2.1 Fractions

If we divide one whole unit into several equal portions, then one or more of these equal portions can be represented by a fraction.

A fraction is composed of the following three parts:

1. Numerator: the number of equal parts of a whole unit.
2. Fraction bar: the division sign, meaning ‘divided by’.
3. Denominator: the total number of equal parts into which the whole unit is divided.

For example, [latex] \displaystyle{\frac{3}{8}} [/latex] is a fraction.

The numerator ‘3’ indicates that the fraction represents 3 equal parts of a whole unit and the denominator ‘8’ indicates that the whole unit is divided into 8 equal parts, as shown above. The fraction bar indicates that the numerator ‘3’ is divided by the denominator ‘8’.

[latex] \displaystyle{\frac{3}{8}} [/latex] is read as “three divided by eight”, “three-eighths”, or “three over eight”. All of these indicate that 3 is the numerator, 8 is the denominator, and the fraction represents 3 of 8 pieces of the whole. The numerator and denominator are referred to as the terms of the fraction.

Note: The denominator of a fraction cannot be zero, since a number cannot be divided into zero equal parts.

Any number that can be represented as a fraction is known as a rational number.

For example,

• [latex] \displaystyle{\frac{2}{3}} [/latex], [latex] \displaystyle{\frac{5}{2}} [/latex], and [latex] \displaystyle{\frac{7}{1}} [/latex] are rational numbers.
• The whole number 7 can be written as the fraction [latex] \displaystyle{\frac{7}{1}} [/latex]; therefore, 7 is also a rational number.

Fractions can be represented on a number line. They are plotted in-between whole numbers.

For example, [latex] \displaystyle{\frac{1}{2}} [/latex] is represented on a number line as:

When one unit is divided into two equal portions, each portion represents one-half of that unit. One of such equal portions is written as [latex] \displaystyle{\frac{1}{2}} [/latex].

Example 2.1-a: Representing Fractions on a Number Line

Represent the following fractions on a number line:

1. [latex] \displaystyle{\frac{2}{3}} [/latex]

2. [latex] \displaystyle{\frac{3}{5}} [/latex]

Types of Fractions

Proper Fractions

A proper fraction is a fraction in which the numerator is less than the denominator.

For example,

• [latex] \displaystyle{\frac{3}{8}} [/latex] is a proper fraction because the numerator, 3, is less than the denominator, 8.

  

Improper Fractions

An improper fraction is a fraction in which the numerator is greater than or equal to the denominator; i.e., the value of the entire fraction is more than 1.

For example,

• [latex] \displaystyle{\frac{7}{4}} [/latex] is an improper fraction because the numerator 7 is greater than the denominator 4
  (i.e., 7 > 4, or [latex] \displaystyle{\frac{7}{4}} [/latex] > 1).

  

Mixed Numbers (or Mixed Fractions)

A mixed number consists of both a whole number and a proper fraction, written side-by-side, which implies that the whole number and the proper fraction are added.

For example,

• [latex] 3{\frac{5}{8}} [/latex] is a mixed number, where 3 is the whole number, and [latex] \displaystyle{\frac{5}{8}} [/latex] is the proper fraction.
  [latex] 3{\frac{5}{8}} [/latex] implies 3 + [latex] \displaystyle{\frac{5}{8}} [/latex]

  

Relationship Between Mixed Numbers and Improper Fractions

Converting a Mixed Number to an Improper Fraction

Note: You can perform the above steps in a single line of arithmetic, as follows:

[latex] 3{\frac{5}{8}} [/latex] = [latex] \displaystyle{\frac{3(8) + 5}{8}} [/latex] = [latex] \displaystyle{\frac{24 + 5}{8}} [/latex] = [latex] \displaystyle{\frac{29}{8}} [/latex]

There is a total of 29 pieces, each piece being one-eighth in size.

