Math Fundamentals for Operators in Training

Math Fundamentals for Operators in

Training Course # 1009 or 1009-V

Updated 1‐2021

State of Tennessee

Math Fundamentals for Operators in Training August 10-12, 2021 Course #1009 or 1009-V

Instructor: Amanda Carter Fleming Training Center [email protected]

Wednesday

8:30 Registration and Welcome 8:45 Order of Operations 11:00 Lunch 12:00 Solving Equations

Thursday

8:30 Review Day 1 8:45 Ratios 10:00 Proportions 1100 Lunch 12:00 Dimensional Analysis

Friday

8:30 Exam Review & Practice 11:15 Lunch 12:30 Exam

Phone: 615-898-6507 Fax: 615-898-8064 E-mail: [email protected]

Fleming Training Center 2022 Blanton Dr.

Murfreesboro, TN 37129

Table of Contents

Section 1 Order of Operations 3

Section 2 Solving Equations 19

Section 3 Ratios 47

Section 4 Proportions 61

Section 5 Dimensional Analysis 79

Section 6 Solutions 103

2

Ord

Sec

der of

ction

f Oper

1

rationns

3

ORDER OF OPERATIONSBasic Math for Operators-in-Training

WHAT IS ORDER OF OPERATIONS?

• A set way to solve an calculation8 16 4

• Which way is the correct way?

8 16 424 4

6

8 16 48 4

12

Section 1 TDEC - Fleming Training Center

Order of Operations4

PEMDAS

• Parenthesis• Exponents• Multiplication/Division• Addition/Subtraction

8 16 424 4

6

8 16 48 4

12

EXAMPLE7 3 4 2 5 6

7 3 4 2 5 6

21 4 2 5 6

84 2 5 6

42 5 6

42 30

12

ParenthesisExponentsMultiplication/DivisionAddition/Subtraction

Since Multiplication and Division are on the same “level,” work left to right

Skip the subtraction because it is on the next “level”

TDEC - Fleming Training Center Section 1

Order of Operations 5

Math Fundamentals

Order of Operations (1)

1. 3(6 + 7)

2. 5 × 3 × 2

3. 72 ÷ 9 + 7

4. 2 + 7 × 5

5. 20 ÷ [4 − (10 − 8)]

6. 40 ÷ 4 − (5 − 3)

7. 9 + 9 + 6 – 5



8. (5 + 16) ÷ 7 – 2

9. 3

10. 2 7

11. 8 5 9

Answers

1. 39 7. 19

2. 30 8. 1

3. 15 9. 6

4. 37 10. 12

5. 10 11. 10

6. 8



Math Fundamentals

Order of Operations (2)

Order of Operations

1. 8 + 96 ÷ 2

2. 42 ÷ 6 – 3

3. 42 – 15 ÷ 5

4. 8 × 6 + 5

5. 45 + 6 × 15

6. 91 ÷ 7 – 9



7. 848 ‐ 2 × 67

8. 8 + 9 – 2 × 3

9. 36 ÷ 3 – 4 × 3 + 24 ÷ 2

10. 55 + 8 × 2 ÷ 2 + 34 – 17 + 19

11. 44 – (18 ÷ 9)

12. (73 + 4) + 14 × 6

13. (9 + 5) x (9 + 5)



14. [(15 + 18) – 33] × 6

15. (10 + 2) + 15 x 3

16. [16 – (14 ÷ 7)] + 82

17. (10 x 7 ‐ 82) + 10

18. (69 – 32) – (0 + 5)

19. 4 x (13 – 3) + 42

20. (10 – 5)2 + [(12 + 6) x 52]



Order of Operations (2) Answers

1. 56

2. 4

3. 39

4. 53

5. 135

6. 4

7. 714

8. 11

9. 12

10. 99

11. 42

12. 161

13. 196

14. 0

15. 57

16. 96

17. 16

18. 55

19. 56

20. 475



Math Fundamentals Order of Operations (3)

1. (14 + 2) x 8 – 4

2. 4 x 3 + (3 + 6)

3. (11 + 5) + 10 x 5

4. (8 + 27 – 5) x 6

5. (10 + 3) x (7 – 5)

6. (12 + 7) x 9 + 2



7. 2 x 3 + (9 + 6)

8. (9 + 3) + 15 x 5

9. (10 + 20 – 6) x 6

10. (14 + 3) x (12 + 5)

11. (14 + (15 - 3)) x 7

12. 12 + ((17 + 4) + 2)



13. (7 + (18 – 3 + 2))

14. ((11 + 4) + 4) + 8

15. (10 + (18 – 3)) x 7

16. 2 + ((13 + 5) + 6)

17. ((10 – 2) x 5) – 10

18. 13 + (10 + (11 – 5))



19. 15 + (5 x (17 – 6))

20. 8 + (14 – 7 – 6))

21. 18 + (5 x (11 – 4)²)

22. ((14 – 2) +14 – 2)²)

23. 14 + (5 x (4 + 3)²)

24. 18 + ((10 + 3) + 2²)



25. (4² + (10 - 2 + 4²))

26. (6² + (20 – 5 + 3²))

27. 18 + ((11 + 7) + 3²)

28. ((5 + 4) ² x 2) + 2²

29. ((18 + 2) + (20 – 4)²)

30. ((10 – 4)² + 6) - 4²



Order of Operations (3) Answers:

1) 124

2) 21

3) 66

4) 180

5) 26

6) 173

7) 21

8) 87

9) 144

10) 289

11) 182

12) 35

13) 24

14) 27

15) 175

16) 26

17) 30

18) 29

19) 70

20) 9

21) 263

22) 576

23) 259

24) 35

25) 40

26) 60

27) 45

28) 166

29) 276

30) 26



18

So

Sec

olving

ction

g Equa

2

ationss

19

Introduction to Equations

Basic Algebra for Operators‐in‐Training

Evaluating Expressions• An expression is a statement of value

– A statement of some type of quantity

10 + 5 370 ‐ x

• The expression gives the area of a

triangle where b is the base of the triangle and h is the height

– If b is 7 cm and h is 4 cm, then we can evaluate the expression

𝑏 ℎ 14 𝑐𝑚


Solving Equations20


• An equation is when expressions are set to be equal to each other

235

28

2 7=143 4 = 0.75


• Equations are “balanced”

