution Feeder Setting (gpd)
Determining Chemical Solution Feeder Setting (mL/min)
Determining Percent of Solutions
Determining Percent Strength of Liquid Solutions

Determining Percent Strength of Mixed Solutions
Dry Chemical Feeder Calibration
Solution Chemical Feeder Calibration
Determining Chemical Usage

Chapter 17

Sedimentation Calculations
Sedimentation
Tank Volume Calculations
Calculating Tank Volume
Detention Time
Surface Overßow Rate
Mean Flow Velocity
Weir Loading Rate (Weir Overßow Rate)
Percent Settled Biosolids
Determining Lime Dosage (mg/L)
Determining Lime Dosage (lb/day)
Determining Lime Dosage (g/min)

Chapter 18

Filtration Calculations
Water Filtration
Flow Rate through a Filter (gpm)
Filtration Rate
Unit Filter Run Volume (UFRV)
Backwash Rate
Backwash Rise Rate
Volume of Backwash Water Required (gal)

Required Depth of Backwash Water Tank (ft)
Backwash Pumping Rate (gpm)
Percent Product Water Used for Backwatering
Percent Mud Ball Volume

Chapter 19

Water Chlorination Calculations
Chlorine Disinfection
Determining Chlorine Dosage (Feed Rate)
Calculating Chlorine Dose, Demand and Residual
Breakpoint Chlorination Calculations
Calculating Dry Hypochlorite Feed Rate
Calculating Hypochlorite Solution Feed Rate

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© 2004 by CRC Press LLC

Calculating Percent Strength of Solutions
Calculating Percent Strength Using Dry Hypochlorite
Calculating Percent Strength Using Liquid Hypochlorite
Chemical Use Calculations

Chapter 20

Fluoridation
Water Fluoridation
Fluoride Compounds
Sodium Fluoride
Sodium Fluorosilicate

Fluorosilicic Acid
Optimal Fluoride Levels
Fluoridation Process Calculations
Percent Fluoride Ion in a Compound
Fluoride Feed Rate
Fluoride Feed Rates for Saturator
Calculated Dosages
Calculated Dosage Problems

Chapter 21

Water Softening
Water Hardness
Calculating Calcium Hardness as CaCO

3

Calculating Magnesium Hardness as CaCO

3

Calculating Total Hardness
Calculating Carbonate and Noncarbonate Hardness
Alkalinity Determination
Determining Bicarbonate, Carbonate, and Hydroxide Alkalinity
Lime Dosage Calculation for Removal of Carbonate Hardness
Calculation for Removal of Noncarbonate Hardness
Recarbonation Calculation
Calculating Feed Rates
Ion Exchange Capacity

Water Treatment Capacity
Treatment Time Calculation (Until Regeneration Required)
Salt and Brine Required for Regeneration

PART III

Wastewater Math Concepts

Chapter 22

Preliminary Treatment Calculations
Screening
Screening Removal Calculations
Screening Pit Capacity Calculations
Grit Removal
Grit Removal Calculations
Grit Channel Velocity Calculation

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Chapter 23

Primary Treatment Calculations
Process Control Calculations
Surface Loading Rate (Surface Settling Rate/Surface Overßow Rate)
Weir Overßow Rate (Weir Loading Rate)
Biosolids Pumping
Percent Total Solids (%TS)
BOD and SS Removed (lb/d)

Chapter 24

Trickling Filter Calculations
Trickling Filter Process Calculations
Hydraulic Loading Rate
Organic Loading Rate
BOD and SS Removed
Recirculation Ratio

Chapter 25

Rotating Biological Contactors (RBCs)
RBC Process Control Calculations
Hydraulic Loading Rate
Soluble BOD
Organic Loading Rate
Total Media Area

Chapter 26

Activated Biosolids
Activated Biosolids Process Control Calculations
Moving Averages
BOD or COD Loading
Solids Inventory
Food-to-Microorganism Ratio (F/M Ratio)
Gould Biosolids Age
Mean Cell Residence Time (MCRT)
Estimating Return Rates from SSV

