2.2 Arithmetic Operations with Fractions

Least or Lowest Common Denominator (LCD)

The Least Common Denominator (LCD) of a set of two or more fractions is the smallest whole number that is divisible by each of the denominators. It is the least common multiple (LCM) of the denominators of the fractions. There are two methods of finding the LCM, as explained in Chapter 1, Section 1.3.

In performing addition and subtraction of fractions, it is necessary to determine equivalent fractions with common denominators. The best choice for a common denominator is the LCD, because it makes any further simplification easier.

Example 2.2-a: Determining the Least Common Denominator

Determine the LCD of [latex] \displaystyle{\frac{4}{9}} [/latex] and [latex] \displaystyle{\frac{7}{15}} [/latex].

Comparing Fractions

Fractions can easily be compared when they have the same denominator. If they do not have the same denominator, first determine the LCD of the fractions and convert them into equivalent fractions with the LCD as the denominator.

Example 2.2-b: Comparing Fractions

Which of the fractions, [latex] \displaystyle{\frac{5}{12}} [/latex] or [latex] \displaystyle{\frac{3}{8}} [/latex], is greater?

Addition of Fractions

The denominator of a fraction indicates the number of parts into which an item is divided. Therefore, addition of fractions requires that the denominators of every fraction be the same. If the demoninators are different, they must first be made the same by determining the LCD and changing each fraction to its equivalent fraction with the LCD as the denominator.

When the fractions have the same denominator, the numerators of each of the fractions may be added. The numerator of the resulting fraction is equal to this sum, and the denominator is equal to the common denominator of the fractions being added.

Express the final answer reduced to lowest terms and as a mixed number, where applicable.

Example 2.2-c: Adding Fractions that have the Same Denominator

Add [latex] \displaystyle{\frac{2}{9}} [/latex] and [latex] \displaystyle{\frac{5}{9}} [/latex].

Solution

[latex] \displaystyle{\frac{2}{9} + \frac{5}{9}} [/latex]

The denominators of the fractions are the same. Adding the numerators and keeping the common denominator,

[latex] \displaystyle{= \frac{2 + 5}{9}} [/latex]

[latex] \displaystyle{= \frac{7}{9}} [/latex]

Therefore, the result from adding [latex] \displaystyle{= \frac{2}{9}} [/latex] and [latex] \displaystyle{= \frac{5}{9}} [/latex] is [latex] \displaystyle{\frac{7}{9}} [/latex].

Example 2.2-d: Adding Fractions that have Different Denominators

Add [latex] \displaystyle{\frac{3}{4}} [/latex] and [latex] \displaystyle{\frac{2}{3}} [/latex].

Example 2.2-e: Adding a Mixed Number and a Proper Fraction

Add [latex] 3\frac{5}{6} [/latex] and [latex] \displaystyle{\frac{4}{9}} [/latex].

Example 2.2-f: Adding Mixed Numbers

Add:

1. [latex] 2\frac{1}{6} [/latex] and [latex] 4\frac{3}{4} [/latex]

2. [latex] 15\frac{2}{3} [/latex] and [latex] 3\frac{3}{5} [/latex]

Subtraction of Fractions

The process for subtraction of fractions is the same as that of addition of fractions. First, determine a common denominator and change each fraction to its equivalent fraction with the common denominator. When the fractions have the same denominator, the numerators of the fractions may be subtracted. The numerator of the resulting fraction is equal to this difference, and the denominator is equal to the common denominator of the fractions being subtracted.

Express the final answer reduced to lowest terms and as a mixed number, where applicable.

Example 2.2-g: Subtracting Fractions that have the Same Denonimator

Subtract [latex] \displaystyle{\frac{3}{8}} [/latex] from [latex] \displaystyle{\frac{7}{8}} [/latex].

Solution

[latex] \displaystyle{\frac{7}{8} - \frac{3}{8}} [/latex]

The denominators of the fractions are the same. Subtracting the numerators and keeping the common denominator,

[latex] \displaystyle{= \frac{7 - 3}{8}} [/latex]

[latex] \displaystyle{= \frac{4}{8} = \frac{1}{2}} [/latex]

Reducing to lowest terms,

[latex] \displaystyle{= \frac{1}{2}} [/latex]

Therefore, the result from subtracting [latex] \displaystyle{\frac{3}{8}} [/latex] from [latex] \displaystyle{\frac{7}{8}} [/latex] is [latex] \displaystyle{\frac{1}{2}} [/latex].