– The quantity on one side of the equal sign is equivalent to the quantity on the other side of the equal sign

25 4 20 5100 100


Solving Equations 21


• Equations are “balanced”

– It is vital to maintain that balance. 25 4 20 5

100 115– To maintain that balance, we whatever we do to

one side of the equation, we must do the same to the other side

25 4 20 5115 115

15

1515

Variables

• So far, we have known all the numbers that we are working with

• In algebra, we start to see and work with variables

– A variable is a symbol that represents different varying values

𝑥 5𝐼𝑓 𝑥 1,

𝑡ℎ𝑒𝑛 𝑥 5 6


Solving Equations22


• But what if we didn’t know what one of those were

25 𝑦 20 5

• We would have to solve the equation to find the value of the unknown (y)

– This is accomplished by getting the unknown by itself on one side of the equal sign.


25 𝑦 20 5• 25 is on the same side of the equal sign as the

unknown

– To get rid of it, we have to perform the opposite function

25 𝑦25

1 𝑦 20 5𝑦 100

• Is this answer correct?




• If y=100, and 25 𝑦 20 5– Does the equation balance out with the new

information?

25 100 20 52500 100

• The equation is no longer balanced so it is untrue


25 𝑦 20 5• How do we solve for the unknown without

losing the balance?

– What we do to one side of the equal sign, we must do to the other side

25 𝑦25

20 525

1 𝑦10025

𝑦 4

• Now, is the answer correct?


Solving Equations24


• If y=4, and 25 𝑦 20 5– Does the equation balance out with the new

information?

25 4 20 5100 100

• The equation is balanced

Solving for X

• Rules for solving for X:

– X on top

– X alone

• Questions to ask:

– Is X alone?

– What is keeping X from being alone?

– What is it doing to X?

– What do we have to do to get rid of it?



Example

𝑥 7 107 7

𝑥 0 10 7

𝑥 17

Example 1

730𝑥

3847

730𝑥

3847

38471

38471

730𝑥

38473847

1

3847 730 𝑥

2,808,310 𝑥

What you do to one side of the equation, must be done to the other side.


Solving Equations26

Example 20.5=

(165)(3)(8.34)

x

0.5 4128.3𝑥

0.54128.3𝑥

𝑥1

𝑥1

0.54128.3𝑥

𝑥1

𝑥 0.5 4128.3

𝑥 0.5

0.54128.3

0.5

𝑥4128.3

0.5

𝑥 8256.6

What you do to one side of the equation, must be done to the other side.

Simplify

Solving for X

• When solving for x involving addition and subtraction, the balance of the equation must still remain.

– What you do to one side you must do to the other



Example 3

115 105 80 𝑥 386

300 𝑥 386

300 𝑥 300 386 300

𝑥 86

Example 4

17 23 7 𝑥 38

47 𝑥 38

47 𝑥 𝑥 38 𝑥

47 38 𝑥

47 38 38 𝑥 38

9 𝑥

Step 1. Simplify

Step 2. Make x positive


Solving Equations28

Solving for X2

• Follow same procedure as solving for X

• Then take the square root

𝑥 15,625

𝑥 15,625

𝑥 125

Example 50.785 𝑥 2826

0.785 𝑥0.785

28260.785

𝑥28260.785

𝑥 3600

𝑥 3600

𝑥 60



Math Fundamentals Solving for the Unknown (1)

1) Clarence has 8 small spoons and 7 big spoons. Write an expression that shows how many spoons Clarence has.

2) Oliver earned 115 extra credit points. Dustin earned 57 fewer extra credit points than Oliver. Write an expression that shows how many extra credit points Dustin earned.

3) Andy had 98 DVDs. His mother bought him X more DVDs. Write an expression that shows how many DVDs Andy has now.

4) Terrell bought 6 boxes of granola bars. There are X granola bars in each box. Write an expression that shows how many granola bars Terrell bought.

5) The jazz band has 5 members. To raise money for a tour, each member sold X raffle tickets. Choose the expression that shows the number of raffle tickets sold.


Solving Equations30

Solve for the unknown value.

Addition

6) 3 + g = 10

7) x + 2 = 3

8) x + 15 = 19 + 22

9) 7 + 10 + x + 7 + 9 = 41

10) x + 93 = 165

Subtraction

11) 3 = k − 2

12) x – 2 = 9

13) x – 93 = 65

14) 9.5 – x = 8.7

15) 115 = x – 7.5



Multiplication

16) 10 = (2)(w)

17) (5)(m) = 10

18) 48 = (6)(m)

19) 8.1 = (3)(x)(1.5)

20) (0.785)(0.33)(0.33)(x) = 0.49

Division

21) 12

22) 6

23) 50

24) 56.5 .


Solving Equations32

25) 114 . .

.

