Apply GCF and LCM to MonomialsMississippi Standard: Apply the concepts of Greatest Common Factor (GCF)and Least Common Multiple (LCM) to monomials with variables.

You can find the Greatest Common Factor (GCF) of two or more monomials byfinding the product of their common prime factors.

ExercisesFind the GCF of each set of monomials.

1. 12x, 40x2 2. 18m, 45mn 3. 14n, 42n2

4. 4st, 10s 5. 5ab, 6b2 6. 14b, 56b2

7. 36a3b, 56ab2 8. 30a3b2, 24a2b 9. 32mn2, 16n, 12n3

Find the LCM of each set of monomials.

10. 20c, 12c 11. 16xy, 3x 12. 36ab, 4b

13. 16a2, 14ab 14. 7x, 12x 15. 21mn, 28n2

16. 20st, 50s2t 17. 75n2, 25n4 18. 10x, 20x2, 40xy

GEOMETRY For Exercises 19 and 20, use the squares shown.

19. What is the GCF of the sides of the squares?

20. What is the LCM of the sides of the squares?

4xy 16x2

Find the GCF of Monomials

Find the GCF of 16xy2 and 30xy.

Find the prime factorization of each monomial.

16xy2 � 2 � 2 � 2 � 2 � x � y � y Circle the common factors.

30xy � 2 � 3 � 5 � x � y

The GCF of 16xy2 and 30xy is 2 � x � y or 2xy.

Find the LCM of Monomials

Find the LCM of 18xy2 and 10y.

Find the prime factorization of each monomial.

18xy2 � 2 � 3 � 3 � x � y � y Circle the common factors.

10y � 2 � 5 � y

2 � 3 � 3 � 5 � x � y � y Multiply the factors, using the common factors only once.

The LCM of 18xy2 and 10y is 2 � 3 � 3 � 5 � x � y � y or 90xy2.

You can find the Least Common Multiply (LCM) of two or more monomials bymultiplying the factors, using the common factors only once.

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 600

Compare Data SetsMississippi Standard: Use a given mean, mode, median, and range tosummarize and compare data sets including investigation of the differenteffects that change in data values have on these measures.

When given the mean, median, mode, and range of data sets, you can often analyze and make comparisons of the data, without knowing the data values.

Ten Tallest Buildings (meters)

MeanMedianModeRange

New York

289281

248, 319133

Texas

260255

none76

Summarize and Compare Data Sets

BUILDINGS The mean, median, mode, and range of the ten tallestbuildings in New York and Texas are given in the table.

Which state has a greater average building height of its ten tallestbuildings?

The mean, or average, height of the ten tallest buildings in New York is 289 meters. The mean height of the ten tallest buildings in Texas is 260 meters. Compare the data.

289 � 260

So, New York has a greater average building height of its ten tallestbuildings than Texas.

Which state has the same building height for more than one of its tentallest buildings?

The mode represents data values that appear most often. In New York, aheight of 248 meters and 319 meters appear most often. In Texas, there are no building heights that appear more than once or most often. So,New York has the same building height for more than one of its ten tallest buildings.

The tallest building in Texas is 305 meters. How tall is the tenth tallestbuilding?

The range of the ten tallest buildings in Texas is 76 meters, so subtract 76 from 305.

305 � 76 � 229

The tenth tallest building in Texas is 229 meters tall.

When a value of a data set changes, you can often determine how the mean, median, mode, and range will be affected without recalculating allthe measures.

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 601

ExercisesBASEBALL For Exercises 1–4, use the information in the table on the number of runs batted in by the season leader for the National League and American League.

1. Which league had a greater average number of runs batted in by its season leaders from 1997–2006?

2. From 1997–2006, the greatest number of runs batted in by the season leader for the National League was 160 runs. What was the fewest number of runs batted in by the season leader?

3. From 1997–2006, the fewest number of runs batted in by the season leaderfor the American League was 137 runs. What was the greatest number ofruns batted in by the season leader?

4. Write a statement comparing the middle number of runs batted in by theNational League and American League season leaders from 1997–2006.

ONLINE For Exercises 5 and 6, use the list below that shows the number ofhours Ms. Wright’s students spent online last week. The mean of the data is6, the median is 7, the mode is 7, and the range is 12.