60

Sludge Volume Index (SVI)
Mass Balance: Settling Tank Suspended Solids
Biosolids Waste Based upon Mass Balance
Oxidation Ditch Detention Time

Chapter 27

Treatment Ponds
Treatment Pond Parameters
Determining Pond Area (Inches)
Determining Pond Volume (Acre-Feet)
Determining Flow Rate (Acre-Feet/Day)
Determining Flow Rate (Acre-Inches/Day)
Treatment Pond Process Control Calculations
Hydraulic Detention Time (Days)
BOD Loading

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© 2004 by CRC Press LLC

Organic Loading Rate
BOD Removal EfÞciency
Population Loading
Hydraulic Loading (Inches/Day) (Overßow Rate)

Chapter 28

Chemical Dosage Calculations
Chemical Dosing
Chemical Feed Rate
Chlorine Dose, Demand, and Residual
Hypochlorite Dosage
Chemical Solutions
Mixing Solutions of Different Strength
Solution Mixtures Target Percent Strength
Solution Chemical Feeder Setting (gpd)
Chemical Feed Pump: Percent Stroke Setting
Chemical Solution Feeder Setting (mL/min)
Chemical Feed Calibration
Average Use Calculations

Chapter 29

Biosolids Production and Pumping Calculations
Process Residuals
Primary and Secondary Solids Production Calculations
Primary ClariÞer Solids Production Calculations
Secondary ClariÞer Solids Production Calculations
Percent Solids
Biosolids Pumping
Estimating Daily Biosolids Production
Biosolids Production in Pounds/Million Gallons
Biosolids Production in Wet Tons/Year
Biosolids Pumping Time

Chapter 30

Biosolids Thickening Calculations
Thickening
Gravity/Dissolved Air Flotation Thickener Calculations
Estimating Daily Biosolids Production
Surface Loading Rate (gpd/day/ft

2

)
Solids Loading Rate (lb/d/ft

2

)
Concentration Factor (CF)
Air-to-Solids Ratio
Recycle Flow in Percent
Centrifuge Thickening Calculations

Chapter 31

Biosolids Digestion
Biosolids Stabilization
Aerobic Digestion Process Control Calculations

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Volatile Solids Loading (lb/ft

3

/day)
Digestion Time (day)
pH Adjustment
Anaerobic Digestion Process Control Calculations
Required Seed Volume (gal)
Volatile Acids to Alkalinity Ratio
Biosolids Retention Time
Estimated Gas Production (ft

3

/d)
Volatile Matter Reduction (%)
Percent Moisture Reduction in Digested Biosolids

Chapter 32

Biosolids Dewatering and Disposal
Biosolids Dewatering
Pressure Filtration Calculations
Plate and Frame Press
Belt Filter Press
Rotary Vacuum Filter Dewatering Calculations
Filter Loading
Filter Yield
Vacuum Filter Operating Time
Percent Solids Recovery
Sand Drying Bed Calculations

Sand Drying Beds Process Control Calculations
Biosolids Disposal
Land Application Calculations
Biosolids to Compost
Composting Calculations

PART IV Laboratory Calculations

Chapter 33

Water/Wastewater Laboratory Calculations
Water/Wastewater Lab
Faucet Flow Estimation
Service Line Flushing Time
Composite Sampling Calculation (Proportioning Factor)
Composite Sampling Procedure and Calculation
Biochemical Oxygen Demand (BOD) Calculations
BOD 7-Day Moving Average
Moles and Molarity
Moles
Normality
Settleability (Activated Biosolids Solids)
Settleable Solids
Biosolids Total Solids, Fixed Solids, and Volatile Solids
Wastewater Suspended Solids and Volatile Suspended Solids
Biosolids Volume Index (BVI) and Biosolids Density Index (BDI)