Assorted

26) (5)(x) + 9 = 9

27) 7 + (6)(x) = 37

28) 4

29) 2

30) 7 𝑓 16 12



Answers

1) 8+7

2) 115‐57

3) 98+x

4) (6)(x)

5) (5)(x)

6) g=7

7) x=1

8) x=26

9) x=8

10) x=72

11) 5=k

12) x=11

13) x=158

14) 0.8=x

15) 122.5=x

16) 5=w

17) m=2

18) 8=m

19) 1.8=x

20) x=5.73

21) 96=t

22) 0.33=e

23) 2=x

24) x=8.06

25) x=0.005

26) x=0

27) x=5

28) m=41

29) 13=x

30) f=4


Solving Equations34

Solving for the Unknown (2)

Basics – finding x 1. 7 + 10 + x + 7 + 9 = 41 2. 16=(2)(x) 3. 142 = (3)(x)+13 4. 10.1 = 9.5 + x 5. x + 15 = 19 + 22 6. 16 = (2)(x)

7. 211 = (15)(x)(0.785) 8. (0.785)(2.5)(2.5)(x) = 5151.56 9. 100 = 50 x 10. 233 = 44 x



11. 56.5 = 3800 (x)(8.34)

12. 10 = x 4 13. 940 = x (0.785)(90)(90)

14. x = (165)(3)(8.34) 0.5

15. 114 = (230)(1.15)(8.34) (0.785)(70)(70)(x)

16. 2 = x 180

17. 46 = (105)(x)(8.34) (0.785)(100)(100)(4)

18. 2.4 = (0.785)(5)(5)(4)(7.48) x


Solving Equations36

19. 19,747 = (20)(12)(x)(7.48) 20. (15)(12)(1.25)(7.48) = 337 x

21. x = 213 (4.5)(8.34)

22. x = 2.4 246

23. 6 = (x)(0.18)(8.34) (65)(1.3)(8.34)

24. (3000)(3.6)(8.34) = 23.4 (0.785)(x)

25. 109 = x (0.785)(80)(80)

26. (x)(3.7)(8.34) = 3620



27. 2.5 = 1,270,000 x 28. 0.59 = (170)(2.42)(8.34) (1980)(x)(8.34) 29. 142 = (2)(x) + 13 30. (3.5)(x) – 62 = 560

Finding x2

31. x2 = 100

32. (2)(x2) = 288 33. 942 = (0.785)(x2)(12) 34. 6358.5 = (0.785)(x2) 35. 835 = 4,200,000 (0.785)(x2) 36. 920 = 3,312,000 x2


Solving Equations38

37. 23.9 = (3650)(3.95)(8.34) (0.785)(x2) 38. (0.785)(D2) = 5024 39. (x2)(10)(7.48) = 10,771.2 40. 51 = 64,000 (0.785)(D2)

41. (0.785)(D2) = 0.54 42. 2.1 = (0.785)(D2)(15)(7.48) (0.785)(80)(80)



Answers 1) 8 2) 8 3) 43 4) 0.6 5) 26 6) 8 7) 17.92 8) 1050 9) 2 10) 5.3 11) 8.06 12) 40 13) 5,976,990 14) 8256.6 15) 0.005 16) 360 17) 1649.42 18) 244.66 19) 11.0 20) 4.99 21) 7993.89

22) 590.4 23) 2816.67 24) 4903.48 25) 547,616 26) 117.31 27) 508,000 28) 0.35 29) 64.5 30) 177.7149 31) 10 32) 12 33) 10 34) 90 35) 80.05 36) 60 37) 80.06 38) 80 39) 12 40) 39.98 41) 0.83 42) 10.94


Solving Equations40

Math Fundamentals

Solving for a Variable

1. Solve for width. Area length width

2. Solve for height.

Area base height

2

3. Solve for diameter. Area 0.785 D

4. Solve for time in minutes.

lbday

grams 60 min hr⁄ 24 hr day⁄454 gram lb⁄ min

5. Solve for diameter. feet π Diameter



6. Solve for flow.

feed rate dose flow 8.34 lb gal⁄

% purity

7. Solve for area. flow rate area velocity

8. Solve for width. flow rate length width velocity

9. Solve for distance.

flow rate 0.785 Diameterdistance

time

10. Solve for pressure. force pressure area

11. Solve for flow.

bhp flow head

3960 % pump


Solving Equations42

12. Solve for % motor efficiency.

mhp flow head

3960 % pump % motor

13. Solve for head.

whp flow head

3960

14. Solve for dose.

mass dose concentration 8.34 lb gal⁄

15. Solve for substance weight.

specific gravity substance weight

water weight

16. Solve for volume2. concentration volume concentration volume

17. Solve for time.

velocity distance

time



18. Solve for diameter. volume 1

3 0.785 D h

19. Solve for height. volume 0.785 D h

20. Solve for width. volume length width height


Solving Equations44

Answers

1. width

2. height

3. .

D

4. min / /

/ ⁄

5. D

6. %

.flow

7. area

8. width

9. distance .

10. pressure

11. % flow

12. % motor%

13. head

14. . ⁄

dose

15. water weight specific gravity substance weight

16. volume

17. time

18. .

D

19. .

height

20. width



46

Section 3

Ratios

47

Three bank robbers have a bag of money that they intend to split evenly in the morning. After they are all asleep one of the robbers wakes up and decides that he doesn’t trust the other two, so he takes his third cut and goes back to sleep. A little later, the second robber wakes up and decides that he doesn’t trust the other two, so he takes his third cut and goes back to sleep. A little later, the third robber wakes up and does the same. In the morning, they all wake up and notice that the bag looks smaller than they remember, but no one says anything. The split what’s there evenly three ways. What proportion/percentage of the money stolen did each robber walk away with?

Ratios

How much of the grids are shaded red? Write each in at least 2 ways.


Ratios48

Ratios


Ratios



Ratios 49

Ratios


Ratios

Shade 3/10 of this grid.


Ratios50

Ratios


Ratios



Ratios 51

Ratios

What percent of the months start with a “J”?

3/12 = 0.25 = 25%

Ratios

What percent of the months start with a “J”?

3/12 = 0.25 = 25%

What is the ratio of months that start with “J” to all the months?

3/12 = 0.25

You might also see it written as3:12 = 1:4


Ratios52

Ratios

Simplifying Ratios

Let’s think of percentages. If 25% of a group of people are married, then you might say that 25 out of every 100 are married.

You could say that 1 out of every 4 are married.

Ratios – with notation

Simplifying Ratios – What does that look like mathematically?

