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Problem: How many milligrams per liter is a 1.4% solution? Solution:

1.4% = — x 1,000,000 mg/L (the weight of 1 L water to 106) = 14,000 mg/L

Problem: Calculate pounds per million gallons for 1 ppm (1 mg/L) of water.

Solution: Because 1 gal of water = 8.34 lb,

1 gal 1 gal x 8.34 lb/gal „ „ . „„„ „„„ ,

106 gal 1,000,000 gal s

Problem: How many pounds of activated carbon (AC) are needed with 42 lb of sand to make the mixture 26% AC?

Solution: Let x be the weight of AC:

10.92

0.74

Problem: A pipe is laid at a rise of 140 mm in 22 m. What is the grade? Solution:

140 mm 140 mm

Problem: A motor is rated as 40 horsepower (hp); however, the output horsepower of the motor is only 26.5 hp. What is the efficiency of the motor?

Solution:

Efficiency = Output (hp) x 100% = 26 5 hp x 100% = 66% Input (hp) 40 hp

3.5 SIGNIFICANT DIGITS

When rounding numbers, the following key points are important:

• Numbers are rounded to reduce the number of digits to the right of the decimal point. This is done for convenience, not for accuracy.

• A number is rounded off by dropping one or more numbers from the right and adding zeroes if necessary to place the decimal point. If the last figure dropped is 5 or more, increase the last retained figure by 1. If the last digit dropped is less than 5, do not increase the last retained figure. If the digit 5 is dropped, round off preceding digit to the nearest even number.

Problem: Round off the following numbers to one decimal.

Problem: Round off 10,546 to 4, 3, 2, and 1 significant figures. Solution:

10,546 = 10,550 to 4 significant figures 10,546 = 10,500 to 3 significant figures

10.546 = 11,000 to 2 significant figures

10.547 = 10,000 to 1 significant figure

When determining significant figures, the following key points are important:

1. The concept of significant figures is related to rounding.

2. It can be used to determine where to round off.

Rule: Significant figures are those numbers that are known to be reliable. The position of the decimal point does not determine the number of significant figures.

Problem: How many significant figures are in a measurement of 1.35? Solution: Three significant figures: 1, 3, and 5.

Problem: How many significant figures are in a measurement of 0.000135?

Solution: Again, three significant figures: 1, 3, and 5. The three zeros are used only to place the decimal point.

Problem: How many significant figures are in a measurement of 103,500?

Solution: Four significant figures: 1, 0, 3, and 5. The remaining two zeros are used to place the decimal point.

Key Point: No answer can be more accurate than the least accurate piece of data used to calculate the answer.

Problem: How many significant figures are in 27,000.0?

Solution: There are six significant figures: 2, 7, 0, 0, 0, 0. In this case, the .0 means that the measurement is precise to 1/10 unit. The zeros indicate measured values and are not used solely to place the decimal point.

3.6 PoWERs AND ExPoNENTs

When working with powers and exponents, the following key points are important:

• Powers are used to identify area, as in square feet, and volume, as in cubic feet.

• Powers can also be used to indicate that a number should be squared, cubed, etc. This later designation is the number of times a number must be multiplied times itself.

• If all of the factors are alike, as 4 x 4 x 4 x 4 = 256, the product is called a power. Thus, 256 is a power of 4, and 4 is the base of the power. A power is a product obtained by using a base a certain number of times as a factor.

• Instead of writing 4 x 4 x 4 x 4, it is more convenient to use an exponent to indicate that the factor 4 is used as a factor four times. This exponent, a small number placed above and to the right of the base number, indicates how many times the base is to be used as a factor. Using this system of notation, the multiplication 4 x 4 x 4 x 4 is written as 44. The 4 is the exponent, showing that 4 is to be used as a factor 4 times.

• These same consideration apply to letters (a, b, x, y, etc.) as well:

Key Point: When a number or letter does not have an exponent, it is considered to have an exponent of one.

The powers of 1:

The powers of 10:

Problem: How is the term 23 written in expanded form?

Solution: The power (exponent) of 3 means that the base number (2) is multiplied by itself three times:

Problem: How is the term (3/8)2 written in expanded form?

