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1.9 Fractions: addition and subtraction with unlike denominators

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to add and subtract fractions with unlike denominators. By the end of the module students should be able to add and subtract fractions with unlike denominators.

Section overview

  • A Basic Rule
  • Addition and Subtraction of Fractions

A basic rule

There is a basic rule that must be followed when adding or subtracting fractions.

A basic rule

Fractions can only be added or subtracted conveniently if they have like denomi­nators.

To see why this rule makes sense, let's consider the problem of adding a quarter and a dime.

1 quarter + 1 dime = 35 cents

Now,

1   quarter = 25 100 1   dime = 10 100 same denominations 35 ¢ = 35 100

25 100 + 10 100 = 25 + 10 100 = 35 100 size 12{ { {"25"} over {"100"} } + { {"10"} over {"100"} } = { {"25"+"10"} over {"100"} } = { {"35"} over {"100"} } } {}

In order to combine a quarter and a dime to produce 35¢, we convert them to quantities of the same denomination.

Same denomination → same denominator

Addition and subtraction of fractions

Least common multiple (lcm) and least common denominator (lcd)

In [link] , we examined the least common multiple (LCM) of a collection of numbers. If these numbers are used as denominators of fractions, we call the least common multiple, the least common denominator (LCD).

Method of adding or subtracting fractions with unlike denominators

To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as a denominator the least common denomina­tor ( LCD) of the original denominators.

Sample set a

Find the following sums and differences.

1 6 + 3 4 size 12{ { {1} over {6} } + { {3} over {4} } } {} . The denominators are not the same. Find the LCD of 6 and 4.

6 = 2 ⋅ 3 4 = 2 2 The LCD = 2 2 ⋅ 3 = 4 ⋅ 3 = 12

Write each of the original fractions as a new, equivalent fraction having the common denomina­tor 12.

1 6 + 3 4 = 12 + 12 size 12{ { {1} over {6} } + { {3} over {4} } = { {} over {"12"} } + { {} over {"12"} } } {}

To find a new numerator, we divide the original denominator into the LCD. Since the original denominator is being multiplied by this quotient, we must multiply the original numerator by this quotient.

12 ÷ 6 = 2 size 12{"12" div 6=2} {} Multiply 1 by 2: 1 times 2 equals 2. The 1 is the original numerator, and the 2, the product of the multiplication is the new numerator.

12 ÷ 4 = 3 size 12{"12" div 4=3} {} Multiply 3 by 3: 3 times 3 equals 9. 3 is the original numerator. 9 is the new numerator.

1 6 + 3 4 = 1 ⋅ 2 12 + 3 ⋅ 3 12 = 2 12 + 9 12 Now the denominators are the same. = 2 + 9 12 Add the numerators and place the sum over the common denominator. = 11 12

1 2 + 2 3 size 12{ { {1} over {2} } + { {2} over {3} } } {} . The denominators are not the same. Find the LCD of 2 and 3.

LCD = 2 ⋅ 3 = 6 size 12{ ital "LCD"=2 cdot 3=6} {}

Write each of the original fractions as a new, equivalent fraction having the common denominator 6.

1 2 + 2 3 = 6 + 6 size 12{ { {1} over {2} } + { {2} over {3} } = { {} over {6} } + { {} over {6} } } {}

To find a new numerator, we divide the original denominator into the LCD. Since the original denominator is being multiplied by this quotient, we must multiply the original numerator by this quotient.

6 ÷ 2 = 3 size 12{"6 " div " 2 "=" 3"} {} Multiply the numerator 1 by 3.

6 ÷ 2 = 3 size 12{"6 " div " 2 "=" 3"} {} Multiply the numerator 2 by 2.

1 2 + 2 3 = 1 ⋅ 3 6 + 2 ⋅ 3 6 = 3 6 + 4 6 = 3 + 4 6 = 7 6  or  1 1 6

5 9 − 5 12 size 12{ { {5} over {9} } - { {5} over {"12"} } } {} . The denominators are not the same. Find the LCD of 9 and 12.

9 = 3 ⋅ 3 = 3 2 12 = 2 ⋅ 6 = 2 ⋅ 2 ⋅ 3 = 2 2 ⋅ 3 LCD = 2 2 ⋅ 3 2 = 4 ⋅ 9 = 36

5 9 − 5 12 = 36 − 36 size 12{ { {5} over {9} } - { {5} over {"12"} } = { {} over {"36"} } - { {} over {"36"} } } {}

36 ÷ 9 = 4 size 12{"36"¸9=4} {} Multiply the numerator 5 by 4.

