1. Factors
LESSON 2-1
Problem of the Day
Find three different odd numbers whose sum is 21. List all possibilities.
1, 3, 17; 1, 5, 15; 1, 7, 13; 1, 9, 11; 3, 5, 13; 3, 7, 11; 5, 7, 9
2-1

2. 1. Vocabulary Review When you multiply two numbers,
Factors
LESSON 2-1
Check Skills You’ll Need
(For help, go to Lesson 1-4.)
1. Vocabulary Review When you multiply two numbers,
the result is called the ? .
Simplify each expression.
2. –10(10) 3. –8(– 7)
4. 5(– 4)(– 2) 5. –1 • 1 • 0
Check Skills You’ll Need
2-1

3. Solutions 1. product 2. –100 3. 56 4. 40 5. 0 Factors
LESSON 2-1
Check Skills You’ll Need
Solutions
1. product –
2-1

4. Identify each number as prime or composite. Explain.
Factors
LESSON 2-1
Additional Examples
Identify each number as prime or composite. Explain.
a. 57
Composite; the sum of the digits is 12, which is divisible by 3.
b. 1,354
Composite; the number is divisible by 2.
c. 43
Prime; the number is divisible only by 43 and 1.
d. 975
Composite; the number is divisible by 5.
Quick Check
2-1

5. Use a factor tree to find the prime factorization of 588.
Factors
LESSON 2-1
Additional Examples
Use a factor tree to find the prime factorization of 588.
The number 588 is divisible by 2 because the units digit is 8.
588
294 2
prime
Begin the factor tree with 2 • 294.
147 2
prime 49 3
prime
Stop when all factors are prime. 7
prime
Quick Check
The prime factorization of 588 is 2 • 2 • 3 • 7 • 7, or 22 • 3 • 72.
2-1

6. Step 1 Find the prime factorization of each number.
Factors
LESSON 2-1
Additional Examples
Quick Check
Find the GCF of 55 and 231.
Step 1 Find the prime factorization of each number. 55 11 5
231 77 3 7
Step 2 Find the product of the common prime factors of each number.
55 = 5 • 11
231 = 3 • 7 • 11
The only common prime factor is 11. The GCF of 55 and 231 is 11.
2-1

7. A band with 36 members is marching with a 32-member
Factors
LESSON 2-1
Additional Examples
Quick Check
A band with 36 members is marching with a 32-member
band. If the two bands are to have the same number of columns,
what is the greatest number of columns in which you could
arrange the two bands?
Begin by finding the factors of 36 and 32.
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
32: 1, 2, 4, 8, 16, 32
The GCF is 4.
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
32: 1, 2, 4, 8, 16, 32
The factors 1, 2, and 4 are common to both numbers.
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
32: 1, 2, 4, 8, 16, 32
So, 4 is the greatest number of columns in which you can arrange the bands.
2-1

8. Write the prime factorization of each number. 1. 24 2. 27 3. 31
Factors
LESSON 2-1
Lesson Quiz
Write the prime factorization of each number.
Find the GCF of each pair of numbers.
4. 4 and and 27
23 • 3 33 31 2 9
2-1

9. Equivalent Forms of Rational Numbers
LESSON 2-2
Problem of the Day
At the Red Valley Sports Camp, 15 kids went horseback riding, 14 played tennis, 23 went hiking, and the rest of the campers stayed in their cabins. If 83 kids were in the camp, how many stayed indoors? 31
2-2

10. Equivalent Forms of Rational Numbers
LESSON 2-2
Check Skills You’ll Need
(For help, go to Lesson 2-1.)
1. Vocabulary Review Name the prime factorization
of 100.
Find the GCF of each pair of numbers.
2. 6, , , , 40
Check Skills You’ll Need
2-2

11. Equivalent Forms of Rational Numbers
LESSON 2-2
Check Skills You’ll Need
Solutions
1. 2 • 2 • 5 • 5
2. 6 = 2 • 3, 12 = 2 • 2 • 3; GCF = 6
3. 8 = 2 • 2 • 2, 12 = 2 • 2 • 3; GCF = 2 • 2 = 4
4. 25 = 5 • 5, 50 = 2 • 5 • 5; GCF = 5 • 5 = 25
5. 36 = 2 • 2 • 3 • 3, 40 = 2 • 2 • 5; GCF = 2 • 2 = 4
2-2