Converting an Improper Fraction to a Mixed Number

Follow these steps to convert an improper fraction to a mixed number:	Example: Convert [latex] \displaystyle{\frac{29}{8}} [/latex] to an improper fraction.
Step 1: Divide the numerator by the denominator.
Step 2: The quotient becomes the whole number and the remainder becomes the numerator of the fraction portion of the mixed number.
Step 3: The denominator of the fraction portion of the mixed number is the same as the denominator of the original improper fraction.	Therefore, [latex] \displaystyle{\frac{29}{8}} [/latex] = [latex] 3{\frac{5}{8}} [/latex]

Equivalent Fractions

When both the numerator and denominator of a fraction are either multiplied by the same number or divided by the same number, the result is a new fraction known as an equivalent fraction. Equivalent fractions have the same value.

That is, the same part (or portion) of a whole unit can be represented by different fractions.

For example,

[latex] \displaystyle{\frac{1}{2}} [/latex], [latex] \displaystyle{\frac{2}{4}} [/latex], [latex] \displaystyle{\frac{3}{6}} [/latex], [latex] \displaystyle{\frac{4}{8}} [/latex], [latex] \displaystyle{\frac{5}{10}} [/latex], ... are equivalent fractions.

Example 2.1-b: Finding Equivalent Fractions by Raising to Higher Terms

Find two equivalent fractions of [latex] \displaystyle{\frac{2}{5}} [/latex] by raising to higher terms.

Solution

(2 portions of 5 equal parts of the whole)

Multiplying both the numerator and denominator by 2,

(4 portions of 10 equal parts of the whole)

Multiplying both the numerator and denominator by 3,

(6 portions of 15 equal parts of the whole)

Therefore, [latex] \displaystyle{\frac{4}{10}} [/latex] and [latex] \displaystyle{\frac{6}{15}} [/latex] are equivalent fractions of [latex] \displaystyle{\frac{2}{5}} [/latex].

Example 2.1-c: inding Equivalent Fractions by Reducing to Lower Terms

Find two equivalent fractions of [latex] \displaystyle{\frac{12}{30}} [/latex] by reducing to lower terms.

Solution

(12 portions of 30 equal parts of the whole)

Dividing both the numerator and denominator by 3,

(4 portions of 10 equal parts of the whole)

Further dividing both the numerator and denominator by 2,

(2 portions of 5 equal parts of the whole)

(2 portions of 5 equal parts of the whole)

Or, the lowest terms can be found by dividing both the numerator and denominator of the original fraction, [latex] \displaystyle{\frac{12}{30}} [/latex], by 6,

Therefore, [latex] \displaystyle{\frac{4}{10}} [/latex] and [latex] \displaystyle{\frac{2}{5}} [/latex] are equivalent fractions of [latex] \displaystyle{\frac{12}{30}} [/latex].

Note: The fraction [latex] \displaystyle{\frac{2}{5}} [/latex] cannot be further reduced, as 2 and 5 do not have any common factors.
Therefore, [latex] \displaystyle{\frac{2}{5}} [/latex] is a fraction in its lowest (or simplest) terms. We will learn more about this further in the section.

Identifying Equivalent Fractions Using Cross Products

If the cross products of two fractions are equal, then the two fractions are equivalent fractions, and vice versa (i.e., if the fractions are equivalent, then their cross products are equal).

That is, if [latex] \displaystyle{\frac{a}{b}} [/latex] = [latex] \displaystyle{\frac{c}{d}} [/latex], then a × d = b × c         a × d and b × c are known as cross products

For example, [latex] \displaystyle{\frac{3}{5}} [/latex] and [latex] \displaystyle{\frac{12}{20}} [/latex] are equivalent fractions because their cross products, 3 × 20 and 5 × 12, are equal:

3 × 20 = 60

5 × 12 = 60

Example 2.1-d: Classifying Fractions as Equivalent or Not Equivalent

Classify the pair of fractions as ‘equivalent’ or ‘not equivalent’ by using their cross products.

