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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to compare fractions. By the end of the module students should be able to understand ordering of numbers and be familiar with grouping symbols and compare two or more fractions.

Section overview

  • Order and the Inequality Symbols
  • Comparing Fractions

Order and the inequality symbols

Our number system is called an ordered number system because the numbers in the system can be placed in order from smaller to larger. This is easily seen on the number line.

A number line showing marks for 0 through 10. An arrow points to the left, labeled smaller. Another arrow points to the right, labeled larger.

On the number line, a number that appears to the right of another number is larger than that other number. For example, 5 is greater than 2 because 5 is located to the right of 2 on the number line. We may also say that 2 is less than 5.

To make the inequality phrases "greater than" and "less than" more brief, mathematicians represent them with the symbols>and<, respectively.

Symbols for greater than>And less than<

>represents the phrase "greater than."
<represents the phrase "less than."

5>2 represents "5 is greater than 2."

2<5 represents "2 is less than 5."

Comparing fractions

Recall that the fraction 4 5 size 12{ { {4} over {5} } } {} indicates that we have 4 of 5 parts of some whole quantity, and the fraction 3 5 size 12{ { {3} over {5} } } {} indicates that we have 3 of 5 parts. Since 4 of 5 parts is more than 3 of 5 parts, 4 5 size 12{ { {4} over {5} } } {} is greater than 3 5 size 12{ { {3} over {5} } } {} ; that is,

4 5 > 3 5 size 12{ { {4} over {5} }>{ {3} over {5} } } {}

We have just observed that when two fractions have the same denominator, we can determine which is larger by comparing the numerators.

Comparing fractions

If two fractions have the same denominators, the fraction with the larger nu­merator is the larger fraction.

Thus, to compare the sizes of two or more fractions, we need only convert each of them to equivalent fractions that have a common denominator. We then compare the numerators. It is convenient if the common denominator is the LCD. The fraction with the larger numerator is the larger fraction.

Sample set a

Compare 8 9 size 12{ { {8} over {9} } } {} and 14 15 size 12{ { {"14"} over {"15"} } } {} .

Convert each fraction to an equivalent fraction with the LCD as the denominator. Find the LCD.

9 = 3 2 15 = 3 5 The LCD = 3 2 5 = 9 5 = 45

8 9 = 8 5 45 = 40 45 14 15 = 14 3 45 = 42 45 alignl { stack { size 12{ { {8} over {9} } = { {8 cdot 5} over {"45"} } = { {"40"} over {"45"} } } {} #size 12{ { {"14"} over {"15"} } = { {"14" cdot 3} over {"45"} } = { {"42"} over {"45"} } } {} } } {}

Since 40 < 42 size 12{"40"<"42"} {} ,

40 45 < 42 45 size 12{ { {"40"} over {"45"} }<{ {"42"} over {"45"} } } {}

Thus 8 9 < 14 15 size 12{ { {8} over {9} }<{ {"14"} over {"15"} } } {} .

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Write 5 6 , 7 10 , size 12{ { {5} over {6} } , { {7} over {"10"} } ,} {} and 13 15 size 12{ { {"13"} over {"15"} } } {} in order from smallest to largest.

Convert each fraction to an equivalent fraction with the LCD as the denominator.

Find the LCD.

6 = 2 3 10 = 2 5 15 = 3 5 The LCD = 2 3 5 = 30 size 12{2 cdot 3 cdot 5="30"} {}

5 6 = 5 5 30 = 25 30 size 12{ { {5} over {6} } = { {5 cdot 5} over {"30"} } = { {"25"} over {"30"} } } {}

7 10 = 7 3 30 = 21 30 size 12{ { {7} over {"10"} } = { {7 cdot 3} over {"30"} } = { {"21"} over {"30"} } } {}

13 15 = 13 2 30 = 26 30 size 12{ { {"13"} over {"15"} } = { {"13" cdot 2} over {"30"} } = { {"26"} over {"30"} } } {}