12. Equivalent Forms of Rational Numbers
LESSON 2-2
Additional Examples
Write in simplest form using the GCF.
138
150
The GCF of 138 and 150 is 6.
Divide the numerator and the denominator by the GCF.
138
150
138 ÷ 6
150 ÷ 6 =
Simplify. The numbers 23 and 25 are relatively prime. = 23 25
Quick Check
2-2

13. Equivalent Forms of Rational Numbers
LESSON 2-2
Additional Examples
Write in simplest form using prime factorization. 60
126
Write the prime factorizations of the numerator and denominator. 60
126
2 • 2 • 3 • 5
2 • 3 • 3 • 7 =
Divide the common factors.
2 • 2 • 3 • 5
2 • 3 • 3 • 7 1 =
Simplify. 10 21 =
Quick Check
2-2

14. Equivalent Forms of Rational Numbers
LESSON 2-2
Additional Examples
Write each batting average as a decimal.
a. Joe made 4 hits in 20 times at bat.
Write the batting average as a fraction. 4 20
Divide the numerator by the denominator.
This is a terminating decimal.
0.2
Joe’s batting average was .200.
b. Pat made 6 hits in 33 times at bat.
Write the batting average as a fraction. 6 33
Use a calculator. This is a repeating decimal.
Pat’s batting average was about .182.
Quick Check
2-2

15. Equivalent Forms of Rational Numbers
LESSON 2-2
Additional Examples
Write as a mixed number.
3.225 =
Write as a fraction with the denominator 1.
3.225 1
Since there are 3 digits to the right of the decimal, multiply the numerator and denominator by 103 or 1,000. =
3,225
1,000
Simplify using the GCF, 25. =
3,225 ÷ 25
1,000 ÷ 25
129 40
Write as a mixed number.
= 3 9 40
Quick Check
2-2

16. Equivalent Forms of Rational Numbers
LESSON 2-2
Lesson Quiz
Write each as a fraction in simplest form. –
3. Write as a decimal.
Write each decimal as a mixed number or fraction in
simplest form. 30 42 12 18 5 7 2 3 – 2 16
0.125 3 4 2 2 5
2-2

17. Comparing and Ordering Rational Numbers
LESSON 2-3
Problem of the Day
Express 4 days, 12 hours in minutes.
6,480 min
2-3

18. Comparing and Ordering Rational Numbers
LESSON 2-3
Check Skills You’ll Need
(For help, go to Lesson 2-2.)
1. Vocabulary Review Explain what the numerator of a
fraction represents.
Use the GCF to write each fraction in simplest form. 12 20 15 55 16 64 50
550
Check Skills You’ll Need
2-3

19. Comparing and Ordering Rational Numbers
LESSON 2-3
Check Skills You’ll Need
Solutions
1. The numerator represents a part of the whole.
12 ÷ 4
20 ÷ 4 = 3 5
15 ÷ 5
55 ÷ 5 = 3 11
16 ÷ 16
64 ÷ 16 = 1 4
50 ÷ 50
550 ÷ 50 = 1 11
2-3

20. Comparing and Ordering Rational Numbers
LESSON 2-3
Additional Examples
Which is greater, or ? 7 18 5 12
Multiples of 18: 18, 36
Multiples of 12: 12, 24, 36
List multiples of each denominator to find their LCD.
Since the LCM of 18 and 12 is 36, the LCD of the fractions is 36.
Multiply the numerator and denominator by 2. 7 18
7 • 2
18 • 2 =
Simplify. 14 36 =
2-3

21. Comparing and Ordering Rational Numbers
LESSON 2-3
Additional Examples
(continued)
Multiply the numerator and denominator by 3. 5 12
5 • 3
12 • 3 =
Simplify. 15 36 =
Since 15 36 14
> , 7 18 . 5 12
Quick Check
2-3

22. Comparing and Ordering Rational Numbers
LESSON 2-3
Additional Examples
The Eagles won 7 out of 11 games while the Seals won 8 out of 12 games. Which team has the better record?
Change each fraction to a decimal. Compare the two decimals.
Divide. Use a calculator.
Eagles: 7 11
Seals: 8 12
Since > 0.636, the Seals have the better record.
Quick Check
2-3