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PART V

Workbook Practice Problems

Chapter 34

Workbook Practice Problems
Basic Math Operations (Problems 1 to 43)
Decimal Operations
Percentage Calculations
Find

x

Ratio and Proportion
Area of Rectangles
Circumference and Area of Circles
Fundamental Operations (Water/Wastewater) (Problems 1 to 342)
Tank Volume Calculations.
Channel and Pipeline Capacity Calculations
Miscellaneous Volume Calculations
Flow, Velocity, and Conversion Calculations
Average Flow Rates
Flow Conversions
General Flow and Velocity Calculations
Chemical Dosage Calculations
BOD, COD, and SS Loading Calculations
BOD and SS Removal (lb/day)
Pounds of Solids Under Aeration
WAS Pumping Rate Calculations

Hydraulic Loading Rate Calculations
Surface Overßow Rate Calculations
Filtration Rate Calculations
Backwash Rate Calculations
Unit Filter Run Volume (UFRV) Calculations
Weir Overßow Rate Calculations
Organic Loading Rate Calculations
Food/Microorganism (F/M) Ratio Calculations
Solids Loading Rate Calculations
Digester Loading Rate Calculations
Digester Volatile Solids Loading Ratio Calculations
Population Loading and Population Equivalent
General Loading Rate Calculations
Detention Time Calculations
Sludge Age Calculations
Solids Retention Time (SRT) Calculations
General Detention Time and Retention Time Calculations
EfÞciency and General Percent Calculations
Percent Solids and Sludge Pumping Rate Calculations
Percent Volatile Solids Calculations
Seed Sludge Calculations
Solution Strength Calculations
Pump and Motor EfÞciency Calculations
General EfÞciency and Percent Calculations
Pumping Calculations

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Density and SpeciÞc Gravity

Force and Pressure Calculations
Head and Head Loss Calculations
Horsepower Calculations
Pump Capacity Calculations
General Pumping Calculations
Basic Electricity Calculations
Water Treatment Calculations (Problems 1 to 457)
Water Sources and Storage Calculations: Well Drawdown
Well Yield
SpeciÞc Yield
Well Casing Disinfection
Deep-Well Turbine Pump Calculations
Pond Storage Capacity
Copper Sulfate Dosing
General Water Source and Storage Calculations
Coagulation and Flocculation Calculations: Unit Process Volume
Detention Time
Calculating Dry Chemical Feeder Setting (lb/day)
Calculating Solution Feeder Setting (gal/day)
Calculating Solution Feeder Setting (mL/min)
Percent Strength of Solutions
Mixing Solutions of Different Strength
Dry Chemical Feeder Calibration
Solution Chemical Feeder Calibration
Chemical Use Calculations
General Coagulation and Flocculation
Sedimentation — Tank Volume
Detention Time
Surface Overßow Rate
Mean Flow Velocity

Weir Loading Rate
Percent Settled Sludge
Lime Dosage
Lime Dose Required (lb/day)
Lime Dose Required (g/min)
General Sedimentation Calculations
Filtration — Flow Rate through a Filter
Filtration Rate (gpm/ft

2

)
Unit Filter Run Volume (UFRV)
Backwash (gpm/ft

2

)
Volume of Backwash Water Required (gal)
Required Depth of Backwash Water Tank (ft)
Percent of Product Water Used for Backwashing
Percent Mud Ball Volume
General Filtration Calculations
Chlorination: Chlorine Feed Rate
Chlorine Dose, Demand, and Residual
Dry Hypochlorite Feed Rate
Hypochlorite Solution Feed Rate
Percent Strength of Solutions

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© 2004 by CRC Press LLC

Mixing Hypochlorite Solutions
General Chlorination Calculations
Fluoridation: Concentration Expressions
Percent Fluoride Ion in a Compound
Calculating Dry Feed Rate (lb/day)
Calculating Fluoride Dosage (mg/L)
Solution Mixtures
General Fluoridation Calculations
Softening: Equivalent Weight/Hardness of CaCO

