1. CHAPTER 2
Review of Fractions

2. Identify types of fractions.
2-1
Learning Outcomes
Identify types of fractions.
Convert an improper fraction to a whole or mixed number.
Convert a mixed or whole number to an improper fraction.
Reduce a fraction to lowest terms.
Raise a fraction to higher terms.

3. A fraction is used to identify parts of a whole.
2-1-1
Identify types of fractions
Section 2-1
Fractions
A fraction is used to identify parts of a whole.
It describes the relationship between the part and the whole.
There are four parts.
One is shaded white, or 1 in 4 which is

4. The number appearing below the fraction line. Numerator
Key Terms…
Section 2-1
Fractions
Denominator
The number appearing below the fraction line.
Numerator
The number appearing above the fraction line.
Fraction line
Horizontal line separating numerator and denominator.
Proper fraction
A fraction with a value less than “1” (ex., )

5. Yes, because the value of the fraction is less than “1”.
HOW TO:
Look at the fraction
Section 2-1
Fractions
2 is the numerator.
3 is the denominator.
Is it a proper fraction?
Yes, because the value of the fraction is less than “1”.

6. What part of the area is shaded purple?
HOW TO:
Identify the fraction
Section 2-1
Fractions
What part of the area is shaded purple?

7. What part of the area is shaded purple?
HOW TO:
Identify the fraction
Section 2-1
Fractions
What part of the area is shaded purple?

8. HOW TO:
Improper fraction
Section 2-1
Fractions
The numerator is greater than or equal to the denominator.
Therefore the fraction is greater than or equal to “1”.
Proper or improper?
Improper
Proper
Improper

9. Convert an improper fraction to a whole or mixed number
2-1-2
Section 2-1
Fractions
Divide the numerator of the improper fraction by the denominator.
If the remainder is zero, the quotient is a whole number.
If the remainder is not zero, the improper fraction will be expressed as a mixed number.
Examples = =
= 12

10. Multiply the denominator of the mixed number by the whole number part.
Convert a whole or mixed number to an improper fraction
2-1-3
Section 2-1
Fractions
Multiply the denominator of the mixed number by the whole number part.
Add the product from the previous step to the numerator of the mixed number.
This is the numerator of the improper fraction.
Use the denominator of the mixed number as the denominator of the improper fraction.

11. Convert to an improper fraction. 10
An Example…
Section 2-1
Fractions
Convert to an improper fraction. 10
The numerator of the fraction is 3.
Multiply the whole number—which is 10—by the denominator—which is 4. The result is 40.
Add the numerator to the product: = 43.
Retain the same denominator.
The improper fraction equivalent is

12. Check if the denominator can be divided by the numerator:
2-1-4
Reduce a fraction to lowest terms
Section 2-1
Fractions
Inspect the numerator and denominator to find any whole number by which both can be evenly divided.
Carry out the operation until there is no one number that both can be evenly divided by.
Check if the denominator can be divided by the numerator:
TIP:
Example:
can be reduced to
when 3 is divided into 15.

13. Check if the denominator can be divided by the numerator:
Reduce these examples to the lowest terms…
Section 2-1
Fractions = = =
TIP:
Check if the denominator can be divided by the numerator:

14. Find the greatest common divisor of two numbers
HOW TO:
Section 2-1
Fractions
The most direct way to reduce a fraction to lowest terms is to use the GCD.
The GCD is the largest number by which the denominator and numerator can be evenly divided.
Example:
The GCD of 15 and 20 is 5. Any number greater than 5 would result in a quotient with a remainder.

15. Find the GCD of 42 and 28. More Examples: HOW TO: Find the GCD
Section 2-1
Fractions
Find the GCD of 42 and 28.
Divide the larger number by the smaller number: 42 divided by 28 = 1 R 14
Divide the divisor (28) by the remainder of the previous operation: (14) = 2 R 0.
When the R = 0, the divisor from that operation (14, in this case) is the GCD.
More Examples:
30, 36 30, , 51
GCD = 6
GCD = 5
GCD = 17

16. 2 = 2 2-1-5 Raise a fraction to higher terms
Section 2-1
Fractions
is equal to
8 / 4 = 2 2
Divide the two denominators. “4” goes into “8” two (2) times.
Multiply “3” by “2” to get the equivalent numerator.
Multiply “4” by “2” to get the equivalent denominator. = 2 2

17. Determine the equivalent fraction in higher terms:
Examples…
Section 2-1
Fractions
Determine the equivalent fraction in higher terms: = = =

18. Add fractions with like (common) denominators.
2-2
Learning Outcomes
Add fractions with like (common) denominators.
Find the least common denominator for two or more fractions.
Add fractions and mixed numbers.
Subtract fractions and mixed numbers.