36 ÷ 12 = 3 size 12{"36"¸"12"=3} {} Multiply the numerator 5 by 3.

5 9 − 5 12 = 5 ⋅ 4 36 − 5 ⋅ 3 36 = 20 36 − 15 36 = 20 − 15 36 = 5 36

5 6 − 1 8 + 7 16 size 12{ { {5} over {6} } - { {1} over {8} } + { {7} over {"16"} } } {} The denominators are not the same. Find the LCD of 6, 8, and 16

6 = 2 ⋅ 3 8 = 2 ⋅ 4 = 2 ⋅ 2 ⋅ 2 = 2 3 16 = 2 ⋅ 8 = 2 ⋅ 2 ⋅ 4 = 2 ⋅ 2 ⋅ 2 ⋅ 2 = 2 4  The LCD is  2 4 ⋅ 3 = 48

5 6 − 1 8 + 7 16 = 48 − 48 + 48 size 12{ { {5} over {6} } - { {1} over {8} } + { {7} over {"16"} } = { {} over {"48"} } - { {} over {"48"} } + { {} over {"48"} } } {}

48 ÷ 6 = 8 size 12{"48"¸6=8} {} Multiply the numerator 5 by 8

48 ÷ 8 = 6 size 12{"48"¸8=6} {} Multiply the numerator 1 by 6

48 ÷ 16 = 3 size 12{"48"¸"16"=3} {} Multiply the numerator 7 by 3

5 6 − 1 8 + 7 16 = 5 ⋅ 8 48 − 1 ⋅ 6 48 + 7 ⋅ 3 48 = 40 48 − 6 48 + 21 48 = 40 − 6 + 21 48 = 55 48  or  1 7 48

Practice set a

Find the following sums and differences.

3 4 + 1 12 size 12{ { {3} over {4} } + { {1} over {"12"} } } {}

5 6 size 12{ { {5} over {6} } } {}

1 2 − 3 7 size 12{ { {1} over {2} } - { {3} over {7} } } {}

1 14 size 12{ { {1} over {"14"} } } {}

7 10 - 5 8 size 12{ { {7} over {"10"} } - { {5} over {8} } } {}

3 40 size 12{ { {3} over {"40"} } } {}

15 16 + 1 2 − 3 4 size 12{ { {"15"} over {"16"} } + { {1} over {2} } - { {3} over {4} } } {}

11 16 size 12{ { {"11"} over {"16"} } } {}

1 32 − 1 48 size 12{ { {1} over {"32"} } - { {1} over {"48"} } } {}

1 96 size 12{ { {1} over {"96"} } } {}

Exercises

A most basic rule of arithmetic states that two fractions may be added or subtracted conveniently only if they have .

The same denominator

For the following problems, find the sums and differences.