1. [latex] \displaystyle{\frac{2}{5}} [/latex] and [latex] \displaystyle{\frac{12}{20}} [/latex]

2. [latex] \displaystyle{\frac{5}{4}} [/latex] and [latex] \displaystyle{\frac{20}{12}} [/latex]

3. [latex] \displaystyle{\frac{3}{8}} [/latex] and [latex] \displaystyle{\frac{9}{24}} [/latex]

Fractions in Lowest (or Simplest) Terms

Dividing both the numerator and denominator of a fraction by the same number, which results in an equivalent fraction, is known as reducing or simplifying the fraction.

For example, we saw in Example 2.1-c that [latex] \displaystyle{\frac{4}{10}} [/latex] and [latex] \displaystyle{\frac{2}{5}} [/latex] are reduced fractions of [latex] \displaystyle{\frac{12}{30}} [/latex].

A fraction in which the numerator and denominator have no factors in common (other than 1) is said to be a fraction in its lowest (or simplest) terms.

Any fraction can be fully reduced to its lowest terms by one of the following two methods:

Method 1: Dividing both the numerator and denominator by the greatest common factor (GCF).

Method 2: Writing the numerator and denominator as products of prime factors and reducing by the common prime factors.

Example 2.1-e: Reducing Fractions to their Lowest Terms

Reduce the following fractions to their lowest terms.

1. [latex] \displaystyle{\frac{40}{45}} [/latex]

2. [latex] \displaystyle{\frac{54}{24}} [/latex]

Reciprocals of Fractions

Two numbers whose product is equal to 1 are known as reciprocals of each other. Every non-zero real number has a reciprocal.

For example,

• [latex] \displaystyle{\frac{2}{3}} [/latex] and [latex] \displaystyle{\frac{3}{2}} [/latex] are reciprocals of each other, because [latex] \displaystyle{\frac{2}{3} × \frac{3}{2} = 1} [/latex]

When the numerator and denominator of a fraction are interchanged, the resulting fraction is the reciprocal of the original fraction.

For example,

• 5 and [latex] \displaystyle{\frac{1}{5}} [/latex] are reciprocals (5 can also be written as [latex] \displaystyle{\frac{5}{1}} [/latex]).
• Similarly, the reciprocal of[latex] \displaystyle{\frac{-2}{5}} [/latex] is [latex] \displaystyle{\frac{5}{-2} = -\frac{5}{2}} [/latex]

The reciprocal of a positive number is always positive and the reciprocal of a negative number is always negative.

Note:

• The reciprocal of a number is not the negative of that number.

  (The reciprocal of [latex] 3 ≠ -3 [/latex]. The reciprocal of [latex] \displaystyle{3 = \frac{1}{3}} [/latex].)

• The reciprocal of a fraction is not an equivalent fraction of that fraction.

  (The reciprocal of [latex] \displaystyle{\frac{2}{5} ≠ \frac{4}{10}} [/latex]. The reciprocal of [latex] \displaystyle{\frac{2}{5} = \frac{5}{2}} [/latex].)

Table 2.1-a: Examples of Numbers with their Negatives and Reciprocals

2.1 Exercises

For Problems 1 to 6, classify the fractions as proper fractions, improper fractions, or mixed numbers.

1. a. [latex] 15\frac{12}{13} [/latex]   b. [latex] \displaystyle{\frac{29}{30}} [/latex]

2. a. [latex] \displaystyle{\frac{16}{35}} [/latex]   b. [latex] 3\frac{2}{9} [/latex]

3. a. [latex] \displaystyle{\frac{21}{22}} [/latex]   b. [latex] \displaystyle{\frac{52}{25}} [/latex]

4. a. [latex] \displaystyle{\frac{19}{16}} [/latex]   b. [latex] 9\frac{7}{8} [/latex]

5. a. [latex] 6\frac{1}{2} [/latex]   b. [latex] \displaystyle{\frac{20}{75}} [/latex]

6. a. [latex] 4\frac{2}{5} [/latex]   b. [latex] \displaystyle{\frac{7}{3}} [/latex]

For Problems 7 to 10, convert the mixed numbers to improper fractions.