19. + + = or Add the numerators. The denominator remains the same.
Add Fractions with like (common) denominators
2-2-1
Section 2-2
Adding and Subtracting Fractions
Add the numerators.
The denominator remains the same.
If necessary, convert an improper fraction to a mixed number. + + = or

20. Adding fractions with different denominators.
Find the least common denominator for two or more fractions
2-2-2
Section 2-2
Adding and Subtracting Fractions
Adding fractions with different denominators.
First find the least common denominator (LCD).
The LCD is the smallest number that can be divided evenly by each original denominator.

21. The common denominator can sometimes be found by inspection.
Find the least common denominator for two or more fractions
Section 2-2
Adding and Subtracting Fractions
The common denominator can sometimes be found by inspection.
Mentally selecting a number that can be evenly divided by each denominator.
Find the LCD for: and
Convert to an equivalent fraction in eighths,
Then add the two fractions together.

22. You can use prime numbers to find the LCD.
Find the least common denominator for two or more fractions
Section 2-2
Adding and Subtracting Fractions
It is not as apparent which number might be the LCD, given different denominators.
Such as 12 and 30.
You can use prime numbers to find the LCD.
A prime number is a number greater than 1 that can be divided evenly by only itself and 1.
TIP:
The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

23. Write the denominators in a row. STEP 1
Find the least common denominator for two or more fractions
HOW TO:
Section 2-2
Adding and Subtracting Fractions
Write the denominators in a row.
STEP 1
Divide each one by the smallest prime number by which any of the numbers can be evenly divided.
STEP 2
MORE

24. Write a new row of numbers using: STEP 3
Find the least common denominator for two or more fractions
HOW TO:
Section 2-2
Adding and Subtracting Fractions
Write a new row of numbers using:
1. The quotients from step 1.
2. Any numbers in the first row that cannot be evenly divided by the first prime number.
STEP 3
MORE

25. Continue this process until you have a row of 1s.
Find the least common denominator for two or more fractions
HOW TO:
Section 2-2
Adding and Subtracting Fractions
Divide again by the smallest prime number by which any of the numbers can be evenly divided.
STEP 4
STEP 5
Continue this process until you have a row of 1s.
MORE

26. Multiply all the prime numbers used to divide the denominators.
Find the least common denominator for two or more fractions
HOW TO:
Section 2-2
Adding and Subtracting Fractions
Multiply all the prime numbers used to divide the denominators.
The product is the least common denominator.
STEP 6

27. 5 2 8 Find the least common denominator for two or more fractions
HOW TO:
Section 2-2
Adding and Subtracting Fractions
Add the fractions:
Denominators 5 2 8 1 4
Prime numbers
, ,
First, find the LCD
Multiply the prime numbers from the first column (2 x 2 x 2 x 5) to get the LCD.
The LCD is 40
MORE

28. Convert the fractions to the equivalent using 40 as the denominator.
Find the least common denominator for two or more fractions
HOW TO:
Section 2-2
Adding and Subtracting Fractions
Add the fractions:
Denominators 5 2 8 1 4
Prime numbers
, ,
becomes
The LCD is 40
becomes
Convert the fractions to the equivalent using 40 as the denominator.
becomes
The LCD is 40
MORE

29. Inspect the fraction to determine if it is expressed in lowest terms.
Find the least common denominator for two or more fractions
HOW TO:
Section 2-2
Adding and Subtracting Fractions
Add the fractions:
, , = = + +
If the numerator is greater than the denominator, it is an improper fraction and can be expressed as a mixed number.
Inspect the fraction to determine if it is expressed in lowest terms.

30. Add the whole-number parts.
2-2-3
Add mixed numbers
Section 2-2
Adding and Subtracting Fractions
Add the whole-number parts.
Add the fraction parts and reduce to lowest terms.
Change improper fractions to whole or mixed numbers.

31. Borrow “1” from the whole number to carry out the operation.
2-2-4
Subtracting mixed numbers
Section 2-2
Adding and Subtracting Fractions
STEP 1
– =
Convert the fraction portion of each mixed number to equivalent fractions.
STEP 2
– =
Borrow “1” from the whole number to carry out the operation. -1
– =
Reduce to lowest terms, if necessary.

32. HOW TO:
Subtracting fractions with like denominators
Section 2-2
Adding and Subtracting Fractions
Subtract the smaller numerator from the greater one—the denominator remains the same. – = =
Reduce to lowest terms, if necessary.

