## Basic Mathematics

3.1 INTRODUCTION

Most calculations required by wastewater operators and engineers (as with many others) start with the basics, such as addition, subtraction, multiplication, division, and sequence of operations. Although many of the operations are fundamental tools within each operator's toolbox, it is important to reuse these tools on a consistent basis to remain sharp in their use. Wastewater operators should master basic math definitions and the formation of problems; daily operations require calculation of percentage, average, simple ratio, geometric dimensions, threshold odor number, force, pressure, and head, and, at the higher levels of licensure, the use of dimensional analysis and advanced math operations.

3.2 BASIC MATH TERMINOLOGY, DEFINITIONS, AND CALCULATION STEPS

The following basic definitions will aid in understanding the material that follows.

• 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.

• 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 x 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 ^ 4.

• A dividend is a number to be divided; a divisor is a number that divides; for example, in 100 ^ 20 = 5, 100 is the dividend, 20 is the divisor, and 5 is the quotient.

• 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 determined for other than circles, 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: feet, inches, meters, yards, etc.

• Pi (n) is a number used in calculations involving circles, spheres, or cones (n = 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.

3.2.1 Calculation steps

Standard methodology used in making mathematical calculations includes the following:

1. If appropriate, make a drawing of the information in the problem.

2. Place the given data on the drawing.

3. Ask "What is the question?" followed by "What are they really looking for?"

4. If the calculation calls for an equation, write it down.

5. Fill in the data in the equation—look to see what is missing.

6. Rearrange or transpose the equation, if necessary.

7. If available, use a calculator.

8. Always write down the answer.

9. Check any solution obtained. Does the answer make sense?

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

3.2.2 Key Math Words

• Less than means to subtract.

### 3.2.3 Calculators

The old saying "Use it or lose it" amply applies to mathematics. Consider a person who first learned to perform long division, multiplication, square roots, adding and subtracting, decimals to fractions, and other math operations using nothing more than pencil and paper and his or her own 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 of "Use it or lose it" makes our point. As with other learned skills, how proficient we remain at 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.

There can be little doubt that 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 working through 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 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, logarithmic capability is 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.

### 3.3 sequence of operations

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.

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 a brief review of the mathematical concepts and applications frequently employed by wastewater operators.

### 3.3.1 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, and (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; thus, 100 - 30 - 15 - 4 = 51 (by steps, 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 x 3) x 5] x 6 = 180 and 5 x [2 x (6 x 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: Whenever no grouping is given, multiplications and divisions are to be performed in the order written, then additions and subtractions in the order written.

3.3.2 Sequence of Operations Examples

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

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

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

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

or by steps:

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

[(3 x 3) x 5] x 6 = 270 and 5 x [3 x (6 x 3)] = 270

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

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

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

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.

### 3.4 PERCENT

The word "percent" means "by the hundred." Percentage is usually designated by the symbol %; thus, 15% means 15 percent or 15/100 or 0.15. These equivalents may be written in the reverse order: 0.15 = 15/100

= 15%. In wastewater treatment, percent is frequently used to express plant performance and for control of biosolids treatment processes. When working with percent, the following key points are important:

• Percents are another way of expressing a part of a whole.

• As mentioned, percent means "by the hundred," so a percentage is the number out of 100. To determine percent, divide the quantity we wish to express as a percent by the total quantity, then multiply by 100:

Part

For example, 22 percent (or 22%) means 22 out of 100, or 22/100. Dividing 22 by 100 results in the decimal 0.22:

When using percentage in calculations (such as when used to calculate hypochlorite dosages and when the percent available chlorine must be considered), the percentage must be converted to an equivalent decimal number; this is accomplished by dividing the percentage by 100. For example, calcium hypochlorite (HTH) contains 65% available chlorine. What is the decimal equivalent of 65%? Because 65% means 65 per hundred, divide 65 by 100: 65/100, which is 0.65.

Decimals and fractions can be converted to percentages. The fraction is first converted to a decimal, then the decimal is multiplied by 100 to get the percentage. For example, if a 50-foot-high water tank has 26 feet of water in it, how full is the tank in terms of the percentage of its capacity?

26 ft

50 ft = 0.52 (decimal equivalent)

Thus, the tank is 52% full.

Problem: The plant operator removes 6500 gal of biosolids from the settling tank. The biosolids contain 325 gal of solids. What is the percent solids in the biosolids?

Solution:

Problem: Convert 65% to decimal percent. Solution:

Percent 65

100 100

Problem: Biosolids contains 5.8% solids. What is the concentration of solids in decimal percent?

Solution:

Note: Unless otherwise noted, all calculations in the text using percent values require the percent to be converted to a decimal before use.

Note: To determine what quantity a percent equals, first convert the percent to a decimal then multiply by the total quantity.

Quantity = Total x Decimal Percent (3.2)

Problem: Biosolids drawn from the settling tank is 5% solids. If 2800 gal of biosolids are withdrawn, how many gallons of solids are removed?

Solution:

Problem: Convert 0.55 to percent. Solution:

To convert 0.55 to 55%, we simply move the decimal point two places to the right.

Problem: Convert 7/22 to a decimal number to a percent. Solution:

Problem: What is the percentage of 3 ppm?

Note: Because 1 liter of water weighs 1 kg (1000 g = 1,000,000 mg), milligrams per liter is parts per million (ppm).

Solution: Because 3 parts per million (ppm) = 3 mg/L:

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