Example 2.2-h: Subtracting Fractions that have Different Denominators

Subtract [latex] \displaystyle{\frac{2}{8}} [/latex] from [latex] \displaystyle{\frac{7}{10}} [/latex].

Example 2.2-i: Subtracting Mixed Numbers

Subtract [latex] 7\frac{2}{3} [/latex] from [latex] 12\frac{1}{2} [/latex].

Multiplication of Fractions

To multiply two or more fractions, first convert any mixed number to its improper fraction. Then, simply multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator.

When multiplying fractions, you can reduce any numerator term with any denominator term. Reduce as much as possible before multiplying the numerators together and the denominators together, in order to keep the numbers as simple as possible.

Express the final answer reduced to lowest terms and as a mixed number, where applicable.

Note: When multiplying mixed numbers, it is incorrect to multiply the whole number parts separately from the fractional parts to arrive at the answer.

Example 2.2-j: Multiplying Fractions

Multiply:

1. [latex] \displaystyle{\frac{3}{2} × \frac{4}{11}} [/latex]

2. [latex] \displaystyle{15 × \frac{2}{5}} [/latex]

3. [latex] 3\frac{1}{8} × 2\frac{4}{5} [/latex]

Division of Fractions

When dividing fractions, as in multiplication, first convert any mixed number to its improper fraction. The division of fractions is done by multiplying the first fraction by the reciprocal of the second fraction. Then, follow the procedure used in multiplication to arrive at the final result.

Express the final answer reduced to lowest terms and as a mixed number, where applicable.

Note:

Dividing by 2 is the same as multiplying by the reciprocal of 2, which is [latex] \displaystyle{\frac{1}{2}} [/latex].

When multiplying or dividing mixed numbers, it is incorrect to multiply or divide the whole number parts separately from the fractional parts to arrive at the answer.

Example 2.2-k: Dividing Fractions

Divide [latex] \displaystyle{\frac{15}{16}} [/latex] by [latex] \displaystyle{\frac{9}{20}} [/latex].

Example 2.2-l: Dividing Mixed Numbers

Divide [latex] 3\frac{3}{20} [/latex] by [latex] 1\frac{4}{5} [/latex].

Complex Fractions

A complex fraction is a fraction in which one or more fractions are found in the numerator or denominator.

For example,

• [latex] \displaystyle{\frac{1}{(\displaystyle{\frac{5}{8}})}} [/latex] is a complex fraction because it has a fraction in the denominator.

• [latex] \displaystyle{\frac{(\displaystyle{\frac{2}{3}})}{6}} [/latex] is a complex fraction because it has a fraction in the numerator.

• [latex] \displaystyle{\frac{(\displaystyle{\frac{2}{5} + \frac{1}{4}})}{3}} [/latex] is a complex fraction because it has two fractions in the numerator.

• [latex] \frac{(\displaystyle{\frac{5}{6}})}{(\displaystyle{\frac{1}{8}})} [/latex] is a complex fraction because it has a fraction in both the numerator and the denominator.

A complex fraction can be simplified by dividing the numerator by the denominator and then, following the rule(s) for dividing fractions.

[latex] \displaystyle{\frac{1}{(\displaystyle{\frac{5}{8}})} = 1 ÷ \frac{5}{8} = 1 × \frac{5}{5} = \frac{8}{5}} [/latex]

[latex] \displaystyle{\frac{(\displaystyle{\frac{2}{3}})}{6} = \frac{2}{3} ÷ 6 = \frac{1}{3} × \frac{1}{3} = \frac{1}{9}} [/latex]

Example 2.2-m: Simplifying Complex Fractions

Add:

1. [latex] \displaystyle{\frac{(\displaystyle{\frac{7}{2}})}{5}} [/latex]

2. [latex] \displaystyle{\frac{6}{(\displaystyle{\frac{9}{8}})}} [/latex]

Powers and Square Roots of Fractions

Powers of fractions are expressed the same way as whole numbers. When the base of a power is a fraction, it is written within brackets.

