Section 1.2 Order of Operations
Mathematical expressions which involve more than one operation appear ambiguous. For example, is or 5 + = 5 + 12 = 17 ? 5 + = = 22 To clarify this question, mathematics has developed the following hierarchy of computations called order of operations.

Page 2

1. 2. 3. 4.
Perform all operations that appear in grouping symbols first. If grouping symbols are nested, do the innermost first. Raise all bases to powers in the order encountered moving from left to right. Perform all multiplications/divisions in the order encountered moving from left to right. Perform all additions/subtractions in the order encountered moving from left to right.
Here grouping symbols means parentheses ( ), brackets [ ], braces { }, etc. An example of a nested expression is (6 + 2 (4 + 1)) 8. The innermost grouping symbol is (4+1) so the result is (6 + 2 5) 8 = (6 + 10) 8 = = 2. Raising a base to a power (also known as an exponent) means repeated multiplication as in 63 = 6 = 216. To perform this calculation on some calculators enter the following keystrokes: =. The key is called a carrot key. If your calculator does not have the carrot key, it probably has the y x exponent key. To perform this same calculation on this type of calculator enter the following keystrokes: 6 y x 3
=. We will cover exponents in greater detail in Section 1.4 - Decimal Fractions.
In the original problem posed above the multiplication of 6 with 2 is performed before the addition of 5. The proper answer is therefore 17. The other interpretation could be achieved by using parentheses ( 5 + 6 ) 2 = = 22. Order of operations is built into all scientific calculators. That is, if you enter the keystrokes in the correct order, the calculator will automatically perform the correct calculation. Therefore 5 + = yields the correct answer of 17. Try your own calculator to see if you get the correct answer. Then you will know if your calculator performs the order of operations correctly or not. In many formulas x occurs as a variable, but then confusion with the times sign can result. To avoid this, alternative symbols for multiplication are used. They are the dot notation and adjacent parentheses as in = = (7)(3) = 21. Some calculators recognize the adjacent parentheses
as multiplication, but some do not. On these calculators the times operation must be inserted between the parentheses. Division is also indicated by a variety of notations. For example, the following all mean 34 divided by 17,

= 34/17 =

34 = = 2. 17

Page 3

In addition to parentheses, brackets and braces, certain symbols act as implied grouping symbols. The most important of these are the fraction bar and the square root symbol. The fraction bar acts to separate the numerator from the denominator. If either or both of the numerator or denominator consist of an expression with operations, these must be performed first before the division indicated by the fraction bar. For example,

7 + = =2. 2+3 5

To perform this computation on the calculator, parentheses need to
be inserted around both numerator and denominator as shown below.
Parentheses are the only grouping symbol the calculator recognizes or uses. The square root symbol also acts as a grouping symbol. Any calculation inside the square root needs to be completed before the root is taken. For example, 25 + 144 = 169 = 13. To perform this computation on the calculator parentheses need to be inserted around the expression inside the square root symbol. On some calculators enter the following keystrokes : On other calculators enter the keystrokes:

25 + 144

Note: On newer calculators like the Casio fx-300MS one enters expressions the way they look , i.e., the square root symbol comes first. On older models like the TI-30Xa

some functions like

come after the expressions they are to evaluate. Give the above
exercise a try to find out how your calculator works. In any case, using parentheses keys when necessary is a good habit to acquire. Failure to do so usually results in wrong answers!

Your Turn!!

Perform the following arithmetical operations. Remember to follow the correct order of operations. 1. 3.
=_________ 2. =_________ 4.

=_________

4(19 12) 2 11

9 + 2 =_________

Page 4

5. 7. 9. 11.