33. Subtracting fractions with different denominators
HOW TO:
Section 2-2
Adding and Subtracting Fractions
As in addition, to subtract fractions you must have a common denominator.
Use the same methods of inspection or prime numbers to determine the LCD.
Carry out the operation.
Reduce to lowest terms as needed. –
= ?
MORE

34. = – = ? – = Subtracting fractions with different denominators HOW TO:
Section 2-2
Adding and Subtracting Fractions
Find the LCD, which is 12. =
Change to an equivalent fraction.
Carry out the operation and reduce to lowest terms, if needed. –
= ? – =

35. Examples…
Section 2-2
Adding and Subtracting Fractions
– =

36. Maria has cups of flour, but only needs cups for her recipe.
An Example…
Section 2-2
Adding and Subtracting Fractions
Maria has cups of flour, but only needs cups for her recipe.
How much will she have left?

37. Julia needs yards of tape to finish a display.
An Example…
Section 2-2
Adding and Subtracting Fractions
Julia needs yards of tape to finish a display.
Bob brought her a yard piece from the supply room.
How much will be left?

38. Multiply fractions and mixed numbers.
2-3
Learning Outcomes
Multiply fractions and mixed numbers.
Divide fractions and mixed numbers.

39. Can this fraction be reduced further?
2-3-1
Multiply fractions and mixed numbers
Section 2-3
Multiplying and Dividing Fractions
To multiply two fractions:
Find the numerator of the product by multiplying the numerators of the fractions.
Find the denominator of the product by multiplying the denominators of the fractions.
Reduce to lowest terms, if needed.
1 x 7 = 7
Can this fraction be reduced further? x
3 x 8 = 24
NO!

40. To keep things simple, if possible, reduce before multiplying.
Multiply fractions and mixed numbers
Section 2-3
Multiplying and Dividing Fractions
To keep things simple, if possible, reduce before multiplying.
TIP: ×
= ?
The 3 in the denominator in the first fraction and the 3 in the numerator in the second fraction cancel each other out and become 1. ×

41. Write the mixed numbers and whole numbers as improper fractions.
HOW TO:
Multiply mixed numbers and whole numbers
Section 2-3
Multiplying and Dividing Fractions
Write the mixed numbers and whole numbers as improper fractions.
Reduce numerators and denominators as appropriate.
Multiply the fractions.
Reduce to lowest terms and/or write as a whole number or mixed number.

42. The “3” can be reduced to “1” and the “15” to “5” before multiplying.
An Example…
Section 2-3
Multiplying and Dividing Fractions ×
= ? = = x ×
The “3” can be reduced to “1” and the “15” to “5” before multiplying. 25 4 =
Product:
Convert to a mixed number:

43. Are products always larger than their factors?
Products and factors
Section 2-3
Multiplying and Dividing Fractions
Are products always larger than their factors?
No. When the multiplier is a proper fraction, the product is less than the original number.
5 ×
= 3 × =
This is also true when the multiplicand is a whole number, fraction or mixed number.

44. What is the reciprocal of ?
2-3-2
Divide fractions and mixed numbers
Section 2-3
Multiplying and Dividing Fractions
The relationship between multiplying and dividing fractions involves a concept called reciprocals.
Two numbers are reciprocals if their product equals 1.
2 is the reciprocal of
What is the reciprocal of ?
The reciprocal is 3

45. Write the numbers as fractions. Find the reciprocal of the divisor.
HOW TO:
Divide fractions or mixed numbers
Section 2-3
Multiplying and Dividing Fractions
Write the numbers as fractions.
Find the reciprocal of the divisor.
Multiply the dividend by the reciprocal of the divisor.
Reduce to lowest terms, and/or write as a whole or mixed number.

46. Convert to an improper fraction:
HOW TO:
Divide fractions or mixed numbers
Section 2-3
Multiplying and Dividing Fractions
÷ = ?
Convert to an improper fraction:
Change to its reciprocal, which is:
Change from division to multiplication.
x = =

47. She can only make 4 appliqués, and she needs more fabric.
An Example…
Section 2-3
Multiplying and Dividing Fractions
Madison Duke makes appliqués. A customer has ordered five appliqués.
Madison has yard of fabric, and each appliqué uses of a yard.
Does she need more fabric? ÷
becomes
× 6
Simplify by dividing 4 and 6 by 2
× 3
The answer is
She can only make 4 appliqués, and she needs more fabric.

48. A home goods store is stacking decorative boxes on shelves.
An Example…
Section 2-3
Multiplying and Dividing Fractions
A home goods store is stacking decorative boxes on shelves.
Each box is inches tall.
The shelf space is 45 inches.
How many boxes will fit on each shelf?
Nine

49. Exercise Set

50. What are these fractions called? improper fractions
EXERCISE SET
2. Give five examples of fractions whose value is greater than or equal to 1.
Examples:
What are these fractions called?
improper fractions

51. Identify each fraction as proper or improper. See Example 1.
EXERCISE SET
Identify each fraction as proper or improper. See Example 1. 4.
improper 6.

