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4.8: Order of Operations with Fractions

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Let’s begin by taking powers of fractions. Recall that

am=aa...am times

Example 1

Simplify: (−3/4)2.

Solution

By definition,

(34)2=(34)(34)  Fact: a2=aa.=3344  Multiply numerators and denominators. Product of even number of negative factors is positive.=916  Simplify.

Exercise

Simplify:

(25)2

Answer

4/25

Example 2

Simplify: (−2/3)3.

Solution

By definition,

(23)3=(23)(23)(23)  Fact: a3=aaa.=222333  Multiply numerators and denominators. Product of odd number of negative factors is negative.=827  Simplify.

Exercise

Simplify:

(16)3

Answer

−1/216

The last two examples reiterate a principle learned earlier.

Odd and Even
  • The product of an even number of negative factors is positive.
  • The product of an odd number of negative factors is negative.

Order of Operations

For convenience, we repeat here the rules guiding order of operations.

Rules Guiding Order of Operations

When evaluating expressions, proceed in the following order.

  1. Evaluate expressions contained in grouping symbols first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.
  2. Evaluate all exponents that appear in the expression.
  3. Perform all multiplications and divisions in the order that they appear in the expression, moving left to right.
  4. Perform all additions and subtractions in the order that they appear in the expression, moving left to right.
Example 3

Simplify: 12+14(13).

Solution

Multiply first, then add.

12+14(13)=12+(112)  Multiply: 14(13)=112.=1626+(112)  Equivalent fractions, LCD = 12.=612+(112)  Simplify numerator and denominator.=712  Add over common denominator.

Exercise

Simplify: 23+34(12)

Answer

−25/24

Example 4

Simplify: 2(12)2+4(12).

Solution

Exponents first, then multiply, then add.

2(12)2+4(12)=2(14)+4(12)  Exponent first: (12)2=14.=12+(21)  Multiply: 2(14)=12 and 4(12)=21.=12+(2212)  Equivalent fractions, LCD = 2.=12+(42)  Simplify numerator and denominator.=32  Add over common denominator.

Exercise

Simplify: 3(13)22(13)

Answer

1

Example 5

Given a = −3/4, b = 1/2, c = 1/3, and d = −1/4, evaluate the expression abcd.

Solution

Recall that it is good practice to prepare parentheses before substituting.

adbc=( )( )( )( )

Substitute the given values into the algebraic expression, then simplify using order of operations.

abcd=(34)(12)(13)(14)  Substitute: 3/4 for a, 1/2 for b,1/3 for c,  and 1/4 for d.=38(112)  Multiply first: (34)(12)=38 and (13)(14)=112.=38+112  Subtract by adding opposite.=3383+12122  Equivalent fractions; LCD = 24.=924+224  Simplify numerators and denominators.=724  Add over common denominator.

Exercise

Given a = −1/2, b = 1/3, and c = −1/5, evaluate a + bc.

Answer

−17/30

Example 6

Given a = −1/4 and b = 1/2, evaluate (a2b2) ÷ (a + b).

Solution

Recall that it is good practice to prepare parentheses before substituting.

(a2b2)÷(a+b)(( )2( )2)÷(( )+( ))

Substitute the given values into the algebraic expression, then evaluate exponents first.

(a2b2)÷(a+b)=((14)2(12)2)÷((14)+(12)) =(11614)÷(14+12)

We must evaluate parentheses first. Inside each set of parentheses, create equivalent fractions and perform subtractions and additions next.

=(1161444)÷(14+1222)=(116416)÷(14+24)=316÷14

Invert and multiply.

=31641=1216

Reduce.

=12÷416÷434

Note: In the last step, you could also reduce by prime factoring numerator and denominator and canceling common factors.

Exercise

Give a = −1/2 and b = −1/3, evaluate ab ÷ (a + b).

Answer

−1/5

Complex Fractions

Complex Fractions

When the numerator and denominator of a fraction contain fractions themselves, such an expression is called a complex fraction.

You can use the standard order of operations to simplify a complex fraction. Recall the advice when a fraction is present.

Fractional Expressions

If a fractional expression is present, simplify the numerator and denominator separately, then divide.

Example 7

Simplify:

12+133432

Solution

We have addition in the numerator, subtraction in the denominator. In each case, we need equivalent fractions with a common denominator.

12+133432=1323+1232  Create equivalent fractions.=36+263464  Simplify numerator and denominator.=1634   Numerator: 36+26=16. Denominator: 3464=34.

The last expression asks us to divide. Invert and multiply.