5 + 13 + 2 1+ 3
=_________ 6. =_________ 8. =_________ 10. =_________ 12.
=_________ =_________ =_________ =_________

32 + 42

4[3 + 2(9-2)] + 2(4 2)

33[ ( 6 + 5 ) - 22 ]

From a board 10 feet long a piece 29 inches was cut off. How long is the piece remaining? Ignore the width of the cut. 13) ________________
In 1993 General Motors had total sales of $133,621,900,000 , while Ford Motor Company had sales of $108,521,000,000. How much more were GMs sales than Fords? 14) If 128 copies of a software package cost $5760, what is the cost per copy? 15) ________________ ________________
A carpenter earns $14 per hour plus time and a half for overtime (more than 40 hours in one week). If she works 58 hours one week, what is her gross pay for that week? 16) ________________
You order 15 CDs at $8 a CD and 24 cassette tapes at $3 a tape. What is the cost of the order? 17) ________________
A truck averages 16 miles per gallon and has a 25 gallon gas tank. What is the furthest distance the truck can travel without stopping for gas? 18) ________________

Page 5

A stairway consists of rises of 6 in and must reach a height of 10 feet. How many rises are needed? 19) ________________
A stairway consists of 4 in rises and treads of 18 in. If the height of the stairs is 4 feet, what is the distance taken up by the stairway on the lower floor? 20) ________________

Page 17

Take the fraction from Step 4 and divide it by 10 raised to the power of the number from Step 1. This number, worked out as a fraction, is the fraction equivalent to the original repeating decimal.
To illustrate the steps convert 0.00666 to a fraction. Step 1. The number of places from the decimal point to the repeating string of 6s is two. Step 2. The result is the decimal 0.666. Step 3. The whole number is 0. Step 4. There is one repeating digit, a 6 , so the result is 0 + =. Step 5. Dividing two thirds by 102 = 100 gives
/ 1 1. So 0.006 =. 102 = 100 = = = / 150
As a more complicated example consider converting 3.1527272727 to a fraction. Step 1. The number of places from the decimal point to the repeating string of 27s is two. Step 2. The result is the decimal 315.272727. Step 3. The whole number is 315. Step 4. There are two repeating digits, 27 , so the result is 315 + Step 5. Dividing the answer of Step 4 by 102 = 100 gives
315 / 100 = = = =3. / 275 275

3 /. = 315 + = 11 /

Using a calculator we can verify that 3
42 = 3 + = 3.15272727. = 3.1527. 275
Often we wish to approximate a decimal number by finding another decimal roughly equal to the first number, but expressed with less digits. This process is called rounding. To round use the following procedure: 1. 2. Determine the decimal place to which the number is to be rounded. Often this is stated in the problem or application. If the digit to the right of this decimal place is less than 5, then replace all digits to the right of this decimal place by zeros or discard them if they are to the right of the decimal point. If the digit to the right of the decimal place is 5 or greater, then increase the digit in this decimal place by 1 and replace all digits to the right of this decimal place by zeros or discard them if they are to the right of the decimal point.

Page 18

As an example, consider rounding 10,547.395 to the different decimal places shown in the following table. 10,547.395 rounded to 2 places 1 place the nearest unit the nearest ten the nearest hundred the nearest thousand Decimal Place of Rounding hundredths place tenths place ones place tens place hundreds place thousands place Result 10,547.40 10,547.4 10,547. 10,550 10,500 11,000

MD) Multiply and Divide, working from left to right if there are more than one of these. AS) Add and Subtract any remaining terms, working left to right.
Evaluate a numeric expression Follow the order of operations to simplify any expression involving numbers and operations to a single value. Example Solution
Evaluate the expression 2. Multiplication is higher than subtraction in the order priority, so we must multiply first and replace it with the result 10. The subtraction is done last.

2 = = 7

Note: The middle dot between two numbers indicates a multiplication, not a decimal point. The times symbol is avoided in algebra because it is too easy to confuse with the variable x. It is also customary to drop the symbol entirely, so whenever there is no symbol between two numbers, the operation is multiplication.
The expression 2 can also be written as 17 5(2). Here, there is no symbol between the number 5 and the number (2), so multiplication is assumed.

Page 31

Example Solution
Evaluate the expression 3(5 + 4). In this case, the group in parentheses must be calculated first. Replace any expression inside parentheses with its answer before continuing with any operations outside the parentheses. 3(5 + 4) = 3(9) = 27
Evaluate the expression 5 + 11(8 5) 16. Again, the group in parentheses must be calculated first. Then do the multiplication, and finish with the addition and subtraction.
5 + 11(8 5) 16 = 5 + 11(3) 16 = 5 + = = 22
Note: Addition and subtraction will always be the last operations performed, unless they are grouped inside parentheses. Example Solution
Evaluate the expression 8 + (5 2) 2. Again, the group in parentheses must be calculated first. 8 + (5 2) 2 = 8 + (3) 2 The raised 2 here is an exponent. Exponents are a short-hand for repeated multiplications. (3)2 means to multiply two copies of the base number 3: (3)(3) = 9. Replace the exponent calculation (3)2 with the result 9, and then continue following PEMDAS: 8 + (5 2) 2 = 8 + (3) 2 = 8+9 = 17