3

Carbonate and Noncarbonate Hardness
Phenolphthalein and Total Alkalinity
Bicarbonate, Carbonate, and Hydroxide Alkalinity
Lime Dosage for Softening
Soda Ash Dosage
Carbon Dioxide for Recarbonation
Chemical Feeder Settings
Ion Exchange Capacity
Water Treatment Capacity
Operating Time
Salt and Brine Required
General Softening Calculations
Wastewater Treatment Calculations (Problems 1 to 574)
Wastewater Collection and Preliminary Treatment
Wet Well Pumping Rate
Screenings Removed

Screenings Pit Capacity
Grit Channel Velocity
Grit Removal
Plant Loadings
General Wastewater Collection and Preliminary Treatment Calculations
Primary Treatment
Trickling Filters
General Trickling Filter Calculations
Rotating Biological Contactors (RBCs)
Activated Sludge
Waste Treatment Ponds
Detention Time
Chemical Dosage
Percent Strength of Solutions
General Chemical Dosage Calculations
Sludge Production and Thickening
Digestion
Sludge Dewatering
Laboratory Calculations (Water and Wastewater) (Problems 1 to 80)
Estimating Faucet Flow
Service Line Flushing Time
Solution Concentration
Biochemical Oxygen Demand (BOD)
Settleability Solids
Molarity and Moles
Sludge Total Solids and Volatile Solids

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Suspended Solids and Volatile Suspended Solids
Sludge Volume Index (SVI) and Sludge Density Index (SDI)
Temperature
Chlorine Residual
General Laboratory Calculations

APPENDIX

Workbook Answer Key
Basic Math Operations (Problems 1 to 43)
Fundamental Operations (Water/Wastewater) (Problems 1 to 342)
Water Treatment Calculations (Problems 1 to 457)
Wastewater Calculations (Problems 1 to 574)
Laboratory Calculations (Water and Wastewater) (Problems 1 to 80)


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Part I

Basic Math Concepts

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1

Introduction

TOPICS

• Math Terminology and Definitions
• Calculators
Anyone who has had the opportunity to work in waterworks and/or wastewater treatment, even for
a short time, learns quickly that water/wastewater treatment operations involve a large number of
process control calculations. All of these calculations are based upon basic math principles. In this
chapter, we introduce basic mathematical terminology and definitions and calculator operations
that water/wastewater operators are required to use, many of them on a daily basis.
What is

mathematics

? Good question. Mathematics is numbers and symbols. Math uses com-
binations of numbers and symbols to solve practical problems. Every day, we use numbers to count.
Numbers may be considered as representing things counted. The money in your pocket or the
power consumed by an electric motor is expressed in numbers. When operators make entries in
the Plant Daily Operating Log, they enter numbers in parameter columns, indicating the operational
status of various unit processes — many of these math entries are required by the NPDES permit
for the plant. Again, we use numbers every day. Because we use numbers every day, we are all
mathematicians — to a point.
In water/wastewater treatment, we need to take math beyond “to a point”. We need to learn,
understand, appreciate, and use mathematics. Not knowing the key definitions of the terms used is
probably the greatest single cause of failure to understand and appreciate mathematics In math-
ematics, more than in any other subject, each word used has a definite and fixed meaning. The
math terminology and definitions section will aid in understanding the material in this book.

MATH TERMINOLOGY AND DEFINITIONS

• An

integer

,



or an

integral number

,



is a whole number; thus 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, and 12 are the first 12 positive integers.
•A

factor

,



or

divisor

, of a whole number is any other whole number that exactly divides

it. Thus, 2 and 5 are factors of 10.
•A

prime number

in math is a number that has no factors except itself and 1. Examples
of prime numbers are 1, 3, 5, 7, and 11.
•A

composite number

is a number that has factors other than itself and 1. Examples of
composite numbers are 4, 6, 8, 9, and 12.
•A

common factor

,



or

common divisor

,



of two or more numbers is a factor that will exactly

divide each of them. If this factor is the largest factor possible, it is called the

greatest
common divisor

. Thus, 3 is a common divisor of 9 and 27, but 9 is the greatest common
divisor of 9 and 27.
•A

multiple

of a given number is a number that is exactly divisible by the given number.
If a number is exactly divisible by two or more other numbers, it is a common multiple
of them. The least (smallest) such number is called the

lowest common multiple

. Thus,
36 and 72 are common multiples of 12, 9, and 4; however, 36 is the lowest common
multiple.