For example, [latex] \displaystyle{(\frac{2}{3})^2} [/latex] is read as “two-thirds squared”.

• This means that [latex] \displaystyle{\frac{2}{3}} [/latex] is used as a factor two times.

• i.e., [latex] \displaystyle{(\frac{2}{3})^2 = \frac{2}{3} × \frac{2}{3} = \frac{4}{9}} [/latex]

A mixed number that is raised to a power is evaluated by first converting it into an improper fraction, and then following the same procedure explained above.

For example, [latex] (1\frac{2}{3})^4 [/latex] is evaluated by first converting [latex] 1\frac{2}{3} [/latex] into an improper fraction.

[latex] (1\frac{2}{3})^4 = \displaystyle{(\frac{1(3) + 2}{3})^4 = (\frac{5}{3})^4 = (\frac{5}{3})(\frac{5}{3})(\frac{5}{3})(\frac{5}{3}) = \frac{625}{81}} [/latex] [latex] = 7\frac{58}{81} [/latex]

Example 2.2-n: Evaluating Powers of Fractions

Evaluate the following powers:

1. [latex] \displaystyle{(\frac{4}{5})^4} [/latex]

2. [latex] (1\frac{1}{2})^5 [/latex]

Square roots of fractions are calculated the same way as square roots of whole numbers, but the numerators and denominators are evaluated separately.

For example, [latex] \displaystyle{\sqrt{\frac{9}{16}}} [/latex] is the same as [latex] \displaystyle{\frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}} [/latex].

Example 2.2-o: Evaluating Square Roots of Fractions

Evaluate the square root: [latex] \displaystyle{\sqrt{\frac{25}{144}}} [/latex]

2.2 Exercises

For Problems 1 to 8, identify the greater fraction in each pair.

1. [latex] \displaystyle{\frac{4}{3}} [/latex] or [latex] \displaystyle{\frac{6}{5}} [/latex]

2. [latex] \displaystyle{\frac{2}{5}} [/latex] or [latex] \displaystyle{\frac{3}{8}} [/latex]

3. [latex] \displaystyle{\frac{5}{4}} [/latex] or [latex] \displaystyle{\frac{7}{6}} [/latex]

4. [latex] \displaystyle{\frac{12}{15}} [/latex] or [latex] \displaystyle{\frac{35}{45}} [/latex]

5. [latex] \displaystyle{\frac{5}{13}} [/latex] or [latex] \displaystyle{\frac{16}{39}} [/latex]

6. [latex] \displaystyle{\frac{8}{7}} [/latex] or [latex] \displaystyle{\frac{13}{12}} [/latex]

7. [latex] \displaystyle{\frac{3}{8}} [/latex] or [latex] \displaystyle{\frac{25}{48}} [/latex]

8. [latex] \displaystyle{\frac{8}{9}} [/latex] or [latex] \displaystyle{\frac{39}{45}} [/latex]

9. Which of the following fractions are greater than [latex] \displaystyle{\frac{3}{4}} [/latex]?

[latex] \displaystyle{\frac{4}{5}} [/latex], [latex] \displaystyle{\frac{7}{9}} [/latex], [latex] \displaystyle{\frac{5}{7}} [/latex], [latex] \displaystyle{\frac{9}{11}} [/latex]

10. Which of the following fractions are less than [latex] \displaystyle{\frac{2}{3}} [/latex]?

[latex] \displaystyle{\frac{5}{8}} [/latex], [latex] \displaystyle{\frac{6}{7}} [/latex], [latex] \displaystyle{\frac{3}{5}} [/latex], [latex] \displaystyle{\frac{7}{9}} [/latex]

For Problems 11 to 18, perform the addition, reduce to lowest terms, and express the answer as a mixed number, whenever possible.