23. Comparing and Ordering Rational Numbers
LESSON 2-3
Additional Examples
Order –0.175, , – , 1.7, –0.95 from least to greatest. 2 3 5 8
Write each fraction as a decimal. 2 3
0.667 5 8
= –0.625 –
Then graph each decimal on a number line.
The order of the points from left to right gives the order of the numbers from least to greatest.
–0.95 < –0.625 < –0.175 < < 1.7
So, –0.95 < – < –0.175 < < 1.7. 2 3 5 8
Quick Check
2-3

24. Comparing and Ordering Rational Numbers
LESSON 2-3
Lesson Quiz
Compare. Use <, >, or =.
5. Order 0.17, , –0.3, 0, and – from least to greatest.
6. A survey found that 75 out of 125 men and 88 out of
136 women prefer comedy films over action films.
Which group prefers comedy over actions films more? 5 12
< 8 15 4 5
> 8 11 15 50 36
120 7 20 = = 1 5 1 4
–0.3, – , 0, 0.17, 1 4 5
women
2-3

25. Adding and Subtracting Rational Numbers
LESSON 2-4
Problem of the Day
Four divers competed in the belly-flop contest. The bigger the splash the better they do. John made a bigger splash than Bo. Allison came in third. Jennifer came in first with the biggest splash. In what order did the divers finish?
Jennifer, John, Allison, Bo
2-4

26. Adding and Subtracting Rational Numbers
LESSON 2-4
Check Skills You’ll Need
(For help, go to Lesson 1-3.)
1. Vocabulary Review Which numbers are integers:
2, 4.5, 0, –6, ?
Simplify each expression.
2. –9 – – (– 2) 1 3
Check Skills You’ll Need
2-4

27. Adding and Subtracting Rational Numbers
LESSON 2-4
Check Skills You’ll Need
Solutions
1. 2, 0, – – –
2-4

28. Adding and Subtracting Rational Numbers
LESSON 2-4
Additional Examples 1 3 3 4
A recipe calls for cup white flour and cup wheat flour. How many total cups of flour are used? 1 3
1 • 4
3 • 4 + 4 =
3 • 3
4 • 3
Write equivalent fractions with
the same denominator. 4 12 = + 9
Simplify. = 13 12
Add the numerators.
You need , or 1 , cups of flour. 13 12 1
Quick Check
2-4

29. Adding and Subtracting Rational Numbers
LESSON 2-4
Additional Examples
Find – . 9 10 3 4
The LCM of 10 and 4 is 20, so the LCD of and is 20. 9 10 3 4 9 10
Write equivalent fractions using the LCD. 3 4 – = 18 20 15
Subtract the numerators.
18 – 15 20 3 =
Quick Check
2-4

30. Adding and Subtracting Rational Numbers
LESSON 2-4
Additional Examples 3 4 2 3
Find
Method 1 Use improper fractions.
Write each mixed number as an improper fraction. = 27 4 3 2 26
Write equivalent fractions using the LCD, 12. = 81 12
104
Add the numerators. =
185 12
Change the improper fraction to a mixed number.
= 15 5 12
2-4

31. Adding and Subtracting Rational Numbers
LESSON 2-4
Additional Examples
(continued)
Method 2 Rewrite the mixed numbers using common denominators. 9 12 3 4 2 = 8
Rewrite each mixed number using the LCD, 12. 17 12
= 14 +
Add the integers and the fractions. 5 12 =
Change the improper fraction to a mixed number.
Add the integers. 5 12
= 15
Quick Check
2-4

32. Adding and Subtracting Rational Numbers
LESSON 2-4
Additional Examples 3 4
On a 50-foot roll of cable, ft are left. How many feet of cable were used?
Let t = the amount used.
Words amount left + amount used = original amount 3 4
Equation t = 50
2-4

33. Adding and Subtracting Rational Numbers
LESSON 2-4
Additional Examples
(continued)
t = 50 3 4
Subtract from each side.
t = 50 – 15 3 4
Rewrite 50 as , or 1. Subtract.
t = – = 34 4 3 1
The amount of cable used was ft. 1 4
Quick Check
2-4