Solution: In this example, (3/8)2 means: Point: When Parentheses are used, the exponent refers to the entire term within the (3/8)2 = 3/8 x 3/8 parentheses.

Note: When a negative exponent is used with a number or term, a number can be reexpressed using a positive exponent:

Another example is

Problem: How is the term 83 written in expanded form?

Note: A number or letter written as, for example, 30 or X0 does not equal 3 x 1 or X x 1, but simply 1.

3.7 AVERAGES (ARITHMETIC MEAN)

Whether we speak of harmonic mean, geometric mean, or arithmetic mean, each represents the "center," or "middle," of a set of numbers. They capture the intuitive notion of a "central tendency" that may be present in the data. In statistical analysis, an "average of data" is a number that indicates the middle of the distribution of data values.

An average is a way of representing several different measurements as a single number. Although averages can be useful in that they tell us "about" how much or how many, they can also be misleading, as we demonstrate below. You will find two kinds of averages in environmental engineering calculations: the arithmetic mean (or simply mean) and the median.

Definition: The mean (what we usually refer to as an average) is the total of values of a set of observations divided by the number of observations. We simply add up all of the individual measurements and divide by the total number of measurements we took.

Problem: The operator of a waterworks or wastewater treatment plant takes a chlorine residual measurement every day; part of the operator's log is shown below. Find the mean.

Monday

0.9 mg/L

Tuesday

1.0 mg/L

Wednesday

0.9 mg/L

Thursday

1.3 mg/L

Friday

1.1 mg/L

Saturday

1.4 mg/L

Sunday

1.2 mg/L

Solution: Add up the seven chlorine residual readings: 0.9 + 1.0 + 0.9 + 1.3 + 1.1 + 1.4 + 1.2. = 7.8. Next, divide by the number of measurements— in this case, 7:

The mean chlorine residual for the week was 1.11 mg/L.

Problem: A water system has four wells with the following capacities: 115 gallons per minute (gpm), 100 gpm, 125 gpm, and 90 gpm. What is the mean?

Solution:

115 gpm + 100 gpm + 125 gpm + 90 gpm = 430 430 - 4 = 107.5 gpm

Problem: A water system has four storage tanks. Three of them have a capacity of 100,000 gal each, while the fourth has a capacity of 1 million gal. What is the mean capacity of the storage tanks?

Solution: The mean capacity of the storage tanks is

100,000 + 100,000 + 100,000 + 1,000,000 = 1,300,000 1,300,000 - 4 = 325,000 gal

Note: Notice that no tank in Example 3.24 has a capacity anywhere close to the mean.

Problem: Effluent biochemical oxygen demand (BOD) test results for the treatment plant during the month of August are shown below:

Test 1 22 mg/L

Test 2 33 mg/L

Test 3 21 mg/L

Test 4 13 mg/L

What is the average effluent BOD for the month of August? Solution:

Problem: For the primary influent flow, the following composite-sampled solids concentrations were recorded for the week:

Monday

310

mg/L

Tuesday

322

mg/L

Wednesday

305

mg/L

Thursday

326

mg/L

Friday

313

mg/L

Saturday

310

mg/L

Sunday

320

mg/L

Total

2206

mg/L

What is the average SS? Solution:

Sum of all measurements

Average SS =

Number of measurements used 2206 mg/L

3.8 RATios

A ratio is the established relationship between two numbers; it is simply one number divided by another number. For example, if someone says, "I'll give you four to one the Redskins over the Cowboys in the Super Bowl," what does that person mean?

Four to one, or 4:1, is a ratio. If someone gives you four to one, it's his or her $4 to your $1. As another more pertinent example, if an average of 3 cubic feet (ft3) of screenings are removed from each million gallons (MG) of wastewater treated, the ratio of screenings removed to treated wastewater is 3:1. Ratios are normally written using a colon (such as 2:1) or as a fraction (such as 2/1).

When working with ratios, the following key points are important to remember.

• One place where fractions are used in calculations is when ratios are used, such as calculating solutions.