11. a. [latex] \displaystyle{\frac{5}{9} + \frac{7}{9}} [/latex]   b. [latex] \displaystyle{\frac{7}{10} + \frac{9}{20}} [/latex]

12. a. [latex] \displaystyle{\frac{5}{8} + \frac{7}{8}} [/latex]   b. [latex] \displaystyle{\frac{7}{12} + \frac{3}{4}} [/latex]

13. a. [latex] \displaystyle{\frac{23}{12} + \frac{1}{3}} [/latex]   b. [latex] 18\frac{5}{7} + 2\frac{2}{5} [/latex]

14. a. [latex] \displaystyle{\frac{4}{3} + \frac{5}{6}} [/latex]   b. [latex] 12\frac{3}{4} + 5\frac{1}{3} [/latex]

15. a. [latex] 11\frac{1}{4} + 5\frac{2}{3} [/latex]   b. [latex] 7\frac{1}{12} + 5\frac{3}{4} [/latex]

16. a. [latex] 9\frac{3}{4} + 6\frac{1}{6} [/latex]   b. [latex] 8\frac{2}{3} + 5\frac{3}{4} [/latex]

17. a. [latex] \displaystyle{\frac{3}{10} + \frac{47}{100} + \frac{241}{1,000}} [/latex]   b. [latex] \displaystyle{\frac{2}{3} + \frac{3}{4} + \frac{5}{8}} [/latex]

18. a. [latex] \displaystyle{\frac{1}{10} + \frac{17}{100} + \frac{39}{1,000}} [/latex]   b. [latex] \displaystyle{\frac{3}{5} + \frac{7}{10} + \frac{9}{15}} [/latex]

For Problems 19 to 24, perform the subtraction, reduce to lowest terms, and express the answer as a mixed number, whenever possible.

19. a. [latex] \displaystyle{\frac{1}{6} - \frac{1}{8}} [/latex]   b. [latex] \displaystyle{\frac{19}{20} - \frac{3}{10}} [/latex]

20. a. [latex] \displaystyle{\frac{2}{3} - \frac{1}{9}} [/latex]   b. [latex] \displaystyle{\frac{9}{12} - \frac{3}{5}} [/latex]

21. a. [latex] \displaystyle{\frac{17}{9} - \frac{5}{6}} [/latex]   b. [latex] 5\frac{2}{3} - 1\frac{5}{12} [/latex]

22. a. [latex] \displaystyle{\frac{5}{3} - \frac{3}{8}} [/latex]   b. [latex] 16\frac{1}{8} - 1\frac{1}{2} [/latex]

23. a. [latex] 8\frac{5}{12} - 4\frac{3}{6} [/latex]   b. [latex] 5\frac{5}{8} - 4\frac{5}{6} [/latex]

24. a. [latex] 8\frac{5}{6} - 5\frac{3}{9} [/latex]   b. [latex] 9\frac{2}{5} - 7\frac{3}{10} [/latex]

For Problems 25 and 26, perform the mixed additions and subtractions, reduce to lowest terms, and express the answer as a mixed number, whenever possible.

25. a. [latex] \displaystyle{\frac{3}{10} - \frac{4}{1,000} + \frac{5}{100}} [/latex]   b. [latex] \displaystyle{\frac{7}{12} + \frac{5}{6} - \frac{2}{3}} [/latex]

26. a. [latex] \displaystyle{\frac{32}{100} - \frac{8}{1,000} + \frac{3}{25}} [/latex]   b. [latex] \displaystyle{\frac{5}{8} + \frac{13}{16} - \frac{3}{4}} [/latex]

For Problems 27 to 32, perform the multiplication, reduce to lowest terms, and express the answer as a mixed number, whenever possible.

27. a. [latex] \displaystyle{\frac{12}{5} × \frac{25}{3}} [/latex]   b. [latex] \displaystyle{\frac{6}{9} × \frac{19}{12}} [/latex]

28. a. [latex] \displaystyle{\frac{16}{5} × \frac{5}{4}} [/latex]   b. [latex] \displaystyle{3 × \frac{7}{9}} [/latex]

29. a. [latex] \displaystyle{\frac{4}{5} × \frac{23}{9}} [/latex]   b. [latex] 11\frac{3}{4} × 1\frac{1}{74} [/latex]

30. a. [latex] \displaystyle{\frac{3}{8} × \frac{5}{11}} [/latex]   b. [latex] 9\frac{3}{5} × 1\frac{29}{96} [/latex]

31. a. [latex] \displaystyle{\frac{15}{27} × \frac{18}{45}} [/latex]   b. [latex] 2\frac{3}{7} × \displaystyle{\frac{25}{45}} [/latex]

32. a. [latex] \displaystyle{\frac{9}{38} × \frac{19}{63}} [/latex]   b. [latex] 2\frac{2}{9} × 1\frac{1}{2} [/latex]

For Problems 33 to 38, perform the division, reduce to lowest terms, and express the answer as a mixed number, whenever possible.