Since 21 < 25 < 26 ,

21 30 size 12{ { {"21"} over {"30"} } } {} < 25 30 size 12{ { {"25"} over {"30"} } } {} < 26 30 size 12{ { {"26"} over {"30"} } } {}

7 10 size 12{ { {7} over {"10"} } } {} < 5 6 size 12{ { {5} over {6} } } {} < 13 15 size 12{ { {"13"} over {"15"} } } {}

Writing these numbers in order from smallest to largest, we get 7 10 size 12{ { {7} over {"10"} } } {} , 5 6 size 12{ { {5} over {6} } } {} , 13 15 size 12{ { {"13"} over {"15"} } } {} .

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Compare 8 6 7 and 6 3 4 .

To compare mixed numbers that have different whole number parts, we need only compare whole number parts. Since 6<8,

6 3 4 < 8 6 7 size 12{6 { {3} over {4} }<8 { {6} over {7} } } {}

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Compare 4 5 8 and 4 7 12 size 12{4 { {5} over {8} } " and "4 { {7} over {"12"} } } {}

To compare mixed numbers that have the same whole number parts, we need only compare fractional parts.

8 = 2 3 12 = 2 2 3 The LCD = 2 3 3 = 8 3 = 24 size 12{2 rSup { size 8{3} } cdot 3=8 cdot 3="24"} {}

5 8 = 5 3 24 = 15 24 size 12{ { {5} over {8} } = { {5 cdot 3} over {"24"} } = { {"15"} over {"24"} } } {}

7 12 = 7 2 24 = 14 24 size 12{ { {7} over {"12"} } = { {7 cdot 2} over {"24"} } = { {"14"} over {"24"} } } {}

Since 14<15,

14 24 < 15 24 size 12{ { {"14"} over {"24"} }<{ {"15"} over {"24"} } } {}

7 12 < 5 8 size 12{ { {7} over {"12"} }<{ {5} over {8} } } {}

Hence, 4 7 12 < 4 5 8 size 12{4 { {7} over {"12"} }<4 { {5} over {8} } } {}

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Practice set a

Compare 3 4 size 12{ { {3} over {4} } } {} and 4 5 size 12{ { {4} over {5} } } {} .

3 4 < 4 5 size 12{ { {3} over {4} }<{ {4} over {5} } } {}

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Compare 9 10 size 12{ { {9} over {"10"} } } {} and 13 15 size 12{ { {"13"} over {"15"} } } {} .

13 15 < 9 10 size 12{ { {"13"} over {"15"} }<{ {9} over {"10"} } } {}

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Write 13 16 size 12{ { {"13"} over {"16"} } } {} , 17 20 size 12{ { {"17"} over {"20"} } } {} , and 33 40 size 12{ { {"33"} over {"40"} } } {} in order from smallest to largest.

13 16 , 33 40 , 17 20 size 12{ { {"13"} over {"16"} } , { {"33"} over {"40"} } , { {"17"} over {"20"} } } {}

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Compare 11 1 6 size 12{"11" { {1} over {6} } } {} and 9 2 5 size 12{9 { {2} over {5} } } {} .

9 2 5 < 11 1 6 size 12{9 { {2} over {5} }<"11" { {1} over {6} } } {}

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Compare 1 9 14 size 12{1 { {9} over {"14"} } } {} and 1 11 16 size 12{1 { {"11"} over {"16"} } } {} .

1 9 14 < 1 11 16 size 12{1 { {9} over {"14"} }<1 { {"11"} over {"16"} } } {}

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Exercises

Arrange each collection of numbers in order from smallest to largest.