• A ratio is usually stated in the form A is to B as C is to D, which can be written as two fractions that are equal to each other:

• Cross-multiplying solves ratio problems; that is, we multiply the left numerator (A) by the right denominator (D) and say that the product is equal to the left denominator (B) times the right numerator (C):

• If one of the four items is unknown, dividing the two known items that are multiplied together by the known item that is multiplied by the unknown solves the ratio. For example, If 2 lb of alum are needed to treat 500 gal of water, how many pounds of alum will we need to treat 10,000 gal? We can state this as a ratio: "2 lb of alum is to 500 gal of water as x lb of alum is to 10,000 gal of water." This is set up in this manner:

1 lb alum _ x lb alum 500 gal water _ 10,000 gal water

Cross-multiplying:

Transposing:

1 x 10,000

To calculate proportion, suppose, for example, that 5 gal of fuel costs $5.40. What will 15 gal cost?

Problem: If a pump will fill a tank in 20 hr at 4 gpm, how long will it take a 10-gpm pump to fill the same tank?

Solution: First, analyze the problem. Here, the unknown is some number of hours. But, should the answer be larger or smaller than 20 hr? If a 4-gpm pump can fill the tank in 20 hr, a larger (10-gpm) pump should be able to complete the filling in less than 20 hr. Therefore, the answer should be less than 20 hours. Now set up the proportion:

x hr 20 hr

4 gpm 10 gpm

Problem: Solve for the unknown value x in the problem given below.

36 x

Solution:

180 4450

Problem: Solve for the unknown value x in the problem given below.

Solution: 2 x

Problem: 1 lb of chlorine is dissolved in 65 gal of water. To maintain the same concentration, how many pounds of chlorine would have to be dissolved in 150 gal of water?

Solution:

Problem: It takes 5 workers 50 hr to complete a job. At the same rate, how many hours would it take 8 workers to complete the job?

Solution:

5 workers _ x hr 8 workers 50 hr

Problem: If 1.6 L of activated sludge (biosolids) with volatile suspended solids (VSS) of 1900 mg/L is mixed with 7.2 L of raw domestic wastewater with BOD of 250 g/L, what is the food-to-microorganisms (F/M) ratio?

Solution:

Amount of BOD

Amount of VSS

= 250 mg/L x 7.2 L = 0.59 = 0 59 = 1900 mg/L x 1.6L = 1 = .

3.9 DIMENSIONAL ANALYSIS

Dimensional analysis is a problem-solving method that uses the fact that any number or expression can be multiplied by 1 without changing its value. It is a useful technique used to check if a problem is set up correctly. In using dimensional analysis to check a math setup, we work with the dimensions (units of measure) only—not with numbers.

An example of dimensional analysis that is common to everyday life is the unit pricing found in many hardware stores. A shopper can purchase a 1-lb box of nails for 980 at a local hardware store, but a nearby warehouse store sells a 5-lb bag of the same nails for $3.50. The shopper will analyze this problem almost without thinking about it. The solution calls for reducing the problem to the price per pound. The pound is selected without much thought because it is the unit common to both stores. The shopper will pay 700 a pound for the nails at the warehouse store but 980 at the local hardware store. Implicit in the solution to this problem is knowing the unit price, which is expressed in dollars per pound ($/lb).

Note: Unit factors may be made from any two terms that describe the same or equivalent amounts of what we are interested in; for example, we know that 1 inch = 2.54 centimeters.

In order to use the dimensional analysis method, we must know how to perform three basic operations.

3.9.1 Basic Operation: Division of Units

To complete a division of units, always ensure that all units are written in the same format; it is best to express a horizontal fraction (such as gal/ft2) as a vertical fraction.

Horizontal to vertical:

The same procedures are applied in the following examples.

ft3/min becomes min s s/min becomes min

3.9.2 Basic Operation: Divide by a Fraction

We must know how to divide by a fraction. For example,

day min

In the above, notice that the terms in the denominator were inverted before the fractions were multiplied. This is a standard rule that must be followed when dividing fractions.

Another example is:

mm mm

"m2"

2 m becomes mm2 x-2

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