Page 32

Evaluate a variable expression
With a variable expression, individual variables may be assigned specific values. In this case, replace each letter with the known value, and then follow the order of operations to simplify the remaining numeric expression to a single value.
Evaluate the expression a 2 2b when a = 5 and b = 3. Think of the variables as place holders. The best way to do this is to replace each letter with empty parentheses. This makes room for the actual value, which then fills in the parentheses. After filling in the given values for a and b, the exponent calculation has the highest priority, followed by the multiplication, and then the subtraction last.

a 2 2b

= ( ) 2 2( ) = (5) 2 2(3) = 25 2(3) = = 19

Simplify (5p4q3)2(3pq)2. Each group must be simplified first, using the exponent on the right parenthesis. (5p4q3)2(3pq)2 = (25p8q6)(9p2q2) = 225p10q8.

Page 46

The following table summarizes the exponent rules introduced in this section. Use each rule only if the form of the expression exactly matches. The exponents n and m may be any numbers. The variables x and y are used generically and could be any letter or number in the base.
Rules of Exponents x n x m = x n+ m

( x y )n = x n y n

= x mn
Simplify the following exponent expressions. 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 2. 4. 6. 8. 10. 12.

x3 ( x 7 ) x2 x6

y y5 y3

(2x3)4

( 3c )

( ab )

14. 16. 18. 20.

( 5x y )

( 3x y )
( p )( p ) ( 4 x y ) ( 3xy z )

Page 47

21. 23.

(8h k ) ( 5hk )

22. 24.

( 3a b )(10a b )

(3ab2 )(3a 2b)( 3ab) 2

5a 3b 5

Section 2.5 Quotient Rule and Negative Exponents
The exponent rules in the previous section cover simplification of variable expressions involving multiplication only. The division operation is closely related to multiplication, and we can also simplify quotients by applying and expanding on the exponent rules. am = a mn an According to the multiplicative property for exponents, aman = am+ n. To multiply numbers with the same base, we add the exponents. Since division is the opposite of multiplication, it makes sense to guess that the rule uses subtraction of the exponents instead. In reality, we are reducing out common factors in the division problem.
Simplify exponential expressions using the property Example Solution

Simplify

x5. x3
Write each exponent form out the long way as a repeated multiplication.
x 5 xxxxx x x x xx xx = = = = x2. 3 x xxx 1 x x x
The common factors of x reduce out since each x/x is 1. Using the rule, we get the x5 same answer, but more efficiently: 3 = x53 = x 2. x So to divide expressions with the same variable base, we subtract the exponents. The quotient property can be applied to both numeric and variable expressions, as long as the numerator and denominator have powers of the same base.

Page 51

Rules of Exponents x n x m = x n+ m xn = x nm m x

x n = 1 xn

n x xn = n y y
Scientific notation Exponents can be used to write very large and very small numbers in a more concise way. Scientific notation is commonly used in many science applications. Astronomy, for example, deals with very large scales, while biology and chemistry often study objects of very small sizes. This section introduces the form of the scientific notation and provides some practice in converting numbers between standard forms and scientific notation. Scientific notation is considered simpler because there is no need to write a long string of 0 digits at the end of a number. Change numbers written in scientific notation to standard (decimal) form A typical number written in scientific notation looks like 3.4 105. The number has a decimal part (3.4, in this case) that is multiplied by a power of 10. We can convert back to standard form simply by multiplying out the product. The exponent is done first, of course:
105 = 10 = 100, 000. A positive power of 10 always gives a 1 followed by a number of zeroes equal to the exponent. Now multiply this by the decimal part 3.4 100,000 is equal to 340,000. Note that it is standard to use for times in scientific notation. This is perhaps the only place we will routinely use the symbol in algebra. Multiplying any number by 10 simply shifts the decimal point one place to the right. For powers of 10, the decimal point is shifted to the right as many places as the value of the exponent.