Or, 1:4


Ratios 53

Ratios

Some ratios are called something else.

For example, the ratio of miles to hours is called your speed.

MPH is miles per hour

Miles/Hour

Ratios

Ratio are not necessarily fractions. Fractions are a comparison of parts to whole. For example, in the grid below, 2/4 or ½ of the grid is red. That is a comparison of red to white (ratio), but it is also a fraction, because it is comparison of red to whole grid.

MPH is not a fraction. It’s a ratio.


Ratios54

Ratios

If you went 500 miles in 10 hours, what is the ratio of miles to hours?

What’s another way to write that?

Ratios

If a 10 pound bag of potatoes is $5.49, what is the ratio of cost to pounds?

What is the ratio of pounds to cost?

What if I wanted to know the cost per one pound?


Ratios 55

Math Fundamentals

Reducing Ratios (1)

1. 14 : 12 ___________

2. 35 : 14 ___________

3. 36 : 81 ___________

4. 6 : 36 ___________

5. 45 : 72 ___________

6. 25 : 15 ___________

7. 28 : 35 ___________

8. 6 : 12 ___________

9. 9 : 27 ___________

10. 70 : 21 __________

11. 10 : 15 ___________

12. 30 : 42 ___________

13. 24 : 42 ___________

14. 15 : 27 ___________

15. 28 : 36 ___________

16. 80 : 70 ___________


Ratios56

17. 15 : 3 ___________

18. 8 : 6 ___________

19. 16 : 6 ___________

20. 20 : 2 ___________

Answers:

1) 7 : 6

2) 5 : 2

3) 4 : 9

4) 1 : 6

5) 5 : 8

6) 5 : 3

7) 4 : 5

8) 1 : 2

9) 1 : 3

10) 10 : 3

11) 2 : 3

12) 5 : 7

13) 4 : 7

14) 5 : 9

15) 7 : 9

16) 8 : 7

17) 5 : 1

18) 4 : 3

19) 8 : 3

20) 10 : 1


Ratios 57

Math Fundamentals

Ratios (2)

Ratios ‐ Answer in the reduced form.

1. For every 7 hamburgers sold at the malt shop there are 3 hotdogs

sold. What is the ratio of hotdogs sold to hamburgers sold?

2. A group of preschoolers has 15 boys and 4 girls. What is the ratio

of boys to all children?

3. A herd of 36 horses has 12 white and the rest are black horses.

What is the ratio of black horses to white horses?

4. Brayden drew 27 hearts, 1 star, and 5 circles. What is the ratio of

stars to all shapes?



6. A jar contains 36 marbles, of which 10 are blue, 17 are red, and

the rest are green. What is the ratio of blue marbles to green

marbles?


Ratios58




What is the ratio of white to black horses?

9. A pattern has 5 blue triangles to every 80 yellow triangles. What is

the ratio of blue triangles to all triangles?

10. Noah drew 22 hearts and 76 circles. What is the ratio of

circles to all shapes?

11. A pattern has 14 blue triangles to every 18 yellow triangles.

What is the ratio of yellow triangles to blue triangles?

12. A pattern has 6 blue triangles to every 42 yellow triangles.

What is the ratio of yellow triangles to blue triangles?

13. A group of preschoolers has 63 boys and 27 girls. What is the

ratio of boys to all children?


Ratios 59

Answers

Ratios

1) 3:7

2) 15:19

3) 2:1

4) 1:33

5) 29:2

6) 10:9

7) 13:3

8) 3:10

9) 1:17

10) 38:49

11) 9:7

12) 7:1

13) 7:10


Ratios60

Section 4

Proportions

61

Proportions

1009 Math Fundamentals forOperators‐in‐Training

Proportions

• A proportion is two ratios that have been set equal to each other

– a proportion is an equation that can be solved

25100

14


Proportions62

Solving Proportions

• Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross‐multiplying, and solving the resulting equation

𝑥100

14

Solving Proportions

𝑥100

14

𝑥 4 100 1

𝑥 44

100 14

𝑥100

4

𝑥 25

Cross multiply

Solve for x


Proportions 63

Example

• There are 16 ducks and 9 geese in a certain park. Suppose that there are 192 ducks. How many geese are there?

We can determine the ratio of ducks to geese:

16 ducks to 9 geese

We need to turn the question into an equation or proportion

16 𝑑𝑢𝑐𝑘𝑠9 𝑔𝑒𝑒𝑠𝑒

192 𝑑𝑢𝑐𝑘𝑠𝑥 𝑔𝑒𝑒𝑠𝑒

Example

• Be sure to set up equation so that the units match on both sides of the equal sign

𝑑𝑢𝑐𝑘𝑠𝑔𝑒𝑒𝑠𝑒

𝑑𝑢𝑐𝑘𝑠𝑔𝑒𝑒𝑠𝑒


Proportions64

Example

16 𝑑𝑢𝑐𝑘𝑠9 𝑔𝑒𝑒𝑠𝑒

192 𝑑𝑢𝑐𝑘𝑠𝑥 𝑔𝑒𝑒𝑠𝑒

16 𝑥 9 192

16 𝑥16

9 19216

𝑥9 192

16

𝑥1728

16

𝑥 108

Cross multiply

There are 16 ducks and 9 geese in a park. If there are 192 ducks, how many geese are there?


