Basics of Operator Math Your Name and contact info
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Western RCAP Rural Community Assistance Corporation (916)
447-2854 www.rcac.org Midwest RCAP Midwest Assistance Program (952)
758-4334 www.map-inc.org Southern RCAP Community Resource Group
(479) 443-2700 www.crg.org Northeast RCAP RCAP Solutions (800)
488-1969 www.rcapsolutions.org Great Lakes RCAP WSOS Community
Action Commission (800) 775-9767 www.glrcap.org Southeast RCAP
Southeast Rural Community Assistance Project (866) 928-3731
www.southeastrcap.org RCAP National Office 1701 K St. NW, Suite 700
Washington, DC 20006 (800) 321-7227 www.rcap.org | [email protected]
Rural Community Assistance Partnership Practical solutions for
improving rural communities
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Funded under an EPA grant Training and Technical Assistance for
Small Drinking Water Systems to Achieve and Maintain Compliance
Through Assessing and Addressing Deficiencies Acknowledgement
3
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At the end of this session you should be able to: Convert
between different units of measure. Be able to determine perimeter,
area, and volume. Use the dosing equation and other simplified
equations. Learning objectives
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Pre-Test
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MODULE 1 MATH BASICS
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Convert between liters and gallons, Centigrade to Fahrenheit.
Determine how much water is in a storage tank or a pipe. How much
paint do you need to paint a tank? How much chemical do you need to
add? Why do you need math?
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Review math concepts Working with fractions Understanding and
using conversion factors Percentages Area and volume Tactics for
solving problems Solving of an unknown Working with word problems
Outline
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Basic Concepts
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Math OperationSymbolExample MultiplicationxQ = V x A Q = V A No
spaceQ = VA ( ) ( )Q = (V) (A) DivisionR = D 2 R = D 2 /R = D/2 per
R equals D per 2
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Different ways to write the same multiplication problem = 0.32
591 7 45
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What do you do first when you have a complex equation? (2 + 3)
2 + 3(4 2) = Order of operation
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Please excuse my dear Aunt Sally Please -Parentheses Excuse -
Exponents My - Multiplication Dear -Division Aunt-Add Sally
-Subtract Order of operation
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Please excuse my dear Aunt Sally (2 + 3) 2 + 3 x (4 2) = (5) 2
+ 3 x (2) = 25 + 6 = 31
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= ((5 x 2) + 6) / 8
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Solve 1. (5 x 3) + 2 (3+2) = 2. (4 + 3 x 2) 4 x 2 = 3.
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Last number is 5 or greater round up Last digit is 4 or less
keep the same Round the following numbers 45.5101.491 45.4101.5
Rounding
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Round to one place after the decimal Round to whole number 56.4
56.5456.48888 115.001144.890.51 Approximate using rounding (no
calculator) What is 50.115 x 1.95 = Practice
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MODULE 2 WORKING WITH FRACTIONS
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Review and confirm your understanding of fractions. Verify that
you can Multiply and divide fractions. Set up and work with
fractions from the formula/ conversion table. Learning
Objective
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Fractions
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Divide the numerator by the denominator Converting from
fractions to decimals
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Any number divided by itself equals 1 1 mile = 5280 feet
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Multiply the numerators Multiply the denominators Divide the
numerator by the denominator to convert to decimal Multiplying
fractions
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12 23
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What is ?
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Practice Provide the result in decimal form.
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MODULE 3 WORKING WITH CONVERSION FACTORS
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Use conversion factors for common units in water treatment and
distribution. Learning objectives
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Some conversions are weight to volume/volume to units: 1 gallon
= 8.34 pounds 1 cubic foot = 7.48 gallons 1 PSI = 2.31 feet head 1
gallon = 231 cubic inches 1 cu. ft. = 62.4 pounds 1 acre-feet =
325,851 gallons Some conversions are subsets of larger units: 1 day
= 1,440 minutes1 day = 24 hours 1 gallon = 4 quarts 1 acre = 43,560
square feet 1 mile = 5,280 feet 1 yard = 3 feet 1 liter = 1,000
milliliters1 gram = 1,000 milligram 1 kilo = 1,000 grams1 metric
ton = 1,000 kilos Conversions
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Other conversions change U.S. Standard to Metric: 1 gallon =
3.785 liters 1 kilo = 2.2 pounds 1 ppm = 1 mg/L 1 hectare = 2.47
acres 1 gpg = 17.1 mg/L 1 liter = 1.057 quarts 1 ounce = 29.6
milliliters 1 meter = 3.28 feet Conversions
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1 hour = 60 minutes Using conversion factor 1 acre foot =
326,000 gallons Since both sides of the equations are equivalent
:
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1 hour = 60 minutes If units are equivalent, it doesnt matter
what is on top 1 acre = 43,560 square feet
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Note: Units cancel out. Step 1 Step 2 Step 3 Step 4 How many
minutes are in 4 hours? 4 hours x -- = __ minutes
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How much does 4 gallons of water weigh in pounds? 4 gallons x
-- = __ pounds
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How many seconds are there in 1 hour? In some cases, you may
need multiple conversions
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o C = 5/9 ( o F - 32) o F = (9/5 x o C) + 32 Or o F = (1.8 x o
C) + 32 Temperature conversions Source: CDPHE 2014
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1. How many nickels are there in $2.00? 2. You have 20 pounds
of water. How many gallons of water is this? 3. How many cubic feet
per second (cfs) are there in 4 million gallons per day (MGD)? 4.
