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Comparing Fractions

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: 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 4545 size 12{ { {4} over {5} } } {} indicates that we have 4 of 5 parts of some whole quantity, and the fraction 3535 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, 4545 size 12{ { {4} over {5} } } {} is greater than 3535 size 12{ { {3} over {5} } } {}; that is,

4 5 > 3 5 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

Example 1

Compare 8989 size 12{ { {8} over {9} } } {} and 14151415 size 12{ { {"14"} over {"15"} } } {}.

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

9 = 32 15 = 3 ⋅ 5 The LCD = 32 ⋅ 5 = 9⋅5=45 9 = 32 15 = 3 ⋅ 5 The LCD=32⋅5=9⋅5=45

8 9 = 8 ⋅ 5 45 = 40 45 14 15 = 14 ⋅ 3 45 = 42 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<4240<42 size 12{"40"<"42"} {},

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

Thus 89<141589<1415 size 12{ { {8} over {9} } < { {"14"} over {"15"} } } {}.

Example 2

Write 56,710,56,710, size 12{ { {5} over {6} } , { {7} over {"10"} } ,} {} and 13151315 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⋅310=2⋅515=3⋅5The LCD= 2⋅3⋅5=30 6=2⋅310=2⋅515=3⋅5The LCD=2⋅3⋅5=30 size 12{2 cdot 3 cdot 5="30"} {}

5 6 = 5 ⋅ 5 30 = 25 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 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 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<25<26,

21302130 size 12{ { {"21"} over {"30"} } } {} < 25302530 size 12{ { {"25"} over {"30"} } } {} < 26302630 size 12{ { {"26"} over {"30"} } } {}

710710 size 12{ { {7} over {"10"} } } {} < 5656 size 12{ { {5} over {6} } } {} < 13151315 size 12{ { {"13"} over {"15"} } } {}

Writing these numbers in order from smallest to largest, we get 710710 size 12{ { {7} over {"10"} } } {}, 5656 size 12{ { {5} over {6} } } {}, 13151315 size 12{ { {"13"} over {"15"} } } {}.

Example 3

Compare 867867 and 634634.

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

Example 4

Compare 458 and 4712458 and 4712 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 = 23 12 = 22⋅3 The LCD = 23⋅3=8⋅3=248 = 23 12 = 22⋅3 The LCD=23⋅3=8⋅3=24 size 12{2 rSup { size 8{3} } cdot 3=8 cdot 3="24"} {}

5 8 = 5 ⋅ 3 24 = 15 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 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 14 24 < 15 24 size 12{ { {"14"} over {"24"} } < { {"15"} over {"24"} } } {}

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

Hence, 4712<4584712<458 size 12{4 { {7} over {"12"} } <4 { {5} over {8} } } {}

Practice Set A

Exercise 1

Compare 3434 size 12{ { {3} over {4} } } {} and 4545 size 12{ { {4} over {5} } } {}.

Solution

34<4534<45 size 12{ { {3} over {4} } < { {4} over {5} } } {}

Exercise 2

Compare 910910 size 12{ { {9} over {"10"} } } {} and 13151315 size 12{ { {"13"} over {"15"} } } {}.

Solution

1315<9101315<910 size 12{ { {"13"} over {"15"} } < { {9} over {"10"} } } {}

Exercise 3

Write 13161316 size 12{ { {"13"} over {"16"} } } {}, 17201720 size 12{ { {"17"} over {"20"} } } {}, and 33403340 size 12{ { {"33"} over {"40"} } } {} in order from smallest to largest.

Solution

1316,3340,17201316,3340,1720 size 12{ { {"13"} over {"16"} } , { {"33"} over {"40"} } , { {"17"} over {"20"} } } {}

Exercise 4

Compare 11161116 size 12{"11" { {1} over {6} } } {} and 925925 size 12{9 { {2} over {5} } } {}.

Solution

925<1116925<1116 size 12{9 { {2} over {5} } <"11" { {1} over {6} } } {}

Exercise 5

Compare 19141914 size 12{1 { {9} over {"14"} } } {} and 1111611116 size 12{1 { {"11"} over {"16"} } } {}.

