Comparison of Rational Numbers

Concept Explanation

Comparison of Rational Numbers

We know how to compare two integers or two fractions and tell which is smaller or which is greater among them. We know that every positive integer is greater than zero and every negative integer is less than zero. Also every positive integer is greater than every negative integer. Let us now see how we can compare two rational numbers.

Comparison of Rational Numbers:

Similar to the comparison of integers, we have the following facts about how to compare the rational numbers.

(i) Every positive rational number is greater than 0.

(ii) Every negative rational number is less than 0.

(iii) Every positive rational number is greater than every negative rational number.

(iv) Every rational number represented by a point on the number line is greater than every rational number represented by points on its left.

(v) Every rational number represented by a point on the number line is less than every rational number represented by points on its right.

How to Compare the Two Rational Numbers?

1. Comparison Using Number line: Two rational numbers can be compared by using a number line. For example to compare two negative rational numbers $-frac{1}{2}$ and $-frac{1}{5}$ using number line. We know that the integer which was on the right side of the other integer, was the greater integer.For example, 5 is to the right of 2 on the number line and $5> 2$ . The integer $-2$ is on the right of $-5$ on the number line and $-2> -5$ . Similarly we will mark the given rational numbers on the number line and then the number on the right side will be greater than the other number.

2. Comparison using the method of Fractions: Two positive rational numbers, like $frac{2}{3}$ and $frac{5}{7}$ can be compared as studied earlier in the case of fractions. In order to compare any two rational numbers, we can use the following steps:

Step I: Obtain the given rational numbers.

Step II: Write the given rational numbers so that their denominators are positive.

Step III: Find the LCM of the positive denominators of the rational numbers obtained in step II.

Step IV: Express each rational number (obtained in step II) with the LCM (obtained in step III) as common denominator.

Step V: Compare the numerators of rational numbers obtained in step having greater numerator is the greater rational number.

Illustration: Which of the given rational numbers is greater?

$frac{3}{-4}; and;frac{-5}{6}$

Solution: To find the greater rational number we will follow the steps:

First we write each of the given numbers with positive denominator.

$One; number = frac{3}{-4}= frac{3 times -1}{-4times-1}=frac{-3}{4}$

$Other; number = frac{-5}{6}$

Next step find the LCM

.L.C.M. of 4 and 6 = 12

Therefore,

$frac{-3}{4}=frac{-3times 3}{4times 3}=frac{-9}{12}$

$and;frac{-5}{6}=frac{-5times 2}{6times 2}=frac{-10}{12}$

Now we will compare the numerators, so we get -9 > -10

Hence

$therefore ;frac{-9}{12}>frac{-10}{12}$

$Rightarrow ;frac{3}{-4}>frac{-5}{6}$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

What is the percentage of the smallest number to the greatest number, in the collection of fractions $frac{5}{6},frac{7}{12},frac{13}{18},frac{23}{24}$ ?
A. $60frac{20}{23}$	B. $39frac{3}{23}$	C. $35frac{3}{18}$	D. None of these
Right Option : A
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Question : 2

Arrange the following numbers in descending order: $fn_jvn -2,frac{4}{-3},frac{-11}{15}, frac{3}{4}$
A. $frac{4}{-3} < -2<frac{-11}{15}<frac{3}{4}$		B. $frac{-11}{15} < -2 < frac{4}{-3} < frac{3}{4}$
C. $-2<frac{4}{-3}<frac{-11}{15}<frac{3}{4}$		D. $frac{4}{-3}<frac{-11}{15}<frac{3}{4}<-2$
Right Option : C
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Question : 3

Arrange the following numbers in descending order: $-2,frac{4}{-5},frac{-11}{20}, frac{3}{4}$
A. $frac{3}{4}>-2>frac{-11}{20}>frac{4}{-5}$		B. $frac{3}{4}>frac{-11}{20}>frac{4}{-5}>-2$
C. $frac{3}{4}>frac{4}{-5}>-2>frac{-11}{20}$		D. $frac{3}{4}>frac{4}{-5}>frac{-11}{20}>-2$
Right Option : B
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