Example

1.is equivalent to 1,520. The decimal point was shifted exactly 3 places to the right. Insert zeros to fill all empty places.

Page 52

Small numbers (close to zero) are written with negative exponents on the base 10. Since a negative power means we need to divide by the factors of 10, the decimal place is shifted to the left instead. = 4 = = 0.0009. The decimal place was shifted 4 places to the left. 10 10, 000
2.becomes 0.0000023. Note that the decimal point was shifted exactly 6 places to the left.
Write number in scientific notation To write a number in scientific notation, move the decimal place far enough that there is exactly one (nonzero) digit left of the decimal point. Then multiply by a power of 10, where the exponent is the number of places you moved the decimal point (use a positive exponent if the decimal moved left, and a negative exponent if the decimal moved right). This works to represent the same number, because multiplying by a power of 10 simply moves the decimal point back to where it belongs. Examples

45,000,000 is the same as 4.7,810,000,000 = 7.540 = 5.4 102
0.0000003 = 3 10-7 (Negative exponents are necessary to write numbers less than 1) 0.000764 = 7.64 10-4
Multiply and divide numbers using scientific notation Using scientific notation can simplify calculations involving large numbers. First, convert numbers to scientific notation, and then use exponent rules to reduce the powers of 10. Example Solution
Simplify (320,000)(50,000,000). (320,000)(50,000,000) = (3.2 105)(5 107) = (3.2)(5) == 16,000,000,000,000 Convert to scientific notation Multiply coefficients Add exponents on the base 10 Move the decimal point back 12 places

Page 53

Simplify the following exponent expressions. 1. 3. 5. 7. 9. w0 x 2 9x 0 2. 4.

8w0 x 9 5x0 + ( 9 x )

(15d )

( 4b )

12 xy xy 1

( 2a )

(4 xy)

( 5x )

12 xy 2

xy 2 z

24 x 4 y x 4 y 3
Write the following numbers in standard (decimal) form. 19. 21. 1.4.20. 22. 9.2.05 107
Write the following numbers in scientific notation. 23. 25. 928,000 0.000092 24. 26. 0.000000103 137,100,000,000,000

Page 54

Chapter 2 Practice Exam
Simplify each of the following. 1. 5 + 4(9) 3. 7 + 3(1 5) 5. 5(4x 2) 7. 2(6w + 9v 5) 2. 5(10 4) 32 4. (2)2 6(2) + 8 6. 4u 9v + 7u 10 5v + 6 8. (3x2 3x + 6) (x2 + 2x 1)
Evaluate each of the following expressions with the given values. 9. Evaluate the polynomial x2 + 2x 1 when x = 5. 10. Evaluate the formula f = Solve the following equations. 11. 5x = 40 13. 5y 2 = 3y 20 15. 10 + 4(3c 1) = 2(8c 5)
F to find the value of f when F = 30 and d = 35. d F

12. 14. 16.

3t 9 = 12 2(4z 3) = z 14 2(7x 2) 5(2x 3) = 6(x + 4)
Scientific Notation: 17. Write 1.in standard (decimal) notation. 18. Write 28,000,000,000 in scientific notation. (1.)( 9.). 19. Simplify: (1.) Round your decimal values to the nearest tenth (1 place) and write the simplified answer in scientific notation. Use the Exponent Rules to simplify each of the following expressions. 20. 22.

m9 m 4

21. ( y 3 )

( 3a b )( 5ab )

A cumulative frequency distribution is a distribution that accumulates, adds-up the frequencies, up to and including a specific class. This type of distribution works well when the data can be ordered in some meaningful way. It is particularly useful when the data is numerical or when the data occurs over time. To create a cumulative frequency distribution we must first create a frequency distribution and then we add a new column. The new column is the cumulative