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• An

even number

is a number exactly divisible by 2; thus, 2, 4, 6, 8, 10, and 12 are even
integers.

• An

odd number

is an integer that is not exactly divisible by 2; thus, 1, 3, 5, 7, 9, and 11
are odd integers.
•A

product

is the result of multiplying two or more numbers together; thus, 25 is the
product of 5

¥

5. Also, 4 and 5 are factors of 20.
•A

quotient

is the result of dividing one number by another; for example, 5 is the quotient
of 20 divided by 4.
•A

dividend

is a number to be divided, and a

divisor

is a number that divides; for example,
in 100 ÷ 20 = 5, the dividend is 100, the divisor is 20, and the quotient is 5.
•

Area

is the area of an object, measured in square units
•

Base

is a term used to identify the bottom leg of a triangle, measured in linear units.
•

Circumference

is the distance around an object, measured in linear units. When deter-
mined for a shape other than a circle, it may be called the

perimeter

of the figure, object,
or landscape.
•

Cubic units

are measurements used to express volume, cubic feet, cubic meters, etc.
•

Depth

is the vertical distance from the bottom of the tank to the top. This is normally
measured in terms of liquid depth and given in terms of sidewall depth (SWD), measured
in linear units.
•

Diameter

is the distance from one edge of a circle to the opposite edge passing through
the center, measured in linear units.
•

Height

is the vertical distance from the base or bottom of a unit to the top or surface.
•

Linear units

are measurements used to express distances (e.g., feet, inches, meters, yards).
•

Pi

(

p

) is a number used in calculations involving circles, spheres, or cones;

p

=

3.14.
•

Radius

is the distance from the center of a circle to the edge, measured in linear units.
•

Sphere

is a container shaped like a ball.
•

Square units

are measurements used to express area, square feet, square meters, acres, etc.
•

Volume

is the capacity of the unit (how much it will hold) measured in cubic units (cubic
feet, cubic meters) or in liquid volume units (gallons, liters, million gallons).
•

Width

is the distance from one side of the tank to the other, measured in linear units.

C

ALCULATION

S

TEPS

Standard methodology used in making mathematical calculations includes:
• Making a drawing of the information in the problem, if appropriate.
• Placing the given data on the drawing.
• Asking, “What is the question?” This is the first thing you should ask along with, “What
are they really looking for?”
• Writing it down, if the calculation calls for an equation.
• Filling in the data in the equation — look to see what is missing.
• Rearranging or transposing the equation, if necessary.
• Using a calculator, if available.
• Writing down the answer, always.
• Checking any solution obtained. Does the answer make sense?

ߜ

Important Point:

Solving word math problems is difficult for many operators. Solving these
problems is made easier, however, by understanding a few key words.