34. Adding and Subtracting Rational Numbers
LESSON 2-4
Lesson Quiz
Find each sum or difference.
– – 4. 4
5. It snowed 2 in. on top of 4 in. of snow already on the ground.
How deep is the snow now? 8 9 5 9 4 5 2 3 7 8 1 2 5 6
– 1 1 4 1 3 7 15 1 3 8 7 12 3 1 2 1 2
7 in.
2-4

35. Multiplying and Dividing Rational Numbers
LESSON 2-5
Problem of the Day
Express 106,457,086,299 in words.
one hundred six billion, four hundred fifty-seven million, eighty-six
thousand, two hundred ninety-nine
2-5

36. Multiplying and Dividing Rational Numbers
LESSON 2-5
Check Skills You’ll Need
(For help, go to Lesson 2-2.)
1. Vocabulary Review A rational number can be written in
the form ? , where a and b are integers, and b = 0.
Simplify each expression. 7 21 12 20 9 81 36 66
Check Skills You’ll Need
2-5

37. Multiplying and Dividing Rational Numbers
LESSON 2-5
Check Skills You’ll Need
Solutions
= =
= = a b
7 ÷ 7
21 ÷ 7 1 3
12 ÷ 4
20 ÷ 4 3 5
9 ÷ 9
81 ÷ 9 1 9
36 ÷ 6
66 ÷ 6 6 11
2-5

38. Multiplying and Dividing Rational Numbers
LESSON 2-5
Additional Examples
Find • – . 5 12 4 7
Multiply the numerators and multiply the denominators. 4 7 5 12
• –
= –
5 • 4
12 • 7
Divide the numerator and demoninator by their GCF, 4.
= –
5 • 4
12 • 7 1 3
Simplify.
= – 5 21
Quick Check
2-5

39. Multiplying and Dividing Rational Numbers
LESSON 2-5
Additional Examples
Find the product –3 • – 1 3 5 6
Estimate: –3 • – –3 • (–3) = 9 1 3 5 6
Write as improper fractions.
–3 • – = – • – 1 3 5 6 10 17
Divide the numerator and denominator by their GCF, 2. =
10 • 17
3 • 6 5 3
Simplify. Write as a mixed number. = 85 9
= 9 4
Check Since is close to 9, the answer is reasonable. 4 9
Quick Check
2-5

40. Multiplying and Dividing Rational Numbers
LESSON 2-5
Additional Examples
One bow takes yards of ribbon. How many bows could you make from a roll of ribbon that is yards long? 3 4 1 2
You can use logical reasoning to solve this problem. You need to find how many -yd pieces there are in yards. 3 4 1 2
Divide by . 3 4 1 2
Write the mixed number as an improper fraction. 25 2
12 ÷ = ÷ 3 4 1
Multiply by the reciprocal of . 25 2 4 3
= •
2-5

41. Multiplying and Dividing Rational Numbers
LESSON 2-5
Additional Examples
(continued)
Divide numerator and denominator by the GCF, 2. 25 2 4 3
= • 1
Multiply. Write the fraction as a mixed number.
= = 16 50 3 2
Since you cannot make of a bow, you can make 16 bows. 2 3
Quick Check
2-5

42. Multiplying and Dividing Rational Numbers
LESSON 2-5
Lesson Quiz
• (– ) 2.
÷ ( ) 4.
5. Solve the equation r =
6. Megan has 3 quarts of punch. One serving is quart.
Does she have enough to serve 15 guests? 5 8 4 5
2 • (–1 ) 1 3 8 1 2 – 5 8 –2 – 1 6 – 3 4
(–1 ) ÷ (1 ) 7 8 1 2 2 9 1 4 –1 2 3 5 6 1 2 1 2 1 4 No
2-5

43. Formulas
LESSON 2-6
Problem of the Day
Twin primes are pairs of prime numbers who have a difference of 2. For example, 43 – 41 = 2. Name the twin primes between 2 and 35.
5, 3; 5, 7; 11, 13; 17, 19; 29, 31
2-6

44. 1. Vocabulary Review According to the order of
Formulas
LESSON 2-6
Check Skills You’ll Need
(For help, go to Lesson 1-1.)
1. Vocabulary Review According to the order of
operations, you multiply and divide before you ? and ? .
Evaluate each expression for w = 2 and t = –3.
2. 4w + t 3. 4(w + t)
4. 4w + 4t 5. – 4t –w
Check Skills You’ll Need
2-6