Solution

1914<111161914<11116 size 12{1 { {9} over {"14"} } <1 { {"11"} over {"16"} } } {}

Exercises

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

Exercise 6

35,5835,58 size 12{ { {3} over {5} } , { {5} over {8} } } {}

Solution

35<5835<58 size 12{ { {3} over {5} } < { {5} over {8} } } {}

Exercise 7

16,2716,27 size 12{ { {1} over {6} } , { {2} over {7} } } {}

Exercise 8

34,5634,56 size 12{ { {3} over {4} } , { {5} over {6} } } {}

Solution

34<5634<56 size 12{ { {3} over {4} } < { {5} over {6} } } {}

Exercise 9

79,111279,1112 size 12{ { {7} over {9} } , { {"11"} over {"12"} } } {}

Exercise 10

38,2538,25 size 12{ { {3} over {8} } , { {2} over {5} } } {}

Solution

38<2538<25 size 12{ { {3} over {8} } < { {2} over {5} } } {}

Exercise 11

12,58,71612,58,716 size 12{ { {1} over {2} } , { {5} over {8} } , { {7} over {"16"} } } {}

Exercise 12

12,35,4712,35,47 size 12{ { {1} over {2} } , { {3} over {5} } , { {4} over {7} } } {}

Solution

12<47<3512<47<35 size 12{ { {1} over {2} } < { {4} over {7} } < { {3} over {5} } } {}

Exercise 13

34,23,5634,23,56 size 12{ { {3} over {4} } , { {2} over {3} } , { {5} over {6} } } {}

Exercise 14

34,79,5434,79,54 size 12{ { {3} over {4} } , { {7} over {9} } , { {5} over {4} } } {}

Solution

34<79<5434<79<54 size 12{ { {3} over {4} } < { {7} over {9} } < { {5} over {4} } } {}

Exercise 15

78,1516,111278,1516,1112 size 12{ { {7} over {8} } , { {"15"} over {"16"} } , { {"11"} over {"12"} } } {}

Exercise 16

314,27,34314,27,34 size 12{ { {3} over {"14"} } , { {2} over {7} } , { {3} over {4} } } {}

Solution

314<27<34314<27<34 size 12{ { {3} over {"14"} } < { {2} over {7} } < { {3} over {4} } } {}

Exercise 17

1732,2548,13161732,2548,1316 size 12{ { {"17"} over {"32"} } , { {"25"} over {"48"} } , { {"13"} over {"16"} } } {}

Exercise 18

535,547535,547 size 12{5 { {3} over {5} } ,5 { {4} over {7} } } {}

Solution

547<535547<535 size 12{5 { {4} over {7} } <5 { {3} over {5} } } {}

Exercise 19

11316,1111211316,11112 size 12{"11" { {3} over {"16"} } ,"11" { {1} over {"12"} } } {}

Exercise 20

923,945923,945 size 12{9 { {2} over {3} } ,9 { {4} over {5} } } {}

Solution

923<945923<945 size 12{9 { {2} over {3} } <9 { {4} over {5} } } {}

Exercise 21

723,856723,856 size 12{7 { {2} over {3} } ,8 { {5} over {6} } } {}

Exercise 22

1916,21201916,2120 size 12{1 { {9} over {"16"} } ,2 { {1} over {"20"} } } {}

Solution

1916<21201916<2120 size 12{1 { {9} over {"16"} } <2 { {1} over {"20"} } } {}

Exercise 23

201516,202324201516,202324 size 12{"20" { {"15"} over {"16"} } ,"20" { {"23"} over {"24"} } } {}

Exercise 24

229,237229,237 size 12{2 { {2} over {9} } ,2 { {3} over {7} } } {}

Solution

229<237229<237 size 12{2 { {2} over {9} } <2 { {3} over {7} } } {}

Exercise 25

5813,59205813,5920 size 12{5 { {8} over {"13"} } ,5 { {9} over {"20"} } } {}

Exercises for Review

Exercise 26

((Reference)) Round 267,006,428 to the nearest ten million.

Solution

270,000,000

Exercise 27

((Reference)) Is the number 82,644 divisible by 2? by 3? by 4?

Exercise 28

((Reference)) Convert 327327 size 12{3 { {2} over {7} } } {} to an improper fraction.

Solution

237237 size 12{ { {"23"} over {7} } } {}

Exercise 29

((Reference)) Find the value of 56+310−2556+310−25 size 12{ { {5} over {6} } + { {3} over {"10"} } - { {2} over {5} } } {}

Exercise 30

((Reference)) Find the value of 838+514838+514 size 12{8 { {3} over {8} } +5 { {1} over {4} } } {}.

Solution

1358 or 10981358 or 1098 size 12{"13" { {5} over {8} } " or " { {"109"} over {8} } } {}

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