Proportions 65

Math Fundamentals

Solving Proportions (1)

1) _____ : 15 = 36 : 45

2) 8 : 12 = _____ : 3

3) 27 : _____ = 9 : 18

4) 24 : 4 = 6 : _____

5) _____ : 6 = 3 : 2

6) 5 : _____ = 20 : 8

7) 25 : 35 = 40 : _____

8) 4 : 12 = 6 : _____

9) 14 : 42 = 2 : _____

10) 10 : _____ = 2 : 8

11) 8 : _____ = 4 : 5

12) _____ : 28 = 6 : 12

13) 3 : 4 = 18 : _____

14) 1 : 2 = _____ : 4

15) 72 : 54 = _____ : 6

16) _____ : 64 = 2 : 8

17) _____ : 2 = 27 : 18

18) 21 : 35 = _____ : 10

19) 8 : 3 = _____ : 21

20) 6 : 3 = _____ : 6


Proportions66

Answers

1) 12 2) 2 3) 54 4) 1 5) 9 6) 2 7) 56 8) 18 9) 6 10) 40

11) 10 12) 14 13) 24 14) 2 15) 8 16) 16 17) 3 18) 6 19) 56 20) 12


Proportions 67

Math Fundamentals

Proportions (2)

1. In a certain class, the ratio of passing grades to failing grades is 7 to 5. How

many of the 36 students failed the course?

2. Shares of stock represent how much of a company a person owns. Puff

incorporated is owned by Peter, Paul and Mary. Peter owns 4,050 shares. Paul

owns 2,510 shares and Mary owns 4,200 shares. Suppose the company made a

profit this year of $1,500,000. If each shareholder gets a proportion of the

total profit that is equal to the proportion of the shares that they own, how

much money does Mary receive? (round to the nearest penny)

3. Ben the camel drinks tea (so classy!). He drinks 350 liters of tea every 2 days.

How many liters of tea does Ben drink every 6 days?

4. According to Greg, perfect cherry pies have a ratio of 240 cherries to 3 pies.

How many cherries does Greg need to make 9 perfect cherry pies?


Proportions68

5. You are throwing a party and you need 5 liters of soda for every 12 guests. If

you have 36 guests, how many liters of soda do you need?

6. You can buy 3 apples at the Quick Market for $1.14. You can buy 5 of the same

apples at Stop and Save for $2.35. Which place is the better buy?

7. A jet travels 410 miles in 5 hours. At this rate, how far could the jet fly in 14

hours? What is the rate of speed of the jet in miles per hour?

8. An ice cream factory makes 350 quarts of ice cream in 5 hours. How many

quarts could be made in 12 hours?

9. A test of a new car results in 580 miles on 20 gallons of gas. How far could you

drive on 40 gallons of gas?


Proportions 69

Answers

1) 15 failed

2) $585,501.86

3) 1050 liters

4) 720 cherries

5) 15 liters

6) Quick Mkt

7) 1148 miles; 82 mph

8) 840 qts

9) 1160 miles


Proportions70

Math Fundamentals

Proportions (3)

1. There are a total of 63 bikes. If the ratio of blue bikes to black bikes is 4 to 5, how many of the bikes are blue?

2. Paul can walk 15 steps in 5 minutes. How long does it take Paul to walk 75 steps at the same speed?

3. Candy is at the balloon shop and sees that 10 balloons cost $0.15. He needs 50 balloons to decorate his room. How much will 50 balloons cost?

4. It takes 8 people to pull a 16 ton truck. How many people would it take to pull a 60 ton truck?

5. If a globe rotates through 150 degrees in 5 seconds, how many degrees does it turn in 30 seconds?

6. If the ratio of white pens to green pens is 2 to 8 and there are a total of 30 pens, how many pens are white?

7. Giles is searching for a sock and discovers that he has 10 socks for every 5 pairs of shoes. If he has 20 socks, how many pairs of shoes does he have?

8. The sum of two numbers is 150. The ratio of the same two numbers is 3:2. Find the bigger number.

9. If the ratio of pink flower to blue flower is 3 to 6 and there are total 72 flowers, how many of them are pink?


Proportions 71

10. There are a total of 18 chairs. If the ratio of the red chairs to brown chairs is 2 to 4, how many of them are red?

11. Rock can read 10 books in 30 minutes. How long does it take Rock to read 15 books, if the speed is consistent?

12. Ricky is at the bakery shop when he sees that 8 pastries cost $160. He needs 16 pastries. How much will 16 pastries cost?

13. It takes 10 people to pull a 50 ton bus. How many people would it take to pull a 100 ton bus?

14. If a ball rotates 110 degrees in 8 seconds, how many degrees does it rotate in 32 seconds?

15. If the ratio of red roses to yellow roses is 4 to 6 and there are a total of 50 roses, how many of them are yellow?

16. Freddy is searching for a shirt and discovers that he has 12 shirts for every 6 pair of jeans. If he has 18 shirts, how many pairs of jeans does he have?

17. The sum of two numbers is 200 and the two numbers are in a ratio of 4:6. Find the larger number.

18. If the ratio of the purple flowers to black flowers is 4 to 8 and there are a total of 108 flowers, how many of the flowers are purple?


Proportions72

19. There is a total of 25 bicycles. If the ratio of gray bicycles to black bicycles is 6 to 9, how many of them are black?

20. Ritz can eat 8 apples in 15 minutes. How long does it take Ritz to eat 16 apples at the same rate?

21. Tom is at McDonalds and he sees that 2 burgers cost $40. He needs 12 burgers. How much will 12 burgers cost?

22. It takes 12 people to pull 30 tons of goods. How many people would it take to pull 60 tons of goods?

23. If a tire rotates through 250 degrees in 15 seconds, how many degrees does it rotate in 45 seconds?

24. If the ratio of purple bikes to red bikes is 8 to 12 and there are a total of 100 bikes, how many of them are purple?

25. Andrew is searching for a cup and discovers that he has 20 plates for every 5 pairs of cups. If he has 40 plates, how many pairs of cups does he have?

26. The sum of two numbers is 80. The ratio of those two numbers is 3:5. Find the larger number.


Proportions 73

27. If the ratio of red hair bands to green hair bands is 5 to 9 with a total of 70 hair bands, how many of them are green?

28. There are a total of 100 balloons. If the ratio of yellow balloons to blue balloons is 8 to 12, how many of them are yellow?