33. a. [latex] \displaystyle{\frac{3}{5} ÷ \frac{3}{4}} [/latex]   b. [latex] \displaystyle{\frac{1}{7} ÷ \frac{3}{5}} [/latex]

34. a. [latex] \displaystyle{\frac{2}{3} ÷ \frac{4}{9}} [/latex]   b. [latex] \displaystyle{\frac{3}{8} ÷ 4} [/latex]

35. a. [latex] \displaystyle{\frac{8}{12} ÷ \frac{2}{4}} [/latex]   b. [latex] 10\frac{1}{4} ÷ 2\frac{27}{48} [/latex]

36. a. [latex] \displaystyle{\frac{10}{15} ÷ \frac{3}{7}} [/latex]   b. [latex] 23\frac{1}{2} ÷ 8\frac{13}{16} [/latex]

37. a. [latex] 5\frac{1}{4} ÷ 7 [/latex]   b. [latex] 15 ÷ 3\frac{1}{3} [/latex]

38. a. [latex] 5\frac{1}{5} ÷ 13 [/latex]   b. [latex] 18 ÷ 4\frac{4}{5} [/latex]

For Problems 39 and 40, express the complex fractions as a proper fraction or a mixed number, where possible.

39. a. [latex] \displaystyle{\frac{1}{\displaystyle{(\frac{11}{2})}}} [/latex]   b. [latex] \displaystyle{\frac{8\frac{4}{9}}{\displaystyle{(\frac{11}{2})}}} [/latex]

40. a. [latex] \displaystyle{\frac{1}{\displaystyle{(\frac{9}{4})}}} [/latex]   b. [latex] \displaystyle{\frac{4\frac{3}{4}}{\displaystyle{(\frac{3}{8})}}} [/latex]

For Problems 41 to 46, evaluate the powers of the fractions.

41. a. [latex] \displaystyle{(\frac{3}{4})^2} [/latex]   b. [latex] \displaystyle{(\frac{2}{9})^2} [/latex]

42. a. [latex] \displaystyle{(\frac{3}{5})^2} [/latex]   b. [latex] \displaystyle{(\frac{6}{7})^2} [/latex]

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

44. a. [latex] \displaystyle{(\frac{3}{4})^3} [/latex]   b. [latex] \displaystyle{(\frac{5}{3})^4} [/latex]

45. a. [latex] (2\frac{1}{4})^2 [/latex]   b. [latex] (1\frac{2}{3})^3 [/latex]

46. a. [latex] (1\frac{1}{3})^2 [/latex]   b. [latex] (3\frac{1}{2})^3 [/latex]

For Problems 47 to 52, evaluate the square roots of the fractions.

47. a. [latex] \displaystyle{\sqrt{\frac{1}{16}}} [/latex]   b. [latex] \displaystyle{\sqrt{\frac{1}{10,000}}} [/latex]

48. a. [latex] \displaystyle{\sqrt{\frac{1}{9}}} [/latex]   b. [latex] \displaystyle{\sqrt{\frac{1}{49}}} [/latex]

49. a. [latex] \displaystyle{\sqrt{\frac{36}{100}}} [/latex]   b. [latex] \displaystyle{\sqrt{\frac{144}{81}}} [/latex]

50. a. [latex] \displaystyle{\sqrt{\frac{4}{25}}} [/latex]   b. [latex] \displaystyle{\sqrt{\frac{81}{16}}} [/latex]

51. a. [latex] \sqrt{1\frac{11}{25}} [/latex]   b. [latex] \sqrt{1\frac{21}{100}} [/latex]

52. a. [latex] \sqrt{3\frac{1}{16}} [/latex]   b. [latex] \sqrt{6\frac{1}{4}} [/latex]

For Problems 53 to 74, express your answers as a proper fraction or a mixed number, where appropriate.

53. Alan walked [latex] \displaystyle{\frac{3}{5}} [/latex] km to his friend’s house and from there, he walked another [latex] \displaystyle{\frac{3}{4}} [/latex] km to his school. How far did Alan walk?

54. Peter spent [latex] \displaystyle{\frac{5}{12}} [/latex] of his money on rent and [latex] \displaystyle{\frac{1}{4}} [/latex] on food. What fraction of his money did he spend on rent and food?

55. A bag contains [latex] 2\frac{3}{5} [/latex] kg of red beans and [latex] 1\frac{1}{8} [/latex] kg of green beans. What is the total weight of the bag?

56. Last night, Amy spent [latex] 3\frac{1}{6} [/latex] hours on her math project and [latex] 2\frac{3}{10} [/latex] hours on her design project. How much time did she spend on both projects altogether?

57. Alexander bought [latex] 4\frac{2}{5} [/latex] litres of milk and drank [latex] 1\frac{2}{3} [/latex] litres of it. How much milk was left?

58. Thomas baked a [latex] 2\frac{1}{2} [/latex] pound cake. He gave [latex] 1\frac{5}{8} [/latex] pounds of it to his friend Yan. How much was left?

59. Cassidy bought [latex] \displaystyle{\frac{5}{8}} [/latex] litres of olive oil and used [latex] \displaystyle{\frac{1}{3}} [/latex] litre of the oil while cooking. What quantity of olive oil was left?

60. Sarah had [latex] \displaystyle{\frac{3}{4}} [/latex] kg of cheese. She used [latex] \displaystyle{\frac{2}{7}} [/latex] kg of the cheese while baking. How many kilograms of cheese was left?

61. Mary spent [latex] \displaystyle{\frac{2}{5}} [/latex] of her money on a school bag. She then spent [latex] \displaystyle{\frac{1}{3}} [/latex] of the remainder on shoes. What fraction of her money was spent on shoes?

62. David spent [latex] \displaystyle{\frac{7}{10}} [/latex] of his money on toys and [latex] \displaystyle{\frac{1}{3}} [/latex] of the remainder on food. What fraction of his money was spent on food?

63. Rose travelled [latex] \displaystyle{\frac{3}{5}} [/latex] of her journey by car and the remaining 20 km by bus. How far did she travel by car?

64. After selling [latex] \displaystyle{\frac{2}{5}} [/latex] of its textbooks, a bookstore had 810 books left. How many textbooks were in the bookstore initially?

65. [latex] 2\frac{3}{4} [/latex] litres of juice weighs [latex] 4\frac{2}{3} [/latex] kg. Determine the weight (in kilograms) of 1 litre of juice.

66. Cheng can walk [latex] 5\frac{1}{4} [/latex] km in [latex] 1\frac{1}{2} [/latex] hours. How many kilometres can he walk in 1 hour?

67. A cake that weighs [latex] \displaystyle{\frac{2}{3}} [/latex] kg is cut into slices weighing [latex] \displaystyle{\frac{1}{12}} [/latex] kg each. How many slices are there?

68. A chain of length [latex] \displaystyle{\frac{7}{8}} [/latex] metres is cut into pieces measuring [latex] \displaystyle{\frac{1}{16}} [/latex] metres each. How many pieces are there?

69. A box of cereal contains 917 grams of cereal. How many bowls of cereal will there be if each serving is [latex] 32\frac{3}{4} [/latex] grams?

70. A bottle of medicine contains 80 mg of medicine. Each dose of the medicine is [latex] \displaystyle{\frac{2}{5}} [/latex] mg. How many doses are there in the bottle?

71. If [latex] \displaystyle{\frac{4}{15}} [/latex] of the 1,800 students in a school enrolled for a mathematics course, how many students did not enroll for the course?

72. Out of 320 bulbs, [latex] \displaystyle{\frac{1}{20}} [/latex] of the bulbs are defective. How many of them are not defective?

73. If a wire that is [latex] 42\frac{3}{4} [/latex] cm long is cut into several [latex] 2\frac{1}{4} [/latex] cm equal pieces, how many pieces would exist?

74. The product of two numbers is 9. If one number is [latex] 3\frac{3}{4} [/latex], what is the other number?