1 2 + 1 6 size 12{ { {1} over {2} } + { {1} over {6} } } {}

1 8 + 1 2 size 12{ { {1} over {8} } + { {1} over {2} } } {}

5 8 size 12{ { {5} over {8} } } {}

3 4 + 1 3 size 12{ { {3} over {4} } + { {1} over {3} } } {}

5 8 + 2 3 size 12{ { {5} over {8} } + { {2} over {3} } } {}

31 24 size 12{ { {"31"} over {"24"} } } {}

1 12 + 1 3 size 12{ { {1} over {"12"} } + { {1} over {3} } } {}

6 7 − 1 4 size 12{ { {6} over {7} } - { {1} over {4} } } {}

17 28 size 12{ { {"17"} over {"28"} } } {}

9 10 − 2 5 size 12{ { {9} over {"10"} } - { {2} over {5} } } {}

7 9 − 1 4 size 12{ { {7} over {9} } - { {1} over {4} } } {}

19 36 size 12{ { {"19"} over {"36"} } } {}

8 15 − 3 10 size 12{ { {8} over {"15"} } - { {3} over {"10"} } } {}

8 13 − 5 39 size 12{ { {8} over {"13"} } - { {5} over {"39"} } } {}

19 39 size 12{ { {"19"} over {"39"} } } {}

11 12 − 2 5 size 12{ { {"11"} over {"12"} } - { {2} over {5} } } {}

1 15 + 5 12 size 12{ { {1} over {"15"} } + { {5} over {"12"} } } {}

29 60 size 12{ { {"29"} over {"60"} } } {}

13 88 − 1 4 size 12{ { {"13"} over {"88"} } - { {1} over {4} } } {}

1 9 − 1 81 size 12{ { {1} over {9} } - { {1} over {"81"} } } {}

8 81 size 12{ { {8} over {"81"} } } {}

19 40 + 5 12 size 12{ { {"19"} over {"40"} } + { {5} over {"12"} } } {}

25 26 − 7 10 size 12{ { {"25"} over {"26"} } - { {7} over {"10"} } } {}

17 65 size 12{ { {"17"} over {"65"} } } {}

9 28 − 4 45 size 12{ { {9} over {"28"} } - { {4} over {"45"} } } {}

22 45 − 16 35 size 12{ { {"22"} over {"45"} } - { {"16"} over {"35"} } } {}

2 63 size 12{ { {2} over {"63"} } } {}

56 63 + 22 33 size 12{ { {"56"} over {"63"} } + { {"22"} over {"33"} } } {}

1 16 + 3 4 − 3 8 size 12{ { {1} over {"16"} } + { {3} over {4} } - { {3} over {8} } } {}

7 16 size 12{ { {7} over {"16"} } } {}

5 12 − 1 120 + 19 20 size 12{ { {5} over {"12"} } - { {1} over {"120"} } + { {"19"} over {"20"} } } {}

8 3 − 1 4 + 7 36 size 12{ { {8} over {3} } - { {1} over {4} } + { {7} over {"36"} } } {}

47 18 size 12{ { {"47"} over {"18"} } } {}

11 9 − 1 7 + 16 63 size 12{ { {"11"} over {9} } - { {1} over {7} } + { {"16"} over {"63"} } } {}

12 5 − 2 3 + 17 10 size 12{ { {"12"} over {5} } - { {2} over {3} } + { {"17"} over {"10"} } } {}

103 30 size 12{ { {"103"} over {"30"} } } {}

4 9 + 13 21 − 9 14 size 12{ { {4} over {9} } + { {"13"} over {"21"} } - { {9} over {"14"} } } {}

3 4 − 3 22 + 5 24 size 12{ { {3} over {4} } - { {3} over {"22"} } + { {5} over {"24"} } } {}

217 264 size 12{ { {"217"} over {"264"} } } {}

25 48 − 7 88 + 5 24 size 12{ { {"25"} over {"48"} } - { {7} over {"88"} } + { {5} over {"24"} } } {}

27 40 + 47 48 − 119 126 size 12{ { {"27"} over {"40"} } + { {"47"} over {"48"} } - { {"119"} over {"126"} } } {}

511 720 size 12{ { {"511"} over {"720"} } } {}

41 44 − 5 99 − 11 175 size 12{ { {"41"} over {"44"} } - { {5} over {"99"} } - { {"11"} over {"175"} } } {}

5 12 + 1 18 + 1 24 size 12{ { {5} over {"12"} } + { {1} over {"18"} } + { {1} over {"24"} } } {}

37 72 size 12{ { {"37"} over {"72"} } } {}

5 9 + 1 6 + 7 15 size 12{ { {5} over {9} } + { {1} over {6} } + { {7} over {"15"} } } {}

21 25 + 1 6 + 7 15 size 12{ { {"21"} over {"25"} } + { {1} over {6} } + { {7} over {"15"} } } {}

221 150 size 12{ { {"221"} over {"150"} } } {}

5 18 − 1 36 + 7 9 size 12{ { {5} over {"18"} } - { {1} over {"36"} } + { {7} over {9} } } {}

11 14 − 1 36 − 1 32 size 12{ { {"11"} over {"14"} } - { {1} over {"36"} } - { {1} over {"32"} } } {}

1, 465 2, 016 size 12{ { {1,"465"} over {2,"016"} } } {}

21 33 + 12 22 + 15 55 size 12{ { {"21"} over {"33"} } + { {"12"} over {"22"} } + { {"15"} over {"55"} } } {}

5 51 + 2 34 + 11 68 size 12{ { {5} over {"51"} } + { {2} over {"34"} } + { {"11"} over {"68"} } } {}