29. Furry can eat 10 mangoes in 5 minutes. How long does it take Furry to eat 18 mangoes at the same speed?

30. Harry is at the Pizza Hut and he sees that 5 pizzas cost $300. He needs 15 pizzas. How much do 15 pizzas cost?

31. It takes 24 people to pull a 50 ton iron rod. How many people would it take to pull a 150 ton iron rod?

32. If a coin rotates through 160 degrees in 6 seconds, how many degrees does it rotate in 60 seconds?

33. If the ratio of blue shirts to green shirts is 5 to 12 with a total of 340 shirts, how many of the shirts are blue?


Proportions74

34. Rex is searching for bread and discovers that he has 40 buns for every 10 cubes of cheese. If he has 80 buns, how many cubes of cheese does he have?

35. The sum of two numbers is 500 with a ratio of 15:10. Find the larger number.

36. If the ratio of silver nail paints to golden nail paints is 4 to 2 and there are total 60 nail paints, how many of them are golden?

37. There are a total of 18 frocks. If the ratio of purple frocks to white frocks is 2 to 4, how many of them are white?

38. Johnny can eat 5 chocolates in 2 minutes. How long does it take Johnny to eat 10 chocolates at the same rate?

39. Gerry is at Dominos and sees that 2 choco‐lava cakes cost $50. He needs 6 choco‐lava cakes. How much will those 6 choco‐lava cakes cost?

40. It takes 16 people to pull a 30 ton cable wire. How many people would it take to pull a 60 ton cable wire?


Proportions 75

41. If a marble rotates through 180 degrees in 9 seconds, how many degrees does it rotate in 18 seconds?

42. If the ratio of black jeans to blue jeans is 6 to 18 with a total of 240 jeans, how many of them are blue?

43. Rex is searching for shoes and discovers that he has 25 pairs of shoes for every 15 pairs of socks. If he has 50 pairs of shoes, how many socks does he have?

44. The sum of two numbers is 300 and the two numbers have a ratio of 20:10. Find the larger number.

45. If the ratio of red roses to pink roses is 5 to 4 and there are a total of 45 roses, how many of them are red?


Proportions76

Answers

1) 28 19) 15 37) 12

2) 25 20) 30 38) 4

3) 0.75 21) 240 39) 150

4) 30 22) 24 40) 32

5) 900 23) 750 41) 360

6) 6 24) 40 42) 180

7) 10 25) 10 43) 30

8) 90 26) 50 44) 200

9) 24 27) 45 45) 25

10) 6 28) 40

11) 45 29) 9

12) 320 30) 900

13) 20 31) 72

14) 440 32) 1600

15) 30 33) 100

16) 9 34) 20

17) 120 35) 300

18) 36 36) 20


Proportions 77

78

Section 5

Dimensional Analysis

79

DIMENSIONAL ANALYSIS

MATHEMATICS MANUAL FOR WATER AND WASTEWATER TREATMENT PLANT OPERATORS

BY FRANK R. SPELLMAN


• Used to check if a problem is set up correctly

• Work with the units of measure, not the numbers

• Step 1:

• Express fraction in a vertical format

𝑔𝑎𝑙 𝑓𝑡⁄ to

• Step 2:

• Be able to divide a fraction

becomes


Dimensional Analysis80


• Step 3:

• Know how to divide terms in the numerator and denominator

• Like terms can cancel each other out

• For every term that is canceled in the numerator, a similar term must be canceled in the denominator

𝑙𝑏𝑑𝑎𝑦

𝑑𝑎𝑦𝑚𝑖𝑛

𝑙𝑏𝑚𝑖𝑛

• Units with exponents should be written in expanded form

𝑓𝑡 𝑓𝑡 𝑓𝑡 𝑓𝑡

EXAMPLE 1

• Convert 1800 ft3 into gallons.

• We need the conversion factor that connects the two units

1 cubic foot of water = 7.48 gal

• This is a ratio, so it can be written two different ways

1 𝑓𝑡7.48 𝑔𝑎𝑙

OR 7.48 𝑔𝑎𝑙

1 𝑓𝑡

• We want to use the version that allows us to cancel out units


Dimensional Analysis 81

EXAMPLE 1

1800 𝑓𝑡1


1800 𝑓𝑡7.48 𝑔𝑎𝑙

• Will anything cancel out?

NO

• Let’s try the other version

1800 𝑓𝑡1

7.48 𝑔𝑎𝑙1 𝑓𝑡

1800 7.481 1


YES

13,464 𝑔𝑎𝑙


OR 7.48 𝑔𝑎𝑙

1 𝑓𝑡

EXAMPLE 2

• Determine the square feet given 70𝑓𝑡 𝑠𝑒𝑐⁄ and 4.5𝑓𝑡 𝑠𝑒𝑐⁄

• Use units to determine set up

• Two ways to write the number

4.5 𝑓𝑡𝑠𝑒𝑐

𝑂𝑅 𝑠𝑒𝑐

4.5 𝑓𝑡• Which way is the right way?

70 𝑓𝑡𝑠𝑒𝑐

𝑠𝑒𝑐4.5 𝑓𝑡

• Will anything cancel?



EXAMPLE 2 CONT’D

• Remember, units function the same as numbers.