5. If the student who spent 12 hours online thought about it more andchanges his or her number of hours to 9 hours, how will this affect themean, median, mode, and range?

6. If Ms. Wright recorded a student who said 1 hour as 11 hours on the list,how will this affect the mean, median, mode, and range when she correctsthe data value to 1 hour and recalculates the measures?

Investigate a Change in Data Value

SCHOOL The table at the right shows the math test scores of Mr. Gomez’s fifth period class. The mean is 81%, the median is 82%, the mode is 85%, and the range is 30%. If after handing back the tests, the student who received a 65% did extra credit to change his test score to 75%, how will this affect the mean, median, mode, and range?

Since the test score is increasing, the mean will also increase.

The median is the middle number. Since 75% is still in the lower part of the test scores, it will not affect the median.

The mode is the number that appears most often. A new score of 75% willresult in three test scores of 75%. However, there will still be five test scoresof 85%, so the mode will stay the same.

The range is the difference between the greatest score and the least score.The greatest score will still be 95%. However, the least score will now be70%. So, the range will be 95% � 70%, or 25%, instead of 30%.

Number of Runs Batted In bySeason Leader, 1997–2006

MeanMedianModeRange

NationalLeague

142.914414732

AmericanLeague

147.714614528

Fifth Period Test Scores (%)Fifth Period Test Scores (%)

90 82 85 70 75

91 95 70 72 72

85 85 70 78 90

65 95 88 82 75

78 80 82 85 85

Source: The World Almanac

7 4 7 9 3 1 5 8 10 0 9 4 0

11 3 1 9 0 7 11 12 7 6 8 4 10

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 602

603 Prerequisite Skills

Multiply and Divide with Scientific NotationMississippi Standard: Multiply and divide numbers written in scientific notation.

You can use scientific notation to simplify computations with very large and/or very small numbers.

To multiply numbers in scientific notation, regroup to multiply the factors and multiply the powers of ten. Then simplify. To multiply the powers of ten, use the

.Product of Powers

Multiplication with Scientific Notation

Evaluate the expression (1.3 � 102)(2.5 � 101).

(1.3 � 102)(2.5 � 101) � (1.3 � 2.5)(102 � 101) Commutative and Associative Properties

� (3.25)(102 � 101) Multiply 1.3 by 2.5.

� 3.25 � 102 � 1 Product of Powers

� 3.25 � 103 Add the exponents.

� 3.25 � 1,000 103 � 1,000

� 3,250 Move the decimal point 3 places.

Evaluate the expression (4.2 � 103)(1.6 � 104).

(4.2 � 103)(1.6 � 104) � (4.2 � 1.6)(103 � 104) Commutative and Associative Properties

� (6.72)(103 � 104) Multiply 4.2 by 1.6.

� 6.72 � 103 � 4 Product of Powers

� 6.72 � 107 Add the exponents.

� 6.72 � 10,000,000 107 � 10,000,000

� 67,200,000 Move the decimal point 7 places.

Product of Powers

Words To multiply powers with the same base, add their exponents.

Symbols Arithmetic Algebra

32 � 35 � 32 � 5 or 37 xa � xb � xa � b

To divide numbers in scientific notation, regroup to divide the factors anddivide the powers of ten. Then simplify. To divide the powers of ten, use the

.Quotient of Powers

Quotient of Powers

Words To divide powers with the same base, subtract their exponents.

Symbols Arithmetic Algebra

�44

8

3� � 48 � 3 or 45 �xx

b

a� � xa � b, x � 0

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 603

ExercisesMultiply or divide. Express using exponents.

1. 51 � 54 2. 65 � 64 3. 102 � 103

4. �22

5

3� 5. �77

6

5� 6. �1100

9

6�

Evaluate each expression. Express the result in scientific notation and standard form.

7. (2.6 � 105)(1.9 � 102) 8. (5.3 � 104)(0.9 � 103)

9. (3.7 � 102)(1.2 � 102) 10. (3.3 � 103)(2.1 � 102)

11. (8.5 � 103)(1.1 � 101) 12. (3.9 � 102)(2.3 � 106)

13. (6.45 � 105)(1.2 � 103) 14. (4.18 � 104)(0.9 � 105)

15. �82.3.77��11003

8� 16. �

86.0.74��11002

5�

17. �91.7.82��11005

9� 18. �

42.6.94��11003

4�

19. �81.3.32��11005

7� 20. �

61..35

��11001

6

0�

21. �14..628��11008

2� 22. �92..04

��11001

8

1�

23. BASEBALL The table shows the 2007 salaries of six Major League Baseball players. About how many times greater is Alex Rodriguez’s salary than Juan Castro’s salary?