52. Write the improper fraction as a whole or mixed number.
EXERCISE SET
Write the improper fraction as a whole or mixed number. 8.
10.
12.

53. Write the mixed number as an improper fraction. 14.
EXERCISE SET
Write the mixed number as an improper fraction.
14.
16.
18.

54. Reduce to lowest terms. Try to use the greatest common divisor. (GCD)
EXERCISE SET
Reduce to lowest terms. Try to use the greatest common divisor. (GCD)
20.
22.
24.

55. Rewrite as a fraction with the indicated denominator. 26. 28.
EXERCISE SET
Rewrite as a fraction with the indicated denominator.

56. EXERCISE SET
30. If 8 students in a class of 30 earned grades of A, what fractional part of the class earned A’s?

57. Find the least common denominator (LCD) for these fractions.
EXERCISE SET
Find the least common denominator (LCD) for these fractions.
32.

58. Find the least common denominator (LCD) for these fractions.
EXERCISE SET
Find the least common denominator (LCD) for these fractions.
34.

59. Find the least common denominator (LCD) for these fractions.
EXERCISE SET
Find the least common denominator (LCD) for these fractions.
36.

60. EXERCISE SET
Add. Reduce to lowest terms and write as whole or mixed numbers if appropriate.
38.
40.

61. EXERCISE SET
Add. Reduce to lowest terms and write as whole or mixed numbers if appropriate.
42.
44.

62. EXERCISE SET
46. Three pieces of lumber measure 5 3/8 feet, 7 ½ feet, and 9 ¾ feet. What is the total length of the lumber?

63. EXERCISE SET
Subtract. Borrow when necessary. Reduce the difference to lowest terms (simplify).

64. EXERCISE SET
Subtract. Borrow when necessary. Reduce the difference to lowest terms (simplify).

65. EXERCISE SET
56. A board 3 5/8 feet long must be sawed from a 6-foot board. How long is the remaining piece?

66. EXERCISE SET
Multiply. Reduce to lowest terms and write as whole or mixed numbers if appropriate.
58.
60.

67. EXERCISE SET
Multiply. Reduce to lowest terms and write as whole or mixed numbers if appropriate.
62. 5

68. EXERCISE SET
64. After a family reunion, 10 2/3 cakes were left. If Shirley McCool took 3/8 of these cakes, how many did she take?

69. Find the reciprocal of the numbers. 66. 8
EXERCISE SET
Find the reciprocal of the numbers.
68.
70.

70. EXERCISE SET
Divide. Reduce to lowest terms and write as whole or mixed numbers if appropriate.
72.

71. EXERCISE SET
Divide. Reduce to lowest terms and write as whole or mixed numbers if appropriate.
74.
76.

72. EXERCISE SET
78. Brienne Smith must trim 2 3/16 feet from a board 8 feet long. How long will the board be after it is cut?
80. Sol’s Hardware and Appliance Store is selling electric clothes dryers for 1/3 off the regular price of $288. What is the sale price of the dryer?

73. EXERCISE SET
82. Certain financial aid students must pass 2/3 of their courses each term in order to continue their aid. If a student is taking 18 hours, how many hours must be passed?
84. A stack of 1 5/8-inch plywood measures 91 inches. How many pieces of plywood are in the stack?

74. Practice Test

75. Write as a mixed number or whole number. 4.
EXERCISE SET
Write the reciprocal. 2.
Write as a mixed number or whole number. 4.

76. Write as an improper fraction. 6.
EXERCISE SET
Write as an improper fraction. 6.
Reduce by using the greatest common divisor (GCD).

77. EXERCISE SET
Perform the indicated operation. Reduce results to lowest terms and write as whole or mixed numbers if appropriate.
12.
14.

78. EXERCISE SET
Perform the indicated operation. Reduce results to lowest terms and write as whole or mixed numbers if appropriate.
16.
18.

79. EXERCISE SET
20. A company that employs 580 people expects to lay off 87 workers. What fractional part of the workers are expected to be laid off?

80. EXERCISE SET
22. If city sales tax is 5 ½% and state sales tax is 2 ¼%, what is the total sales tax rate for purchases made in the city?

81. EXERCISE SET
24. The top-rated television series Cupcake Wars requires the two finalists to use their cupcake recipes to create a display for a large event. Lindsey Morton’s Cinnamon Sugar Graham cupcake recipe for 36 cupcakes requires 1 ¼ cup of sugar. How much sugar is required for 900 cupcakes for her display?