𝑓𝑡 𝑓𝑡 𝑓𝑡 𝑓𝑡

• Therefore

70 𝑓𝑡𝑠𝑒𝑐

𝑏𝑒𝑐𝑜𝑚𝑒𝑠 70 𝑓𝑡 𝑓𝑡 𝑓𝑡

𝑠𝑒𝑐

70 𝑓𝑡 𝑓𝑡 𝑓𝑡𝑠𝑒𝑐

𝑠𝑒𝑐4.5 𝑓𝑡


70 1 1 4.5

15.56 𝑓𝑡

Metric System & TemperatureFor Water and Wastewater Plant Operators by Joanne Kirkpatrick Price



Metric Units

King Henry Died By Drinking Chocolate Milk

Metric Units

Kilo Hecto DecaBasic Unit Deci Centi Milli

King Henry Died By Drinking Chocolate Milk

1000Xlarger

100Xlarger

10X larger

MeterLiter

Gram1 unit

10X smaller 100X smaller 1000X smaller

MULTIPLY numbers by 10 if you are getting smaller

DIVIDE number by 10 if you are getting bigger



Problem 1 Convert 2500 milliliters to liters

Converting milliliters to liters requires a move of three place values to the left

Therefore, move the decimal point 3 places to the left

2 5 0 0. = 2.5 Liters3 2 1

Problem 2 Convert 0.75 km into cm

From kilometers to centimeters there is a move of 5 value places to the right

0. 7 5 = 75,000 cm1 2 3 4 5



Examples Convert 1.34 Liters to mL.

1.34 L = 1,340 mL

Convert 76,897 m into km.76897 m = 76.897 km

Convert 34,597 cg into kg.34597 cg = 0.34597 kg

1 2 3

3 2 1

5 4 3 2 1

Metric Conversion

When converting any type of measures •To convert from a larger to smaller metric unit you always multiply•To convert from a smaller to larger unit you always divide



Linear Measure

1 cen meter = 0.3937 inches

1 meter = 3.281 feet

1 meter = 1.0936 yards

1 kilometer = 0.6214 miles

Square Measure

1 cm2 = 0.155 in2

1 m2 = 10.76 2

1 m2 = 1.196 yd2

Cubic Measure

1cm3 = 0.061 in3

1 m3 = 35.3 3

1 m3 = 1.308 yd3

Capacity

1 Liter = 61.025 in3

1 Liter = 0.0353 3

1 Liter = 0.2642 gal

1 gram (g) = 15.43 grains

Weight

1 gram = 0.0353 ounces

1 kilogram = 2.205 pounds

1 inch = 2.540 cm

1 foot = 0.3048 m

1 yard = 0.9144 m

1 mile = 1.609 km

1 in2 = 6.4516 cm2

1 2 = 0.0929 m2

1 yd2 = 0.8361 m2

1 in3 = 16.39 cm3

1 3 = 0.0283 m3

1 yd3 = 0.7645 m3

1 in3 = 0.0164 L

1 3 = 28.32 L

1 gal = 3.79 L

1 grain = 0.0648 g

1 ounce = 28.35 g

1 pound = 454 g

Metric Conversion Equa ons



Math Fundamentals General Conversions (1)

1.) 325 ft3 = gal

2.) 2512 kg = lb

3.) 2.5 miles = ft

4.) 1500 hp = kW

5.) 2.2 ac-ft = gal

6.) 2100 ft2 = ac

7.) 92.6 ft3 = lb

8.) 17,260 ft3 = MG

9.) 0.6% = mg/L

10.) 30 gal = ft3

11.) A screening pit must have a capacity of 400 ft3. How many lbs is this?

12.) A reservoir contains 50 ac-ft of water. How many million gallons of water does it contain?



mg/L & % 13.) 340 mg/L = %

14.) 0.6% = mg/L

15.) 120 mg/L = %

16.) 0.025% = mg/L

17.) 1.5% = mg/L

18.) 5000 mg/L = %

19.) The suspended solids concentration of the return activated sludge is 6800 mg/L. What is the concentration expressed as a percent?

20.) A concentration of 195 mg/L is equivalent to a concentration of what percent?

Metric/English Conversions

21.) 20 feet = meters

22.) 50 L = gal

23.) 70 cm = in



24.) 35 m = feet

25.) 600 mL = gal

26.) 1 lb = mg

27.) 1000 mL = L

28.) 2.7 gal = mL

Linear Measurement

29.) ¼ mile = feet

30.) 4200 feet = miles

31.) 17 feet = meters

32.) 122 inches = feet

33.) 30 meters = inches

34.) 0.6 feet = inches

35.) 492 inches = feet

36.) The total weir length for a sedimentation tank is 142 feet 7 inches. Express this length in terms of feet only.



37.) A one-eighth mile section of pipeline is to be replaced. How many feet of pipeline is this?

38.) 2.7 miles of pipe is how many inches? Flow Conversions

39.) 3.6 cfs = gpm

40.) 1820 gpm = gpd

41.) 45 gps = cfs

42.) 8.6 MGD= gpm

43.) 2.92 MGD = gpm

44.) 385 cfm = gpd

45.) 1,662,000 gpd = gpm

46.) 3.77 cfs = MGD

47.) The flow through a pipeline is 8.4 cfs. What is the flow in gpd?

48.) A treatment plant receives a flow of 6.31 MGD. What is the flow in gpm?



Answers

1.) 2,431 gal 2.) 5,533.04 lb 3.) 13,200 ft 4.) 1,119 kW 5.) 717,200 gal 6.) 0.05 ac 7.) 5,778.24 lb 8.) 0.13 MG 9.) 6,000 mg/L 10.) 4.01 ft3 11.) 24,960 lb 12.) 16.3 MG 13.) 0.034% 14.) 6,000 mg/L 15.) 0.012% 16.) 250 mg/L 17.) 15.000 mg/L 18.) 0.5 % 19.) 0.68% 20.) 0.02 mg/L 21.) 6.1 m 22.) 13.2 gal 23.) 27.56 in 24.) 114.75 ft

25.) 0.16 gal 26.) 454,000 mg 27.) 1 L 28.) 10,219.5 mL 29.) 1,320 ft 30.) 0.8 mi 31.) 5.19 m 32.) 10.17 ft 33.) 1,180.33 in 34.) 7.2 in 35.) 41 36.) 142.58 ft 37.) 660 ft 38.) 171,072 in 39.) 1,615.68 gpm 40.) 2,620,800 gpd 41.) 6.02 cfs 42.) 5964.09 gpm 43.) 2,025.02 gpm 44.) 4,146,912 gpd 45.) 1,154.17 gpm 46.) 2.44 MGD 47.) 5,428,684.8 gpd 48.) 4,381.94 gpm