What is 4 o C in o F? What is 20 o F in o C? 5. A quarter in
football is 15 minutes. The maximum time between plays is 40
seconds. What would be the minimum numbers of plays that could be
run in a game? You try
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1. Convert 20 gpm to MGD 2. Covert 6000 cf to gallons 3.
Convert 7 days into seconds 4. Covert 120 feet of static head into
psi Additional problems
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MODULE 4 PERCENTAGES
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Percent means parts of 100 Symbol: % Examples Tank is 1/2 full:
50% Tank is 1/4 full: 25% Percentages
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Divide the part by the whole and multiply by 100 To determine
percentages
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Percent (%) divided by 100 will give a decimal value
Percentages can be converted to decimals
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Part per million to percentage 1 ppm = 1 part per million
parts
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If you read 85 meters in a day, and you have 2,100 customers,
what percentage of your customers meters did you read? How many
meters would be 10% of your 2,100 customer meters? whole x percent
(decimal form) = part
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If you read 85 meters in a day, and you have 2,100 customers,
what percentage of customers meters did you read? How many meters
would be 10% of your 2,100 customer meters? 2,100 customers x 0.10
= 210 customers
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How much 65% calcium hypochlorite (HTH) is required to obtain 7
pounds of chlorine? The part is 7 pounds, which is 65% of the whole
Convert the percentage to a decimal Chlorine
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How many pounds of chlorine do you have in one gallon of
Chlorox (7.85% chlorine)? Percentage solution *Note: The weight of
8.34 lb/gallon is for pure water. The weight of Chlorox is slightly
greater.
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1. Write percentages in decimal form a. 10% b. 1% c. 0.1% 2.
Your system has 6,435 valves. Your goal is to turn each valve in
three years. What percentage and how many valves would you need to
turn in one year? What percentage and how many valves would you
need to turn each week? Problems
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MODULE 5 AREA AND VOLUME
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Understand and apply basic geometry Describe and calculate the
perimeter, area, and volume of a rectangle Describe and calculate
the circumference, area, and volume of a cylinder Apply unit
conversion factors Use the Formula/Conversion Table Learning
Objectives
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Examples of area: square feet (ft), square meters (m), acres
and hectares. TRIANGLE CIRCLESQUARE or RECTANGLE 2-Dimensional
Shapes
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CYLINDERCUBECONE Examples of volume: cubic feet (ft), cubic
meters (m), and acre feet (af). 3-Dimensional Shapes
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Feet may be abbreviated ft ft 2 = ft x ft (area) = square feet
ft 3 = ft x ft x ft (volume) = cubic feet Math note
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Width Dimension #2 Length Dimension #1 Parts of a Square or
Rectangle
What would be the perimeter? If the length was 10 feet and the
width was 20 feet? Perimeter = Length + Width + Length + Width
Perimeter = 2 x (Length + Width) Width Dimension #2 Length
Dimension #1
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What is the perimeter of a football field? The length of a
football field is 120 yards The width of a football field is 53.3
yards
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2-Dimensional problems How much paint do I need to cover the
outside of a tank? What is the surface area of a contact basin?
What is the flow rate? (velocity times area) Q = VA Area
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Area = (Length)(Width) Width Length Area=Length times
width
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What is the area if: Length is 4 feet and width is 10
feet?
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Volume of a cube = (Length)(Width)(Height) Volume = Length x
Width x Height (or Depth) Length Width Depth or height Volume
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What would be the volume if: Length = 2 feet Height = 4 feet
Width = 4 feet 2 ft 4 ft V = Length x Width x Height V= 2 ft x 4 ft
x 4 ft V = 16 ft 3
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Take the Rubiks cube provided. Determine: Perimeter Area Volume
(Note: Units will be in blocks) Then add a second cube to make a
rectangle. Determine perimeter, area, and volume. Activity
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What would be the volume of a basin that is 10 feet wide, 20
feet long, and 10 feet deep? How many gallons would the basin hold
if full? Problem
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Radius Diameter Circumference Parts of a Circle
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The radius is equal to half the diameter Diameter = 2 x Radius
Radius = Diameter / 2
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Circumference Question: How would you write this equation if
you had the radius measurement? Circumference = (diameter ) (pi) is
equal to 3.14 Circumference = 3.14 (diameter)
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Circumference Circumference = 2 x x radius
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Circumference = diameter x What would be the circumference if
the diameter was 10 feet?