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© 2004 by CRC Press LLC

K

EY

W

ORDS

• The term

of

means to multiply
• The term

and

means to add
• The term

per

means to divide
• The term

less than

means to subtract

CALCULATORS

You have heard the old saying, “Use it or lose it.” This saying amply applies to mathematics.
Consider the person who first learns to perform long division, multiplication, square root, adding
and subtracting, converting decimals to fractions, and other math operations using nothing more
than pencil and paper and brain power. Eventually, this same person is handed a pocket calculator
that can produce all of these functions and much more simply by manipulating certain keys on a
keyboard. This process involves little brainpower — nothing more than punching in correct numbers
and operations to achieve an almost instant answer. Backspacing to the previous statement (“use
it or lose it”) makes our point. As with other learned skills, our proficiency in performing a learned
skill is directly proportionate to the amount of time we spend using the skill — whatever that might
be. We either use it or we lose it. The consistent use of calculators has caused many of us to forget
how to perform basic math operations with pencil and paper — for example, how to perform long
division.
Without a doubt, the proper use of a calculator can reduce the time and effort required to
perform calculations; thus, it is important to recognize the calculator as a helpful tool, with the
help of a well-illustrated instruction manual, of course. The manual should be large enough to read,
not an inch by an inch by a quarter of an inch in size. It should have examples of problems and
answers with illustrations. Careful review of the instructions and practice using example problems
are the best ways to learn how to use the calculator.
Keep in mind that the calculator you select should be large enough so that you can use it. Many
of the modern calculators have keys so small that it is almost impossible to hit just one key. You
will be doing a considerable amount of work during this study effort — make it as easy on yourself
as you can.
Another significant point to keep in mind when selecting a calculator is the importance of
purchasing a unit that has the functions you need. Although a calculator with a lot of functions
may look impressive, it can be complicated to use. Generally, the water/wastewater plant operator
requires a calculator that can add, subtract, multiply, and divide. A calculator with a parentheses

function is helpful, and, if you must calculate geometric means for fecal coliform reporting, for
example, then logarithmic capability is also helpful.
In many cases, calculators can be used to perform several mathematical functions in succession.
Because various calculators are designed using different operating systems, you must review the
instructions carefully to determine how to make the best use of the system.
Finally, it is important to keep a couple of basic rules in mind when performing calculations:
• Always write down the calculations you wish to perform
• Remove any parentheses or brackets by performing the calculations inside first

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© 2004 by CRC Press LLC

2

Sequence of Operations

TOPICS

• Sequence of Operations — Rules
• Sequence of Operations — Examples
Mathematical operations such as addition, subtraction, multiplication, and division are usually
performed in a certain order or sequence. Typically, multiplication and division operations are done
prior to addition and subtraction operations. In addition, mathematical operations are also generally
performed from left to right using this hierarchy. The use of parentheses is also common to set
apart operations that should be performed in a particular sequence.



ߜ

Note:

It is assumed that the reader has a fundamental knowledge of basic arithmetic and math
operations. Thus, the purpose of the following section is to provide only a brief review of the mathe-
matical concepts and applications frequently employed by water/wastewater operators.

SEQUENCE OF OPERATIONS — RULES

Rule 1:

In a series of

additions

, the terms may be placed in any order and grouped in any
way; thus, 4 + 3 = 7 and 3 + 4 = 7; (4 + 3) + (6 + 4) = 17, (6 + 3) + (4 + 4) = 17, and
[6 + (3 + 4) + 4] = 17.

Rule 2:

In a series of

subtractions

, changing the order or the grouping of the terms may
change the result; thus, 100 – 30 = 70, but 30 –100 = –70; (100 – 30) – 10 = 60, but 100 –
(30 – 10) = 80.

Rule 3:

When no grouping is given, the subtractions are performed in the order written
from left to right (e.g., 100 – 30 – 15 – 4 = 51) or by steps, (e.g., 100 – 30 = 70, 70 –
15 = 55, 55 – 4 = 51).

Rule 4:

In a series of

multiplications

, the factors may be placed in any order and in any
grouping; thus, [(2

¥

3)

¥

5]

¥

6 = 180 and 5

¥

[2

¥

(6

¥

3)] = 180.

Rule 5:

In a series of

divisions

, changing the order or the grouping may change the result;
thus, 100 ÷ 10 = 10 but 10 ÷ 100 = 0.1 and (100 ÷ 10) ÷ 2 = 5 but 100 ÷ (10 ÷ 2) = 20.
Again, if no grouping is indicated, the divisions are performed in the order written from
left to right; thus, 100 ÷ 10 ÷ 2 is understood to mean (100 ÷ 10) ÷ 2.

Rule 6:

In a series of mixed mathematical operations, the convention is as follows: When-
ever no grouping is given, multiplications and divisions are to be performed in the order
written, then additions and subtractions in the order written.

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© 2004 by CRC Press LLC

SEQUENCE OF OPERATIONS — EXAMPLES

ߜ

In a series of additions, the terms may be placed in any order and grouped in any way.