7. a. [latex] 3\frac{2}{5} [/latex]   b. [latex] 7\frac{5}{8} [/latex]

8. a. [latex] 2\frac{2}{7} [/latex]   b. [latex] 3\frac{1}{8} [/latex]

9. a. [latex] 4\frac{3}{7} [/latex]   b. [latex] 9\frac{5}{6} [/latex]

10. a. [latex] 5\frac{4}{5} [/latex]   b. [latex] 6\frac{3}{4} [/latex]

For Problems 11 to 14, convert the improper fractions to mixed numbers.

11. a. [latex] \displaystyle{\frac{23}{7}} [/latex]   b. [latex] \displaystyle{\frac{34}{3}} [/latex]

12. a. [latex] \displaystyle{\frac{19}{7}} [/latex]   b. [latex] \displaystyle{\frac{45}{8}} [/latex]

13. a. [latex] \displaystyle{\frac{26}{4}} [/latex]   b. [latex] \displaystyle{\frac{29}{5}} [/latex]

14. a. [latex] \displaystyle{\frac{23}{3}} [/latex]   b. [latex] \displaystyle{\frac{31}{6}} [/latex]

For Problems 15 to 22, classify the pair of fractions as ‘equivalent’ or ‘not equivalent’ by first converting the mixed numbers to improper fractions.

15. [latex] \displaystyle{\frac{47}{8}} [/latex] and [latex] 5\frac{7}{8} [/latex]

16. [latex] \displaystyle{\frac{44}{5}} [/latex] and [latex] 4\frac{4}{5} [/latex]

17. [latex] \displaystyle{\frac{41}{4}} [/latex] and [latex] 10\frac{3}{4} [/latex]

18. [latex] 11\frac{5}{7} [/latex] and [latex] 7\frac{5}{7} [/latex]

19. [latex] \displaystyle{\frac{17}{8}} [/latex] and [latex] 2\frac{3}{8} [/latex]

20. [latex] \displaystyle{\frac{54}{7}} [/latex] and [latex] 7\frac{5}{7} [/latex]

21. [latex] \displaystyle{\frac{45}{11}} [/latex] and [latex] 4\frac{3}{11} [/latex]

22. [latex] \displaystyle{\frac{37}{9}} [/latex] and [latex] 4\frac{1}{9} [/latex]

For Problems 23 to 30, classify the pair of fractions as ‘equivalent’ or ‘not equivalent’ by first converting the improper fractions to mixed numbers.

23. [latex] \displaystyle{\frac{43}{6}} [/latex] and [latex] 7\frac{5}{6} [/latex]

24. [latex] \displaystyle{\frac{15}{4}} [/latex] and [latex] 3\frac{1}{4} [/latex]

25. [latex] \displaystyle{\frac{45}{7}} [/latex] and [latex] 6\frac{3}{7} [/latex]

26. [latex] \displaystyle{\frac{18}{5}} [/latex] and [latex] 3\frac{3}{5} [/latex]

27. [latex] \displaystyle{\frac{34}{8}} [/latex] and [latex] 4\frac{1}{8} [/latex]

28. [latex] 3\frac{8}{9} [/latex] and [latex] \displaystyle{\frac{35}{9}} [/latex]

29. [latex] 7\frac{3}{9} [/latex] and [latex] \displaystyle{\frac{67}{9}} [/latex]

30. [latex] \displaystyle{\frac{41}{12}} [/latex] and [latex] 3\frac{5}{12} [/latex]

For Problems 31 to 36, (i) reduce the fractions to their lowest terms and (ii) write their reciprocals.