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3.5 MG storage tank (20 feet tall) has a radius of 161 ft. What
is the circumference? What would be the circumference of:
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Area Area of a circle= (radius) = 3.14 x radius x radius or
Area of a circle= 0.785 (diameter 2 ) = 0.785 x diameter x diameter
Area
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What would be the area of a circle that has a radius of 10
feet? What would be the area of a circle that has a diameter of 10
feet? Area = (radius) or Area = 0.785 (diameter 2 )
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Depth or Height Area Length Volume of a Cylinder
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Volume = (0.785) (diameter 2 ) (Height) Volume = ( ) (radius 2
) (Height) Volume of a Cylinder Depth or Height
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What would be the volume (in gallons) of a storage tank that
has a diameter of 10 feet and a height of 5 feet? Volume = (0.785)
(diameter 2 ) (Height) Depth or Height
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You have a storage tank that has a diameter of 100 feet and a
height of 10 feet. How much fence would you need to go around the
tank (assume the fence is 10 feet away from the tank)? How paint
would you need to paint the outside of the tank if one can of paint
covers 100 square feet? How much water does the tank hold (in
gallons) if completely full? If 75% full? Problem
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MODULE 6 WORKING WITH EQUATIONS
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Get the unknown you are trying to solve for alone on one side
of the equation. Solving for an unknown Dosing equation Flow rate Q
= A x V
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If you know flow rate and area, find the velocity: Solving for
an unknown 1. Start by writing the equation 2. Get the letter you
need alone Flow rate = Area x Velocity Q = A x V
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If you know the amps and ohms and are trying to find volts:
Solving for an unknown 1. Write down the equation 2. Multiple both
sides of the equation by ohms.
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Important point: Do the same thing to both sides of the
equation. Solving for an unknown
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What happens if what you are trying to solve for is in the
denominator?
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Dosing equation
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If you know feed rate and MGD and wanted to determine the dose,
divide both sides of the equation by (capacity, MGD)(8.34
lbs/gal).
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To find the quantity above the horizontal line, multiply the
pie wedges below the line together. To solve for one of the pie
wedges below the horizontal line, cover that pie wedge, then divide
the remaining pie wedge(s) into the quantity above the horizontal
line. Given units must match the units shown in the pie wheel.
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1. You know force and pressure. Find the area. 2. You know the
detection time and volume of your basin. Find the flow rate. 3. You
know the area. Find the diameter. 4. You know feed rate and dosage.
Determine the Capacity in MGD. Your Turn Force = (Pressure) (Area)
Area of a circle = 0.785 (diameter) 2
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1.Make a drawing of the information. 2.Place data on the
drawing. 3.Write down What do I need to determine? 4.Write down any
equations that you are going to need. 5.Make conversions if
necessary. 6.Fill in the data in the equation. 7.Make the
calculation and write down the answer. 8.Ask yourself, does this
make sense? Word Problems
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A blue tank at 500 feet above sea level is 10 feet high and has
a diameter of 20 feet. The tank is half full. What is the volume of
the tank? Note: Watch out for information you dont need
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A blue tank at 500 feet above sea level is 10 feet high and has
a diameter of 20 feet. The tank is half full. What is the volume of
the tank? Note: Watch out for information you dont need
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Problem 1: A new section of 8 inch diameter pipe is to be
filled with water for testing. If the length of the pipe is 3,500
feet, how many gallons of water will be needed to fill the
pipe?
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Problem 2: A 12 inch water main has to be disinfected to 50
ppm. The main is 4,000 feet long and has 1 valve installed every
500 feet for isolation. How many pounds of chlorine will be needed
to disinfect the entire main?
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Problem 3: A cylindrical cistern with a radius of 5 feet and a
height of 10 feet is filled with water. If 3 pounds of chemical is
dissolved in the water, what will be the dosage in milligrams per
liter? a. 112 mg/L b. 75 mg/L c. 61 mg/L d. 27 mg/L
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Problem 4: A settling basin is 120 feet long, 20 feet wide, and
has a water depth of 15 feet. What is the detention time, in hours,
in the basin when the flow is 9 MGD?
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Problem 5: A chlorine dosage of 0.35 mg/L is applied at the
pump station preceding the plant. The 36 inch force main is 23,000
feet long and the flow is 5,560 gpm. What is the theoretical
contact time for the chlorine prior to entering the plant? a. 1
hour 48 minutes b. 2 hours 24 minutes c. 3 hours 38 minutes d. 4
hours 4 minutes
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ABC Formula and Conversion Factors Sheet
http://www.abccert.org/pdf_docs/ABCWTFTC090613_AB C-C2EPBrand.pdf
AWWA www.awwa.orgwww.awwa.org Math for Distribution System
Operators, 2007 Math for Water Treatment Operators, 2007 Basic
Science Concepts and Applications, 4 th ed. 2010 Rural Community
Assistance Corporation, www.rcac.org Resources
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Practice
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MODULE 7 PRACTICE PROBLEMS
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Post-test Post-test will be handed out. 10 questions try to
finish in 12 minutes. Use formula sheet and a calculator.