E

XAMPLES

3 + 6 = 10 and 6 + 4 = 10
(4 + 5) + (3 + 7) = 19, (3 + 5) + (4 + 7) = 19, and [7 + (5 + 4)] + 3 = 19

ߜ



In a series of subtractions, changing the order or the grouping of the terms may change the result.

E

XAMPLES

100 – 20 = 80, but 20 – 100 = –80
(100 – 30) – 20 = 50, but 100 – (30 – 20) = 90

ߜ



When no grouping is given, the subtractions are performed in the order written — from left to right.

E

XAMPLES

100 – 30 – 20 – 3 = 47
or by steps, 100 – 30 = 70, 70 – 20 = 50, 50 – 3 = 47

ߜ



In a series of multiplications, the factors may be placed in any order and in any grouping.

E

XAMPLES

[(3

¥

3)

¥

5]

¥

6 = 270 and 5

¥

[3

¥

(6

¥

3)] = 270

ߜ



In a series of divisions, changing the order or the grouping may change the result.

E

XAMPLES

100 ÷ 10 = 10, but 10 ÷ 100 = 0.1
(100 ÷ 10) ÷ 2 = 5, but 100 ÷ (10 ÷ 2) = 20

ߜ

If no grouping is indicated, the divisions are performed in the order written — from left to right.

E

XAMPLES

100 ÷ 5 ÷ 2 is understood to mean (100 ÷ 5) ÷ 2

ߜ



In a series of mixed mathematical operations, the rule of thumb is that, whenever no grouping is
given, multiplications and divisions are to be performed in the order written, then additions and
subtractions in the order written.

Example 2.1

Problem

Perform the following mathematical operations to solve for the correct answer:
24 2 6
62
2
+
()
+¥
()
+

+
Ê
Ë
ˆ
¯
= _________
_

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© 2004 by CRC Press LLC

Solution

• Mathematical operations are typically performed going from left to right within an
equation and within sets of parentheses.
• Perform all math operations within the sets of parentheses first:
(Note that the addition of 6 and 2 was performed prior to dividing.)
• Perform all math operations outside of the parentheses. In this case, from left to right.
6 + 12 + 4 = 22

Example 2.2

Problem

Solve the following equation:
(4 – 2) + (3

¥

3) – (15 ÷ 3) – 8 = ____________

Solution

• Perform math operations inside each set of parentheses:
4 – 2 = 2
3

¥

3 = 9
15 ÷ 3 = 5
• Perform addition and subtraction operations from left to right.
• The final answer is 2 + 9 – 5 – 8 = –2.
There may be cases where several operations will be performed within multiple sets of parentheses.
In these cases, we must perform all operations within the innermost set of parentheses first and move
outward. We must continue to observe the hierarchical rules throughout the problem. Brackets [ ]
may indicate additional sets of parentheses.

Example 2.3

Problem

Solve the following equation:
[2

¥

(3 + 5) – 5 + 2]

¥

3 = ____________
246
2612
62
2
8
2
4
+=
¥=
+
==

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© 2004 by CRC Press LLC

Solution

• Perform operations in the innermost set of parentheses.
3 + 5 = 8
• Rewrite the equation.
[2

¥

8 – 5 + 2]

¥

3 =
• Perform multiplication prior to addition and subtraction within the bracket.
[16 – 5 + 2]

¥

3 =
[11 + 2]

¥

3 =
[13]

¥

3 =
• Perform multiplication outside the brackets.
13

¥

3 = 39

Example 2.4

Problem

Solve the following equation:
7 + [2 (3 + 1) – 1]

¥

2 = ____________

Solution

7 + [2 (4) – 1]

¥

2 =
7 + [8 – 1]

¥

2 =
7 + [7]

¥

2 =
7 + 14 = 21

Example 2.5

Problem

Solve the following equation:
[(12 – 4) ÷ 2] + [4

¥

(5 – 3)] = ____________

Solution

[(8) ÷ 2] + [4

¥

(2)] =
[4] + [8] =
4 + 8 = 12

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© 2004 by CRC Press LLC