24. ASTRONOMY The Sun burns about 4.4 � 106 tons of hydrogen per second. How much hydrogen does the Sun burn in one year? (Hint: one year � 3.16 � 107 seconds)

25. OCEANS The area of the Pacific Ocean is 6.0 � 107 square miles. The area of the Atlantic Ocean is 2.96 � 107 square miles. About how many times greater is the area of the Pacific Ocean than the Atlantic Ocean?

2007 Major League Baseball Salaries

Player

Juan CastroCoco CrispNomar GarciaparraChipper JonesKazuo MatsuiAlex Rodriguez

Source: USA Today

Team

Cincinnati RedsBoston Red SoxLos Angeles DodgersAtlanta BravesColorado RockiesNew York Yankees

Salary(dollars)

9.25 � 105

3.83 � 106

8.52 � 106

1.23 � 107

1.5 � 106

2.27 � 107

Division with Scientific Notation

Evaluate the expression �92.4.15��11003

6�.

�92.4.15��11003

6� � ��

92.4.15

����1100

6

3�� Associative Property

� 4.5 � ��1100

6

3�� Divide 9.45 by 2.1.

� 4.5 � 106 � 3 Quotient of Powers

� 4.5 � 103 Subtract the exponents.

� 4.5 � 1,000 103 � 1,000

� 4,500 Move the decimal point 3 places.

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 604

The Density PropertyMississippi Standard: Develop a logical argument to demonstrate the‘denseness’ of rational numbers.

Examine the number line below. Find another integer that lies between theintegers �2 and 3.

The integers �1, 0, 1, and 2 all lie between �2 and 3 on the number line.On the number line above, find a number that lies between 1 and 2. Their

average, 1�12

�, is one number that lies between 1 and 2.

0 1 2 3 4-2 -1

Find a Number Between Two Given Numbers

Find a number that lies between �13

� and �12

� on the number line below.

One number would be their average.

�12

���13

� � �12

�� � �12

���26

� � �36

�� Rewrite �13

� and �12

� with a common denominator.

� �12

���56

�� Add the numerators.

� �152� Multiply.

The rational number, �152�, lies between �

13

� and �12

�.

Find a number that lies between �7 and �6.5.

One number would be their average.

�12

�[�7 � (�6.5)] � �12

�(�13.5) Add �7 and �6.5.

� �6.75 Multiply.

The rational number, �6.75, lies between �7 and �6.5.

0 156

12

13

16

23

The process above of finding another number between any two given numberscan be continued indefinitely. This suggests the .density property

You can use the density property to solve real-world problems.

Density Property for Rational Numbers

Words Between every pair of distinct rational numbers, there are infinitely many rational numbers.

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 605

ExercisesIdentify a number that lies between points A and B on each number line.

1. 2.

3. 4.

Identify a number that lies between each pair of numbers.

5. 6 and 7 6. �10 and �9 7. �34

� and 1

8. �2 and �1�12

� 9. 4�23

� and 4�34

� 10. �5 and �4�13

�

11. �4 and �3 12. 8.25 and 8.75 13. 15.5 and 16

14. SCHOOL For reading class, Dylan is recording the number of hours he reads

each week. This week, Dylan needs to read between 1�12

� and 2 hours. What

is a possible time between 1�12

� and 2 hours that Dylan can read?

15. CROSS COUNTRY For cross-country practice, the coach told the runners

they needed to run between 5�12

� and 5�34

� miles. Give a possible distance between

5�12

� and 5�34

� miles that a runner can run.

16. Demonstrate the density property for rational numbers with severalexamples of your own.

Apply the Density Property

BAKING Genevieve’s grandmother gave her a family recipe for apple pie.Her grandmother does not use an exact amount of sugar, but told

Genevieve to use somewhere between 1�14

� and 1�12

� cups of sugar. If

Genevieve wants to use an exact amount of sugar that is somewhere

between 1�14

� cups and 1�12

� cups, how much sugar can she use?

One possible amount is their average.

�12

��1�14

� � 1�12

�� � �12

��1�14

� � 1�24

�� Rename �12

� as �24

�.

� �12

��2�34

�� Add the whole numbers and add the fractions.

� �12

���141�� Rewrite 2�

43� as an improper fraction.

� �181� or 1�

38

� Simplify.

So, Genevieve can use 1�38

� cups of sugar.

0 1 2 3 4 5

A B

-4 -3 -2 -1 10

A B

A B

110

210

310

410

510

610

0 0.5 1 1.5 2 2.5

A B

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 606

Identify Properties

Name the property shown by each statement.

(7 � 3x) � 2x � 7 � (3x � 2x)

Associative Property

0 � 5a � 0

Zero Property of Multiplication

Algebraic PropertiesMississippi Standard: Apply algebraic properties in problem-solving.

Review the properties in the table below. These properties can be applied when problem-solving.

Properties

The order in which numbers are added or multiplied does not change thesum or product.

Examples 6 � 7 � 7 � 6 a � b � b � a3 � 8 � 8 � 3 a � b � b � a

The way in which numbers are grouped when added or multiplied does notchange the sum or product.

Examples (2 � 7) � 4 � 2 � (7 � 4) (a � b) � c � a � (b � c)(3 � 4) � 5 � 3 � (4 � 5) (a � b) � c � a � (b � c)

To multiply a sum by a number, multiply each addend by the numberoutside the parentheses.

Examples 2(7 � 4) � 2 � 7 � 2 � 4 a(b � c) � ab � ac(5 � 6)3 � 5 � 3 � 6 � 3 (b � c)a � ba � ca

The sum of any number and 0 is the number.

Examples 7 � 0 � 7 a � 0 � a

The product of any number and 0 is 0.

Examples 9 � 0 � 0 a � 0 � 0

The product of any number and 1 is the number.

Examples 3 � 1 � 3 a � 1 � a

Multiplicative Identity

Zero Property of Multiplication

Additive Identity

Distributive Property

Associative Property

Commutative Property

Use Properties to Simplify Expressions

Simplify each expression. Justify each step.

4 � (x � 13)

4 � (x � 13) � 4 � (13 � x) Commutative Property

� (4 � 13) � x Associative Property

� 17 � x Add 4 and 13.

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 607

6(x � 7)

6(x � 7) � 6(x) � 6(7) Distributive Property

� 6x � 42 Multiply.

ExercisesName the property shown by each statement.

1. 3n � m � m � 3n 2. 0 � 18d � 18d

3. (7y � 8) � 10y � 7y � (8 � 10y) 4. 20xy � 1 � 20xy

5. 3(6a � 7b) � 3 � 6a � 3 � 7b 6. 82 � 0 � 0

Simplify each expression. Justify each step.

7. 1 � (6 � x) 8. 5(6a) 9. 11 � (6 � n)

10. 5(x � 8) 11. 15(4w) 12. 9(x � 2)

13. 9 � 2y � 11 � 5y 14. 4(x � 7) � 2x 15. 11n � 7(2 � 3n)

16. ANIMALS A zebra can run up to 40 miles per hour. An elephant can run up to x miles per hour. Write and simplify an expression to find how many more miles azebra will run in six hours than an elephant.

17. CELL PHONES Seven friends have similar cell phone plans. The price of each plan is $x. Three of the seven friends pay an extra $4 per month for unlimitedtext messaging. Write and simplify an expression that represents the total cost ofthe seven plans.

Apply Properties to Problem Solving

MUSEUMS Three friends are going to the science museum. The cost of admission is $x each. It will cost an additional $4 to view a movie on the 3-D screen. Write and simplify an expression that represents the total cost for the three friends.The cost of admission plus the movie can be represented by (x + 4).

Multiply this cost by the number of friends, 3(x + 4).

3(x � 4) � 3(x) � 3(4) Distributive Property

� 3x � 12 Multiply.

So, the total cost for the three friends is $3x � $12.

MUSEUMS Refer to Example 5. A fourth friend will meet the group of friends at the museum but will not go to the movie. Write and simplify an expression that represents the total cost for the four friends.The cost for the fourth friend is $x. Add this to $3x � $12.

3x � 12 � x � 3x � x � 12 Commutative Property

� 4x � 12 Add.

So, the total cost for the four friends is $4x � $12.

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 608

Make Predictions from Circle Graphs and HistogramsMississippi Standard: Use proportions, estimates, and percentages toconstruct, interpret, and make predictions about a population based onhistograms or circle graph representations of data from a sample.

You can make predictions about a given set of data displayed in a circle graph or histogram. Use percentages to make predictions about data displayed in a circle graph.

Predict from a Circle Graph

The circle graph shows the results of a survey ofthe students in the 8th grade at Oakwood JuniorHigh. If there are 560 students at Oakwood JuniorHigh, how many would you predict to choosereality as their favorite type of television show?

The section of the graph representing students whochose reality is 40% of the circle. So find 40% of 560.

To find 40% of 560, you can use either method.

METHOD 1 Write the percent as a decimal.

40% of 560 � 40% 560 Write a multiplication expression.

� 0.40 560 Write 40% as a decimal.

� 224 Multiply.

METHOD 2 Write the percent as a fraction.

40% of 560 � 40% 560 Write a multiplication expression.

� �14000

� �5610

� Write 40% as a fraction. Write 560 as �5160�.

� 224 Multiply.

So, about 224 students at Oakwood Junior High would choose reality as their favorite type of television show.

4%Other

Favorite Type of Television Show

40%Reality

28%Comedy

16%Cartoon

5%Drama

7%Fiction

Predict from a Histogram

The histogram shows the winning times of themen’s 400-meter run in the summer Olympicsfrom 1896 to 2004. Predict the range of speedsthat a runner finishing in first place is mostlikely to be in the next summer Olympics?Explain your reasoning.

The bar at 43.0–44.9 seconds is much higher thanthe others and represents the most winning times.So, the winning speed of the runner in the nextsummer Olympics will most likely be in the 43.0–44.9 second range.

43.0–44.9

45.0–46.9

47.0–48.9

49.0–50.9

51.0–52.9

53.0–54.9

2

4

6

8

10

12

0

Num

ber o

f Win

ners

Time (seconds)

Summer Olympic Men’s 400-Meter RunWinning Times, 1896–2004

Source: The World Almanac

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 609

ExercisesCARS For Exercises 1–3, use the circle graph that shows the mostpopular luxury car colors.

1. If a car dealership sold 50 luxury cars in March, predict howmany were white.

2. If a car dealership sold 250 luxury cars in January throughJune, predict how many were black.

3. If a parking garage has 85 luxury cars parked on a given day,predict how many are silver/gray.

VACATION For Exercises 4 and 5, use the circle graph thatshows the results of a survey of the favorite summertimeactivities of 7th grade students at Parson Junior High.

4. If there are 275 students at Parson Junior High, predicthow many would choose visiting an amusement park astheir favorite summertime activity.

5. If there are 150 students at Parson Junior High, predicthow many would choose swimming or going to camp astheir favorite summertime activity.

HISTORY For Exercises 6 and 7, use the histogram that shows the age of U.S. presidents at theirinauguration.

6. Predict the 5-year age range that the next U.S.president will most likely be in at theirinauguration.

7. Predict the 10-year age range that the next U.S.president will most likely be in at theirinauguration.

SCHOOL For Exercises 8 and 9, use the histogram thatshows the test scores of Mrs. Jeng’s first period mathclass. Mrs. Jeng teaches three math classes of thesame level in the morning.

8. Predict the range that students in Mrs. Jeng’ssecond period math class will most likely score.

9. Predict the range that students in Mrs. Jeng’s thirdperiod math class will least likely score.

5%Other

Most PopularLuxury Car Colors

26%Silver/Gray

28%White12%

Black

9%Red

9%Blue

11%LightBrown

55–5

9

50–5

4

45–4

9

40–4

4

2

4

6

8

10

12

14

16

0

Num

ber o

f Pre

side

nts

60–6

4

65–6

9

Age at Inaguration

U.S. Presidents Age at Inauguration

81–9071–8061–7051–60

2

4

6

8

10

12

14

0

Num

ber o

f Stu

dent

s

91–100Score

Mrs. Jeng’s First Period Test Scores

5%Other

Favorite Summertime Activity

32%Amusement

Park

24%Swim

20%Camp

6%Read

13%Beach

600-610 MAC3-PS-874050 5/9/08 12:16 PM Page 610