=16÷(34)  A complex fraction means divide.=16(43)  Invert and multiply.

Like signs (two negatives) give a positive product. Multiply numerators and denominators, then reduce.

=418  Like signs yields positive answer. Multiply numerators and denominators.=4÷218÷2  Divide both numerator and denominator by 2.=29  Simplify.

Alternatively, one could prime factor and cancel to reduce to lowest terms; that is,

418=22233  Prime factor.=22233  Cancel common factors.=29  Simplify.

Exercise

Simplify:

141314+13

Answer

−1/7

Clearing Fractions

An alternate technique for simplifying complex fractions is available.

Clearing Fractions from Complex Fractions

You can clear fractions from a complex fraction using the following algorithm:

  1. Determine an LCD1 for the numerator.
  2. Determine an LCD2 for the denominator.
  3. Determine an LCD for both LCD1 and LCD2.
  4. Multiply both numerator and denominator by this “combined” LCD.

Let’s apply this technique to the complex fraction of Example 7.

Example 8

Simplify:

12+133432

Solution

As we saw in the solution in Example 7, common denominators of 6 and 4 were used for the numerator and denominator, respectively. Thus, a common denominator for both numerator and denominator would be 12. We begin the alternate solution technique by multiplying both numerator and denominator by 12.

12+133432=12(12+13)12(3432)  Multiply numerator and denominator by 12.=12(12)+12(13)12(34)12(32)  Distribute the 12.=6+4918  Multiply: 12(1/2)=6, 12(1/2)=4.12(3/4)=9, and 12(3/2)=18.=29  Simplify.=29  Like signs yield positive.

Exercise

Simplify: 23+154512

Answer

−14/9

Application — Trapezoid

A trapezoid is a special type of quadrilateral (four-sided polygon).

Trapezoid

A quadrilateral with one pair of parallel opposite sides is called a trapezoid.

Screen Shot 2019-09-04 at 4.20.28 PM.png

The pair of parallel sides are called the bases of the trapezoid. Their lengths are marked by the variables b1 and b2 in the figure above. The distance between the parallel bases is called the height or altitude of the trapezoid. The height is marked by the variable h in the figure above.

Mathematicians use subscripts to create new variables. Thus, b1 (“b sub 1”) and b2 (“b sub 2”) are two distinct variables, used in this case to represent the length of the bases of the trapezoid.

By drawing in a diagonal, we can divide the trapezoid into two triangles (see Figure 4.14).

Screen Shot 2019-09-04 at 4.20.54 PM.png
Figure 4.14: Dividing the trapezoid into two triangles.

We can find the area of the trapezoid by summing the areas of the two triangles.

  • The shaded triangle in Figure 4.14 has base b1 and height h. Hence, the area of the shaded triangle is (1/2)b1h.
  • The unshaded triangle in Figure 4.14 has base b2 and height h. Hence, the area of the unshaded triangle is (1/2)b2h.

Summing the areas, the area of the trapezoid is

Area of Trapezoid=12b1h+12b2h.

We can use the distributive property to factor out a (1/2)h.

Area of a Trapezoid

A trapezoid with bases b1 and b2 and height h has area

A=12h(b1+b2).

That is, to find the area, sum the bases, multiply by the height, and take one-half of the result.

Example 9

Find the area of the trapezoid pictured below.

Screen Shot 2019-09-04 at 4.25.28 PM.png

Solution

The formula for the area of a trapezoid is

A=12h(b1+b2)

Substituting the given bases and height, we get

A=12(3)(414+212).

Simplify the expression inside the parentheses first. Change mixed fractions to improper fractions, make equivalent fractions with a common denominator, then add.

A=12(3)(174+52)=12(3)(174+5222)=12(3)(174+104)=12(31)(274)

Multiply numerators and denominators.

=818

This improper fraction is a perfectly good answer, but let’s change this result to a mixed fraction (81 divided by 8 is 10 with a remainder of 1). Thus, the area of the trapezoid is

A=1018 square inches.

Exercise

A trapezoid has bases measuring 6 and 15 feet, respectively. The height of the trapezoid is 5 feet. Find the area of the trapezoid.

Answer

5212 square feet

Exercises

In Exercises 1-8, simplify the expression.

1. (73)3

2. (12)3

3. (53)4

4. (35)4

5. (12)5

6. (34)5

7. (43)2

8. (85)2


9. If a = 7/6, evaluate a3.

10. If e = 1/6, evaluate e3.

11. If e = −2/3, evaluate −e2.

12. If c = −1/5, evaluate −c2.

13. If b = −5/9, evaluate b2.

14. If c = 5/7, evaluate c2.

15. If b = −1/2, evaluate −b3.

16. If a = −2/9, evaluate −a3.


In Exercises 17-36, simplify the expression.