Math Fundamentals

Converting English to Metric (2)

1. 17 yds2 = ________ m2

2. ________ pounds = 7 kilograms

3. ________ mph = 5 kmph

4. 4.5 gallons = ________ liters

5. 3 miles = ________ km

6. ________ yd2 = 10 m2

7. 16 feet = ________ meters

8. ________ in = 16.5 cm

9. 25 gallons = ________ mL

10. 20.5 mph = ________ kmph

11. ________ ft3 = 3.5 m3

12. ________ yd2 = 9 m2



13. ________ lbs = 17.5 kg

14. 11.5 in = ________ cm

15. ________ ft3 = 1.5 m3

16. ________ ft3 = 12,000 mL

17. 18.5 ft3 = ________ mL

18. ________ yds2 = 14.5 m2

19. 8.5 in = ________ cm

20. 20 yd2 = ________ acres

21. 5 gallons = ______ mL

22. 30 psi = ______ ft of head

23. 41.7 lbs = ______ gallons of water

24. 3628.8 g = ______ lbs

25. 15,840 feet = ______ miles



26. 35 ft³ = ______ gal

27. 2244 gpm = ______ ft³/sec of water

28. 8 MGD = ______ gpm

29. 312 gallons of water = ______ ft³ of water

30. 161.7 ft of head = ______ psi

31. 5,000 gpm = ______ MGD

32. 7 ft³/sec = ______ gpm

33. 5000 gallons of water = ______ ft³

34. 6 miles = ______ feet

35. 1 day = ______ minutes



Answers:

1. 14.29 m2

2. 15.42 lb

3. 3.11 mph

4. 17.03 L

5. 4.83 Km

6. 11.92

7. 4.88 m

8. 6.5 in

9. 94,625 mL

10. 33.0 km

11. 123.53 ft2

12. 10.71 yd2

13. 38.58 lb

14. 29.21 cm

15. 52.94 ft3

16. 0.42 ft3

17. 523.77 L

18. 17.26 yd2

19. 21.59 cm

20. 0.004 ac

21. 18,925 mL

22. 69.28 ft

23. 5 gal

24. 7.99 lb

25. 3 mi

26. 261.8 gal

27. 5 cfs

28. 5,552 gpm

29. 41.71 ft3

30. 70 psi

31. 7.2 MGD

32. 3,141.6 gal/min

33. 668.45 ft3

34. 31,680 ft

35. 1440 min



Math Fundamentals

Conversions (3)

1.) Convert 723 gallons to liters

2.) Convert 17oC to degrees Fahrenheit.

3.) How many feet are in 2.5 miles?

4.) Convert 56 grains per gallon to mg/L.

5.) Convert 56 ft3/s to gallons per minute.

6.) Convert 34oC to degrees Fahrenheit.

7.) Calculate 42.0% of 7,310.

8.) Convert 72 ppm to percent.

9.) A solution was found to be 7.6% hypochlorite. How many milligrams per

liter of hypochlorite are in the solution?

10.) Convert 8.77 acre-ft to gallons.

11.) Convert 1.98 acres to square feet.



12.) Convert 81 ft3 to gallons and liters.

13.) Convert 212oF to degrees Celsius.

14.) Convert 1472 L to gallons.

15.) Convert 0.25 miles to meters.

16.) Convert a chlorine solution of 2.5 ppm to percent.

17.) Convert 2,367 g to pounds.

18.) Convert 3.45 MGD to cubic feet per second.

19.) Convert 63.5% to ppm.

20.) What percent is 12,887 of 475, 258?

Convert the following:

21.) 451oF to degrees Celsius

22.) 8,711,400 gal to acre-feet.



23.) 35 cfs to gpm

24.) 8 lb/sec to lb/day

25.) 45 gal/min to ft3/day

26.) 927 cfm to gps

27.) 0.3 MGD to gal/hr

28.) 89 cfd to cfs

29.) 93 gal/sec to MGD

30.) 2 ft3/min to gal/day

31.) 17 gal/day to lb/min

32.) 1.7 acre-foot to gal

33.) 7800 mg/L to lbs/gal

34.) 890 lb/day to cfm



35.) 10,600 gpd to ft3/sec

36.) 900 grams to lbs

37.) 29.78 lb/hr to gpd

38.) 790 mL to gal

39.) 830 m/min to ft/day

40.) 379 km/day to mph



Conversion Answers:

1.) 2736.6 L

2.) 62.6oF

3.) 13,200 ft

4.) 957.6 mg/L

5.) 25,132.8 gpm

6.) 93.2oF

7.) 3,070.2

8.) 0.0072%

9.) 76,000 mg/L

10.) 2,859,020 gal

11.) 86,248.8 ft2

12.) 2,293.3 L

13.) 100oC

14.) 388.9 gal

15.) 402.6 m

16.) 0.00025%

17.) 5.2 lb

18.) 5.3 cfs

19.) 635,000 mg/L

20.) 2.7%

21.) 232.8oC

22.) 26.7 ac-ft

23.) 15,708 gpm

24.) 691,200 lb/day

25.) 8,663.1 cfd

26.) 115.6 gps

27.) 12,500 gal/hr

28.) 0.001 cfs

29.) 8.04 MGD

30.) 21,542.4 gpd

31.) 0.1 lb/min

32.) 554,200 gal

33.) 0.07 lb/gal

34.) 0.01 cfm

35.) 0.02 cfs

36.) 2.0 lb

37.) 85.7 gpd

38.) 0.2 gal

39.) 3,918,688.5 ft/day

40.) 9.8 mi/hr



102

Section 6

Solutions

103

Documents

Math Fundamentals for Operators in Training