3 5 , 5 8 size 12{ { {3} over {5} } , { {5} over {8} } } {}

3 5 < 5 8 size 12{ { {3} over {5} }<{ {5} over {8} } } {}

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1 6 , 2 7 size 12{ { {1} over {6} } , { {2} over {7} } } {}

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3 4 , 5 6 size 12{ { {3} over {4} } , { {5} over {6} } } {}

3 4 < 5 6 size 12{ { {3} over {4} }<{ {5} over {6} } } {}

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7 9 , 11 12 size 12{ { {7} over {9} } , { {"11"} over {"12"} } } {}

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

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

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1 2 , 5 8 , 7 16 size 12{ { {1} over {2} } , { {5} over {8} } , { {7} over {"16"} } } {}

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1 2 , 3 5 , 4 7 size 12{ { {1} over {2} } , { {3} over {5} } , { {4} over {7} } } {}

1 2 < 4 7 < 3 5 size 12{ { {1} over {2} }<{ {4} over {7} }<{ {3} over {5} } } {}

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3 4 , 2 3 , 5 6 size 12{ { {3} over {4} } , { {2} over {3} } , { {5} over {6} } } {}

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3 4 , 7 9 , 5 4 size 12{ { {3} over {4} } , { {7} over {9} } , { {5} over {4} } } {}

3 4 < 7 9 < 5 4 size 12{ { {3} over {4} }<{ {7} over {9} }<{ {5} over {4} } } {}

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7 8 , 15 16 , 11 12 size 12{ { {7} over {8} } , { {"15"} over {"16"} } , { {"11"} over {"12"} } } {}

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3 14 , 2 7 , 3 4 size 12{ { {3} over {"14"} } , { {2} over {7} } , { {3} over {4} } } {}

3 14 < 2 7 < 3 4 size 12{ { {3} over {"14"} }<{ {2} over {7} }<{ {3} over {4} } } {}

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17 32 , 25 48 , 13 16 size 12{ { {"17"} over {"32"} } , { {"25"} over {"48"} } , { {"13"} over {"16"} } } {}

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5 3 5 , 5 4 7 size 12{5 { {3} over {5} } ,5 { {4} over {7} } } {}

5 4 7 < 5 3 5 size 12{5 { {4} over {7} }<5 { {3} over {5} } } {}

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11 3 16 , 11 1 12 size 12{"11" { {3} over {"16"} } ,"11" { {1} over {"12"} } } {}

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9 2 3 , 9 4 5 size 12{9 { {2} over {3} } ,9 { {4} over {5} } } {}

9 2 3 < 9 4 5 size 12{9 { {2} over {3} }<9 { {4} over {5} } } {}

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7 2 3 , 8 5 6 size 12{7 { {2} over {3} } ,8 { {5} over {6} } } {}

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1 9 16 , 2 1 20 size 12{1 { {9} over {"16"} } ,2 { {1} over {"20"} } } {}

1 9 16 < 2 1 20 size 12{1 { {9} over {"16"} }<2 { {1} over {"20"} } } {}

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20 15 16 , 20 23 24 size 12{"20" { {"15"} over {"16"} } ,"20" { {"23"} over {"24"} } } {}

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2 2 9 , 2 3 7 size 12{2 { {2} over {9} } ,2 { {3} over {7} } } {}

2 2 9 < 2 3 7 size 12{2 { {2} over {9} }<2 { {3} over {7} } } {}

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5 8 13 , 5 9 20 size 12{5 { {8} over {"13"} } ,5 { {9} over {"20"} } } {}

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Exercises for review

( [link] ) Round 267,006,428 to the nearest ten million.

270,000,000

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( [link] ) Is the number 82,644 divisible by 2? by 3? by 4?

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( [link] ) Convert 3 2 7 size 12{3 { {2} over {7} } } {} to an improper fraction.

23 7 size 12{ { {"23"} over {7} } } {}

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( [link] ) Find the value of 5 6 + 3 10 2 5 size 12{ { {5} over {6} } + { {3} over {"10"} } - { {2} over {5} } } {}

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( [link] ) Find the value of 8 3 8 + 5 1 4 size 12{8 { {3} over {8} } +5 { {1} over {4} } } {} .

13 5 8 or 109 8 size 12{"13" { {5} over {8} } " or " { {"109"} over {8} } } {}

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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