17. (12)(16)(78)(79)

18. (34)(12)(35)(14)

19. (98)2(32)(73)

20. (32)2(78)(12)

21. (12)(74)(12)2

22. (15)(94)(74)2

23. 761779

24. 498589

25. 34+97(76)

26. 32+14(98)

27. (13)2+(78)(13)

28. (29)2+(23)(12)

29. 59+5979

30. 12+9813

31. (56)(38)+(79)(34)

32. (74)(65)+(25)(83)

33. 4329(34)

34. 1315(43)

35. (59)(12)+(16)2

36. (14)(16)+(56)2


37. Given a = −5/4, b = 1/2, and c = 3/8, evaluate a + bc.

38. Given a = −3/5, b = 1/5, and c = 1/3, evaluate a + bc.

39. Given x = −1/8, y = 5/2, and z = −1/2, evaluate the expression x + yz.

40. Given x = −5/9, y = 1/4, and z = −2/3, evaluate the expression x + yz.

41. Given a = 3/4, b = 5/7, and c = 1/2, evaluate the expression abc.

42. Given a = 5/9, b = 2/3, and c = 2/9, evaluate the expression abc.

43. Given x = −3/2, y = 1/4, and z = −5/7, evaluate x2yz.

44. Given x = −3/2, y = −1/2, and z = 5/3, evaluate x2yz.

45. Given a = 6/7, b = 2/3, c = −8/9, and d = −6/7, evaluate ab + cd.

46. Given a = 4/9, b = −3/2, c = 7/3, and d = −8/9, evaluate ab + cd.

47. Given w = −1/8, x = −2/7, y = −1/2, and z = 8/7, evaluate wxyz.

48. Given w = 2/7, x = −9/4, y = −3/4, and z = −9/2, evaluate wxyz.

49. Given x = 3/8, y = 3/5, and z = −3/2, evaluate xy + z2.

50. Given x = −1/2, y = 7/5, and z = −3/2, evaluate xy + z2.

51. Given u = 9/7, v = 2/3, and w = −3/7, evaluate uvw2.

52. Given u = 8/7, v = −4/3, and w = 2/3, evaluate uvw2.

53. Given a = 7/8, b = −1/4, and c = −3/2, evaluate a2 + bc.

54. Given a = −5/8, b = 3/2, and c = −3/2, evaluate a2 + bc.

55. Given u = 1/3, v = 5/2, and w = −2/9, evaluate the expression uvw.

56. Given u = −1/2, v = 1/4, and w = −1/4, evaluate the expression uvw.


In Exercises 57-68, simplify the complex rational expression.

57. 83+769214

58. 78+198916

59. 34+4319+53

60. 986574+12

61. 75+5214+12

62. 56+2335+23

63. 32237423

64. 89+342316

65. 124757+16

66. 32583412

67. 37131367

68. 58655438


69. A trapezoid has bases measuring 338 and 512 feet, respectively. The height of the trapezoid is 7 feet. Find the area of the trapezoid.

70. A trapezoid has bases measuring 212 and 678 feet, respectively. The height of the trapezoid is 3 feet. Find the area of the trapezoid.

71. A trapezoid has bases measuring 214 and 738 feet, respectively. The height of the trapezoid is 7 feet. Find the area of the trapezoid.

72. A trapezoid has bases measuring 318 and 612 feet, respectively. The height of the trapezoid is 3 feet. Find the area of the trapezoid.

73. A trapezoid has bases measuring 234 and 658 feet, respectively. The height of the trapezoid is 3 feet. Find the area of the trapezoid.

74. A trapezoid has bases measuring 214 and 718 feet, respectively. The height of the trapezoid is 5 feet. Find the area of the trapezoid.


Answers

1. 34327

3. 62581

5. 132

7. 169

9. 343216

11. 49

13. 2581

15. 18

17. 4372

19. 30564

21. 58

23. 2318

25. 34

27. 1372

29. 8081

31. 1348

33. 32

35. 14

37. 1716

39. 118

41. 1128

43. 177

45. 43

47. 1728

49. 9940

51. 3349

53. 7364

55. 89

57. 4657

59. 7564

61. 785

63. 2629

65. 4523

67. 1611

69. 31116

71. 331116

73. 14116


This page titled 4.8: Order of Operations with Fractions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by David Arnold.

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