65 204 size 12{ { {"65"} over {"204"} } } {}

8 7 − 16 14 + 19 21 size 12{ { {8} over {7} } - { {"16"} over {"14"} } + { {"19"} over {"21"} } } {}

7 15 + 3 10 − 34 60 size 12{ { {7} over {"15"} } + { {3} over {"10"} } - { {"34"} over {"60"} } } {}

1 5 size 12{ { {1} over {5} } } {}

14 15 − 3 10 − 6 25 + 7 20 size 12{ { {"14"} over {"15"} } - { {3} over {"10"} } - { {6} over {"25"} } + { {7} over {"20"} } } {}

11 6 − 5 12 + 17 30 + 25 18 size 12{ { {"11"} over {6} } - { {5} over {"12"} } + { {"17"} over {"30"} } + { {"25"} over {"18"} } } {}

607 180 size 12{ { {"607"} over {"180"} } } {}

1 9 + 22 21 − 5 18 − 1 45 size 12{ { {1} over {9} } + { {"22"} over {"21"} } - { {5} over {"18"} } - { {1} over {"45"} } } {}

7 26 + 28 65 − 51 104 + 0 size 12{ { {7} over {"26"} } + { {"28"} over {"65"} } - { {"51"} over {"104"} } +0} {}

109 520 size 12{ { {"109"} over {"520"} } } {}

A morning trip from San Francisco to Los Angeles took 13 12 size 12{ { {"13"} over {"12"} } } {} hours. The return trip took 57 60 size 12{ { {"57"} over {"60"} } } {} hours. How much longer did the morning trip take?

At the beginning of the week, Starlight Publishing Company's stock was selling for 115 8 size 12{ { {"115"} over {8} } } {} dollars per share. At the end of the week, analysts had noted that the stock had gone up 11 4 size 12{ { {"11"} over {4} } } {} dollars per share. What was the price of the stock, per share, at the end of the week?

$ 137 8 or $ 17 1 8 size 12{$ { {"137"} over {8} } " or "$"17" { {1} over {8} } } {}

A recipe for fruit punch calls for 23 3 size 12{ { {"23"} over {3} } } {} cups of pineapple juice, 1 4 size 12{ { {1} over {4} } } {} cup of lemon juice, 15 2 size 12{ { {"15"} over {2} } } {} cups of orange juice, 2 cups of sugar, 6 cups of water, and 8 cups of carbonated non-cola soft drink. How many cups of ingredients will be in the final mixture?

The side of a particular type of box measures 8 3 4 size 12{8 { {3} over {4} } } {} inches in length. Is it possible to place three such boxes next to each other on a shelf that is 26 1 5 size 12{"26" { {1} over {5} } } {} inches in length? Why or why not?

No; 3 boxes add up to 26 1′′ 4 , which is larger than 25 1′′ 5 .

Four resistors, 3 8 size 12{ { {3} over {8} } } {} ohm, 1 4 size 12{ { {1} over {4} } } {} ohm, 3 5 size 12{ { {3} over {5} } } {} ohm, and 7 8 size 12{ { {7} over {8} } } {} ohm, are connected in series in an electrical circuit. What is the total resistance in the circuit due to these resistors? ("In series" implies addition.)

A copper pipe has an inside diameter of 2 3 16 size 12{2 { {3} over {"16"} } } {} inches and an outside diameter of 2 5 34 size 12{2 { {5} over {"34"} } } {} inches. How thick is the pipe?

No pipe at all; inside diameter is greater than outside diameter

The probability of an event was originally thought to be 15 32 size 12{ { {"15"} over {"32"} } } {} . Additional information decreased the probability by 3 14 size 12{ { {3} over {"14"} } } {} . What is the updated probability?

Exercises for review

( [link] ) Find the difference between 867 and 418.

449

( [link] ) Is 81,147 divisible by 3?

( [link] ) Find the LCM of 11, 15, and 20.

660

( [link] ) Find 3 4 size 12{ { {3} over {4} } } {} of 4 2 9 size 12{4 { {2} over {9} } } {} .

( [link] ) Find the value of 8 15 − 3 15 + 2 15 size 12{ { {8} over {"15"} } - { {3} over {"15"} } + { {2} over {"15"} } } {} .

7 15 size 12{ { {7} over {"15"} } } {}

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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