Page 89

frequencies. The first entry in the cumulative column is the frequency for the first class. The second entry in the cumulative column is the total of the first two frequencies. The third entry in the cumulative column is the total of the first three frequencies. We keep going until we have added together all of the frequencies. This number should check with the number of data values we began with when we started the problem.
In the last example we created a grouped frequency distribution for the temperatures of 108 people who visited a clinic and had their temperature taken by a student nurse. What we would now like to do is to create a cumulative frequency distribution for that data set. We can use what we have created in the last example by adding a new column to the end of the previous table called cumulative frequency. The first row of the table is the same as frequency. The second row is the sum of the first and second frequencies. The third row is the sum of the first three frequencies and so on until we get to the end of the table.
Temperature 97.0 97.9 98.0 98.9 99.0 99.9 100.0 100.9 101.0 101.9 102.0 102.9 103.0 103.9 104.0 104.9 Tally Frequency Cumulative Frequency 107 108
||||| ||||| ||||| || ||||| ||||| ||||| | ||||| ||||| ||||| ||||| ||||| ||||| | ||||| ||| ||||| ||||| | ||||| ||||| ||||| ||||| |||| |
The last entry in the cumulative frequency column should be the total number of data items that we began with when we started the problem.

Page 103

"increasing", while if the value of y goes down as x moves from left to right, y is "decreasing". This follows the same reasoning that we saw in the time-series graph and discussion above.
Suppose we have the formula, y = 2 x , if we substitute, plug-in, various values of x into this formula results in the following table.
x -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 y = 2x 2 - 3 15.00 9.50 5.00 1.50 -1.00 -2.50 -3.00 -2.50 -1.00 1.50 5.00 9.50 15.00
Using this table we construct the graph shown below.

Page 104

Notice that our graph results in a series of points. We could have connected each pair of consecutive points with a straight line but it made more sense to connect them with a curved segment so that the graph is smooth. A graph of this type is called a curve. The graph has a local minimum when x = 0.00.
Section 4.2.7 Circle Graphs
A circle graph which is sometimes called a pie graph is a circle that is divided into sections or wedges according to the percentages of frequencies in each category of the distribution. We can use pie graphs to help us represent the data in a relative frequency distribution. Pie graphs are used most often with categorical or attributive data. To construct a pie graph we first must create the relative frequency distribution of percentages of data within each category or class. Then we take each percentage and multiply by 360o to determine about how large each wedge or section of the pie is going to be. We then draw the wedge and label it.
The N. Y. Times Almanac in 2002 listed the following percentages of world wide energy use from different sources. Construct a pie graph of the relative frequency distribution.
Energy Type Petroleum Coal Dry Natural Gas Hydroelectric Nuclear Other (Wind, Solar, etc)
Percentage used 39.8% 23.2% 22.4% 7.0% 6.4% 1.2%
We first change the percentages to decimal and multiply by 360o to determine the size of each wedge. Then we can draw the graph.
Energy Type Petroleum Coal Dry Natural Gas Hydroelectric Nuclear Other (Wind, Solar, etc) Percentage used 39.8% 23.2% 22.4% 7.0% 6.4% 1.2% Degrees 23 4

a. b. c. d. e. f. g. h. i.
When x = 1, estimate the value of y. When x = 0, estimate the value of y. When x = 3, estimate the value of y. When x = 2, estimate the value of y. When x = 2.5, estimate the value of y. For what intervals of x values is y increasing? For what intervals of x values is y decreasing? For what values of x does y have a local minimum? For what values of x does y have a local maximum?
7. Below is shown the graph of velocity (positive means motion to the right, negative means motion to the left) versus time of a car travelling between two stop lights. The velocity is in units of mph and the time is in units of seconds.

Page 108

a. b. c. d.
How fast was the car moving at the first stop light? How fast was the car moving at the second stop light? Estimate the cars greatest speed. How long after it left the first stop light did the car travel to the right?
8. For a given circuit the amount of applied voltage, V, and the current in amps, I, are related by means of the follows table.

V Voltage I Current

4.0 0.5

8.0 1.0

12.0 1.5

16.0 2.0

20.0 2.5

24.0 3.0

28.0 3.5

32.0 4.0

36.0 4.5
a. Construct a graph of Voltage versus Current from the following table: b. From your graph estimate V when I is 5.0 amps.
9. The cost of electric power is given by the equation C = $0.0558E, where E is the amount of electrical energy consumed in kilowatt hours (Kwhr). a. Generate a graph of C versus E, for 0 < E < 1000 Kwhr. b. Estimate the value of C when E is 600 Kwhr.