31. a. [latex] \displaystyle{\frac{44}{12}} [/latex]   b. [latex] \displaystyle{\frac{42}{70}} [/latex]

32. a. [latex] \displaystyle{\frac{30}{20}} [/latex]   b. [latex] \displaystyle{\frac{48}{84}} [/latex]

33. a. [latex] \displaystyle{\frac{75}{105}} [/latex]   b. [latex] \displaystyle{\frac{144}{48}} [/latex]

34. a. [latex] \displaystyle{\frac{56}{48}} [/latex]   b. [latex] \displaystyle{\frac{84}{21}} [/latex]

35. a. [latex] \displaystyle{\frac{131}{84}} [/latex]   b. [latex] \displaystyle{\frac{54}{126}} [/latex]

36. a. [latex] \displaystyle{\frac{36}{63}} [/latex]   b. [latex] \displaystyle{\frac{60}{96}} [/latex]

For Problems 37 to 44, classify the pair of fractions as ‘equivalent’ or ‘not equivalent’.

37. [latex] \displaystyle{\frac{6}{10}} [/latex] and [latex] \displaystyle{\frac{9}{15}} [/latex]

38. [latex] \displaystyle{\frac{6}{12}} [/latex] and [latex] \displaystyle{\frac{15}{30}} [/latex]

39. [latex] \displaystyle{\frac{12}{18}} [/latex] and [latex] \displaystyle{\frac{18}{27}} [/latex]

40. [latex] \displaystyle{\frac{8}{10}} [/latex] and [latex] \displaystyle{\frac{15}{12}} [/latex]

41. [latex] \displaystyle{\frac{35}{15}} [/latex] and [latex] \displaystyle{\frac{28}{12}} [/latex]

42. [latex] \displaystyle{\frac{15}{12}} [/latex] and [latex] \displaystyle{\frac{36}{45}} [/latex]

43. [latex] \displaystyle{\frac{16}{24}} [/latex] and [latex] \displaystyle{\frac{25}{30}} [/latex]

44. [latex] \displaystyle{\frac{20}{25}} [/latex] and [latex] \displaystyle{\frac{24}{30}} [/latex]

For Problems 45 to 50, determine the missing values.

45. a.[latex] \displaystyle{\frac{42}{36} = \frac{14}{?}} [/latex]   b. [latex] \displaystyle{\frac{42}{36} = \frac{?}{30}} [/latex]

46. a.[latex] \displaystyle{\frac{4}{9} = \frac{?}{27}} [/latex]   b. [latex] \displaystyle{\frac{4}{9} = \frac{20}{?}} [/latex]

47. a.[latex] \displaystyle{\frac{45}{75} = \frac{?}{25}} [/latex]   b. [latex] \displaystyle{\frac{45}{75} = \frac{18}{?}} [/latex]

48. a.[latex] \displaystyle{\frac{9}{12} = \frac{18}{?}} [/latex]   b. [latex] \displaystyle{\frac{9}{12} = \frac{?}{4}} [/latex]

49. a.[latex] \displaystyle{\frac{25}{15} = \frac{?}{3}} [/latex]   b. [latex] \displaystyle{\frac{25}{15} = \frac{35}{?}} [/latex]

50. a.[latex] \displaystyle{\frac{3}{2} = \frac{12}{?}} [/latex]   b. [latex] \displaystyle{\frac{3}{2} = \frac{?}{12}} [/latex]

For Problems 51 to 58, express the answer as a fraction reduced to its lowest terms.

51. What fraction of 1 hour is 25 minutes?

52. What fraction of 1 year is 4 months?

53. Out of 35 students in a math class, 15 received an ‘A’ in their final exam. What fraction of the students in the class received an ‘A’ grade?

54. Karen cut a pizza into 16 equal slices and served 12 slices to her friends. What fraction of the pizza was served?

55. In a finance math course with 490 students, 70 students failed the final exam. What fraction of the students passed the final exam in this course?

56. In a survey of 272 people, 68 people responded ‘yes’ and the remaining responded ‘no’. What fraction of the people responded ‘no’?

57. In a community of 6,000 people, 1,800 were 60 years or older. What fraction of the people were below the age of 60 years?

58. Out of the 480 units in a condominium tower, 182 are rented. What fraction of the units are not rented?