45. 2. 4(2) + (–3) = 8 + (–3) = 5 3. 4[2 + (–3)] = 4(–1) = –4
Formulas
LESSON 2-6
Check Skills You’ll Need
Solutions
1. add; subtract
2. 4(2) + (–3) = 8 + (–3) = [2 + (–3)] = 4(–1) = –4
4. 4(2) + 4(–3) = 8 + (–12) = –4 5. –4(–3) – 2 =12 – 2 =10
2-6

46. Use the formula for the area of a trapezoid.
Formulas
LESSON 2-6
Additional Examples
Find the area of a trapezoid with height of 6 cm and bases of 5.2 cm and 7.5 cm.
A = h (b1 + b2)
Use the formula for the area of a trapezoid. 1 2
= (6.0) ( ) Substitute. 1 2
2-6

47. = (6.0)(12.7) Add within the parentheses.
Formulas
LESSON 2-6
Additional Examples
(continued) 1 2
= (6.0)(12.7) Add within the parentheses.
= 3(12.7) Multiply from left to right.
= 38.1 Simplify.
The area of the trapezoid is 38.1 cm2.
Quick Check
2-6

48. d = rt Use the distance formula.
Formulas
LESSON 2-6
Additional Examples
Find the time it takes a sled-dog team to go 95 miles if their average rate is 19 mph.
The problem gives distance and rate. Use the distance formula, d = rt where d is the distance traveled, r is the rate of travel, and t is the time spent traveling.
d = rt Use the distance formula.
95 = 19 • t Substitute 95 for d and 19 for r.
Divide each side by 19 to isolate t on the right. 95 19
19 • t =
5 = t
Simplify. 95 19 t =
Quick Check
It took the sled-dog team 5 hours to go 95 miles.
2-6

49. Which formula can be used to find the diameter
Formulas
LESSON 2-6
Additional Examples
Which formula can be used to find the diameter
d of a circle, given the circumference C?
Use the circumference formula for a circle.
C =  d  C
Divide each side by to isolate the variable d.
 d =
= d Simplify.  C
The formula for the diameter of a circle is d = .  C
Quick Check
2-6

50. 1. Find the area of a triangle whose base is 18 cm and height is 3 cm.
Formulas
LESSON 2-6
Lesson Quiz
1. Find the area of a triangle whose base is 18 cm and
height is 3 cm.
2. Amina purchased a circular glass tabletop. The radius of
the tabletop is 6.5 inches. Find the area of the tabletop.
Use A = r 2 and let  = 3.
3. Solve for w in the formula V = wh.
4. Tyrone drove 1570 miles in 4 days. Find the average
distance he drove each day.
27 cm2
square inches
w = v h
392.5 miles
2-6

51. Formulate a set of 5 different numbers whose median is 95 and
Powers and Exponents
LESSON 2-7
Problem of the Day
Formulate a set of 5 different numbers whose median is 95 and
whose mean is 100.
Sample Answer
90, 92, 95, 110, 113
2-7

52. 1. Vocabulary Review A ? is an integer that divides
Powers and Exponents
LESSON 2-7
Check Skills You’ll Need
(For help, go to Lesson 2-1.)
1. Vocabulary Review A ? is an integer that divides
another integer with a remainder of 0.
Find the GCF.
2. 12, , , 48
5. 120, , 256
Check Skills You’ll Need
2-7

53. Solutions 1. factor 2. 4 3. 6 4. 16 5. 24 6. 16 Powers and Exponents
LESSON 2-7
Check Skills You’ll Need
Solutions
1. factor
2-7

54. 2 is a factor 3 times, and 7 is a factor 2 times. 23 • 72
Powers and Exponents
LESSON 2-7
Additional Examples
Write using exponents.
2 • 2 • 2 • 7 • 7
2 is a factor 3 times, and 7 is a factor 2 times.
23 • 72
Quick Check
2-7

55. Simplify the expression.
Powers and Exponents
LESSON 2-7
Additional Examples
Simplify the expression.
(–2)6
(–2)6 = (–2)(–2)(–2)(–2)(–2)(–2)
The base is –2.
Multiply.
= 64
Quick Check
2-7

56. Simplify the expression.
Powers and Exponents
LESSON 2-7
Additional Examples
Simplify the expression.
–(2)6
The base is 2.
–26 = –(2 • 2 • 2 • 2 • 2 • 2)
Multiply.
= –64
Quick Check
2-7

57. Simplify the expression.
Powers and Exponents
LESSON 2-7
Additional Examples
Simplify the expression.
38 – (3 • 2)2
38 – (3 • 2)2 = 38 – (6)2
Work inside the grouping symbols.
= 38 – 36
Simplify the power.
= 2
Subtract.
Quick Check
2-7

58. Use the expression to find the radius of a doorway that
Powers and Exponents
LESSON 2-7
Additional Examples
s2 + h2 2h
Use the expression to find the radius of a doorway that
has the dimensions s = 3 ft and h = 1 ft.
s2 + h2 2h
2 • 1 =
Substitute 3 for s and 1 for h.
9 + 1
2 • 1 =
The fraction bar acts as a grouping symbol. Simplify the powers. 10 2 =
Simplify above and below the fraction bar.
Divide.
= 5
The radius of the doorway is 5 ft.
Quick Check
2-7

59. 1. Write a • a • a • b • b using exponents.
Powers and Exponents
LESSON 2-7
Lesson Quiz
1. Write a • a • a • b • b using exponents.
2. Simplify (–4)3. 3. Simplify –25.
4. Simplify (–8 • 5)2 – Evaluate 10 – (5x)2 for x = –2.
6. Find the volume of a child’s wading pool that has a
diameter of 6 feet and a height of 1 foot. Use the
formula V =  r 2h. Use  = 3.
a3b2
–64
–32
1,519
–90
27 cubic feet
2-7

60. Evaluate the following expressions. Write the answers in lowest terms.
Scientific Notation
LESSON 2-8
Problem of the Day
Evaluate the following expressions. Write the answers in lowest terms.
a – = ? b = ? 9 10 1 2 3 4 1 10 2 5 17 20
2-8

61. 1. Vocabulary Review An expression using a base and an
Scientific Notation
LESSON 2-8
Check Skills You’ll Need
(For help, go to Skills Handbook page 634.)
1. Vocabulary Review An expression using a base and an
exponent is a ? .
Multiply.
2. 2  10
 100
 1,000
 10,000
 1,000,000
Check Skills You’ll Need
2-8

62. Scientific Notation
LESSON 2-8
Check Skills You’ll Need
Solutions
1. power
4. 1,500.0
5. 18,030.0
6. 2,390,000.0
2-8

63. Move the decimal 10 places to the right. Insert zeros as necessary. .
Scientific Notation
LESSON 2-8
Additional Examples
At one point, the distance from Earth to the moon is  1010 in. Write this number in standard form.
Move the decimal 10 places to the right. Insert zeros as necessary. .
 1010 =
= 15,134,310,000
At one point, the distance from Earth to the moon is 15,134,310,000 in.
Quick Check
2-8

64. The decimal point moves 5 places to the left. .
Scientific Notation
LESSON 2-8
Additional Examples
The diameter of the planet Jupiter is about 142,800 km. Write this number in scientific notation.
142,800 = 1 42,800.
The decimal point moves 5 places to the left. . 1.
= 1.428  105
Use 5 as the exponent of 10.
The diameter of the planet Jupiter is about  105 km.
Quick Check
2-8

65. Write 4.86 x 10–3 in standard form.
Scientific Notation
LESSON 2-8
Additional Examples
Write 4.86 x 10–3 in standard form.
4.86 x 10–3 = 
Move the decimal point
3 places to the left to make
4.86 less than 1. = 
Quick Check
2-8

66. Write 0.0000059 using scientific notation.
LESSON 2-8
Additional Examples
Write using scientific notation. =
Move the decimal point
6 places to the right to get
a factor greater than 1 but
less than 10.
=  5.9 x 10–6
Use 6 as the exponent of 10.
Quick Check
2-8

67. 1. Write 7.304  102 in standard form.
Scientific Notation
LESSON 2-8
Lesson Quiz
1. Write  102 in standard form.
2. Write 41,700,000,000 in scientific notation.
3. Write 3.03 x 10–5 in standard form.
4. Write using scientific notation.
730.4
4.17  1010
1.27  10–6
2-8