Page 109

Section 4.3 Descriptive Statistics
Descriptive statistics is the body of methods used to represent and summarize sets of numerical data. Descriptive statistics provides a means of describing how a set of measurements (for example, peoples weights measured in pounds) is distributed. Descriptive statistics involves two different kinds of summary statistics. These summary statistics are called measures of center and measures of spread. Section 4.3.1 Measures of Center Measures of center involve a typical data value about which all the other data values are distributed. The most commonly used measures of center are the mean, the median and the mode. The mean is sometimes called the arithmetic average and it is the sum of all the data values divided by the number of data values. If the data is arranged in order, high to low or low to high, then the median is the middle data value with as many data values above it as below it. The mode is the data value that occurs the most often (has the highest frequency). The mode is only meaningful for large sets of data where it is likely for the same score to be measured more than once. If we were to graph a large data set and the mean, median, and the mode were all the same, our graph would look like the graph below on the left. A graph of this type is called symmetric: one side of the graph looks just like the other side of the graph. Symmetric graphs result from distributions of data in which the mean, median and mode are all equal. If any of these three are

. The variable i is called a dummy summation index and
is just used to signify that various values of x are being summed. The designation i = 1 beneath the sigma indicates where the sum begins and the n above the sigma indicates where the sum ends. In statistics it is nearly always the case that sums begin at 1 and end at n, so the following abbreviated symbols are often employed.
= x1 + x 2 + x3 +. + x n = xi = xi = x.
Section 4.3.4 Mean Calculations
The mean is the arithmetic average and it is usually indicated by x (called x bar). Sometimes a second symbol for the mean is used, called the population mean, when all possible data values have been measured. The population mean is represented by the Greek letter lower case mu, . The mean can be computed by adding together all of the data values and dividing by the number of data values that were added together. In terms of a formula, we write the mean calculation as xi. x= n EXAMPLE: In the years the United States had 18 space shuttle missions. The duration of these missions in days is given below. Compute the mean of the space shuttle missions. 6 11
n 8 + 9 + 9 + 14 + 8 + 8 + 10 + 7 + 6 + 9 + 7 + 8 + 10 + 14 + 11 + 8 + 14 + 11 x= x= 18 x = 9.5

Page 113

If there is a lot of data, we may want to form a frequency distribution before we begin to do any computations. The benefits are that the computations we will have to do are reduced because we rely on the fact that multiplication is repeated addition. We do have another formula for the mean that we will use when we have a frequency distribution. xi fi x= fi If the data is ungrouped we use each data value for xi in the formula. If the data is grouped then we have to determine the midpoint of each class and use that for xi. The midpoint is the average of the upper and lower limits of the class
A 15 question practice drivers education test was administered to a group of people planning on taking the written potion of the drivers license examination. The number of correct questions a person got is shown below.

Correct 15 Frequency 5

Compute the mean number correct.
We note that in the formula x =
that the numerator requires we sum the product of the
xs times their corresponding fs. To do this we will create a new column to record the products. The denominator requires us to sum all the frequencies so that we can divide the two sums. Our new table looks like

Page 134

c. The answer is subject to interpretation. There are many answers possible in the context of what type of information is being sought after from the data. 2. Garbage weights a. Grouped frequency distribution.
Weight 4.1 11.0 11.1 18.0 18.1 25.0 25.1 32.0 32.1 39.0 39.1 46.0 46.1 53.0 53.1 60.0 Tally
|||| ||||| ||||| ||||| ||||| ||||| || ||||| ||||| | ||||| |||| ||||| | |||| |

Frequency 4 1

b. Cumulative frequency distribution
Weight Tally Frequency Cumulative Frequency 1 62
4.1 11.0 11.1 18.0 18.1 25.0 25.1 32.0 32.1 39.0 39.1 46.0 46.1 53.0 53.1 60.0 3. M & Ms
a. Frequency Distribution
Color Blue Brown Green Orange Red Yellow Tally ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| |||| ||||| ||| ||||| |||| ||||| ||||| ||||| ||||| ||| ||||| ||||| ||||| ||||| ||||| | Frequency 23 26

Page 135

b. Relative frequency distribution
Color Tally Frequency Relative Frequency 0.13 0.30 0.07 0.08 0.20 0.23
Blue ||||| ||||| ||||| Brown ||||| ||||| ||||| ||||| ||||| ||||| |||| Green ||||| ||| Orange ||||| |||| Red ||||| ||||| ||||| ||||| ||| Yellow ||||| ||||| ||||| ||||| ||||| | 4. a. Grouped frequency distribution
Age Tally ||||| ||||| ||||| ||||| ||||| ||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| || || || ||

Frequency 2 2

Weight Tally Frequency Cumulative Frequency 2 76
||||| ||||| ||||| ||||| ||||| ||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| || || || ||
Relative frequency distribution
Weight Tally Frequency Relative Frequency 0.37 0.39 0.16 0.03 0.03 0.03

Page 136

Section 4.2 Solution Set:
1b. Silver (Ag) 1c.Lead (Pb) 2a.

Balls Sold in One Day

Baseball Basketball Football Golf ball Soccer ball Tennis ball Type

Frequency

Page 137
Tennis ball 18% Soccer ball 15%
Baseball 13% Basketball 15% Football 17%

Golf ball 22%

Estimated Cost per Vehicle
Cumulative Frequency 10 0

Page 138

Mimimum Wage
$6.00 $5.00 Wage $4.00 $3.00 $2.00 $1.00 $0.2005 Year
4b. 2000 or 2005 4c. 5a. ~ 29 m/s 5b. ~ 5.1 sec 5c. ~9.8 m/s2 5d ~98 m/s 6a. y = 1 6b. y = 6 6c. y = 15 6d. y = 10 6e. y = 5 6f. 2 < x < 0 and x > 2 6g. x < 2 and 0 < x < 2 6h. x = 2 and x = 2 6i. x = 0 7a. 7b. 7c. 7d. 0 mph 0 mph ~ 45 mph 40 sec

These conversions have been built into many calculators. Since each of you may have different calculators, please ask your instructor for help if youre not sure how to use your calculator.
Facts about measuring angles Since there are 360 in a circle, the angular measure of a straight line is 180 In a square or right angle there are 90. Two angles whose sum is 180 make a straight line and are called supplementary.
If two lines intersect as shown below: EAD + CAE = EAD + BAD = 180 and EAD + CAE = EAD + BAD = 180. Since EAD is on both sides of this equation, we conclude that CAE = BAD and similarly that CAB = EAD. These equal angles formed by two intersecting lines are called vertex ( or vertical ) angles.

Page 185

EXAMPLE: Consider the following diagram:
SOLUTION: Once the measure of d is known, the remaining three can be determined. Since they are vertex angles, b = d = 14439'35". Since a is the supplement to d, a = 180 14439'35". So 14439'35" = 17959'60" 14439'35". So 180 - 14439'35" = 3520'25" = c. This same calculation can also be done on a on most scientific calculators using a few keystrokes.
Consider a pair of parallel lines crossed by a third line (called the transversal) as shown below. If we imagine that point B is superimposed, or in other words, moved so that it is directly over point F by moving segment CD onto the line through EG , the corresponding angles EFB and CBA are equal. Similarly, EFH = CBF, ABD = BFG, and FBD = HFG. From the equality of vertex angles, the alternating interior (alternate sides of the transversal, inside the two parallel lines) are equal, i.e., EFB = FBD and CBF = BFG. Similarly, the alternating exterior (alternate sides of the transversal, outside the two parallel lines) are equal, i.e., EFH= ABD and CBA = HFG.
Polygons Polygons are closed figures in the plane whose sides are line segments.

Page 186

Triangles
The simplest polygon is the three-sided triangle. The points at the corners A , B , and C are the vertices of the triangle, and the angles BAC , BCA , and ABC are called the interior angles of the triangle. The triangle is often then labeled as triangle ABC.
Important Fact: The interior angles of a triangle always add up to to 180. EXAMPLE: In the triangle below the missing angle 1 is calculated as follows : SOLUTION: 1 = 180 97E13' 3059' = 5148'
What else do we know about triangles? The first question we need to consider is what determines a triangle. Every triangle has three sides and three angles, so six numbers (three angle measures and three side lengths) are associated with every triangle. Two triangles are called congruent if one can be superimposed (or in other words placed on top of the other ) so that the triangles fit exactly on the other. In other words, the triangles have exactly the same shape and size. Essentially, triangles that are congruent are the same. There are three rules for congruency when using triangles: