We will learn the comparison of rational numbers. We know how to compare two integers and also two fractions. 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.

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 paints on its right.

How to compare the two rational
numbers?

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.

Solved examples on comparison of rational numbers:

**1.** Which of the two rational numbers \(\frac{3}{5}\) and \(\frac{-2}{3}\) is greater?

**Solution:**

Clearly \(\frac{3}{5}\) is a positive rational number and \(\frac{-2}{3}\) is a negative rational number. We know that every positive rational number is greater than every negative rational number.

Therefore, \(\frac{3}{5}\) > \(\frac{-2}{3}\).

**2.**
Which of the numbers \(\frac{3}{-4}\) and \(\frac{-5}{6}\) is greater?

**Solution:
**

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

One number = \(\frac{3}{-4}\) = \(\frac{3 × (-1)}{(-4) × (-1)}\) = \(\frac{-3}{4}\).

The other number = \(\frac{-5}{6}\).

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

Therefore, \(\frac{-3}{4}\) = \(\frac{(-3) × 3}{4 × 3}\) = \(\frac{-9}{12}\) and \(\frac{-5}{6}\) = \(\frac{(-5) × 2}{6 × 2}\) = \(\frac{-10}{12}\)

Clearly, \(\frac{-9}{12}\) > \(\frac{-10}{12}\)

Hence, \(\frac{3}{-4}\) > \(\frac{-5}{6}\).

**3.** Which of the two rational numbers \(\frac{5}{7}\) and \(\frac{3}{5}\) is greater?

**Solution:**

Clearly, denominators o f the given rational numbers are positive. The denominators are 7 and 5. The LCM of 7 and 5 is 35. So, we first express each rational number with 35 as common denominator.

Therefore, \(\frac{5}{7}\) = \(\frac{5 × 7}{7 × 7}\) = \(\frac{25}{49}\) and \(\frac{3}{5}\) = \(\frac{3 × 7}{5 × 7}\) = \(\frac{21}{35}\)

Now, we compare the numerators of these rational numbers.

Therefore, 25 > 21

⇒ \(\frac{25}{49}\) > \(\frac{21}{35}\) ⇒ \(\frac{5}{7}\) > \(\frac{3}{5}\).

**4.** Write of the two rational numbers \(\frac{-4}{9}\)
and \(\frac{5}{-12}\) is greater?

**Solution:**

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

Clearly, denominator of \(\frac{-4}{9}\) is positive. The denominator of \(\frac{5}{-12}\) is negative.

So, we express it with positive denominator as follows:

\(\frac{5}{-12}\) = \(\frac{5 × (-1)}{(-12) × (-1)}\) = \(\frac{-5}{12}\), [Multiplying the numerator and denominator by -1]

Now, LCM of denominators 9 and 12 is 36.

We write the rational numbers so that they have a common denominator 36 as follows:

\(\frac{-4}{9}\) = \(\frac{(-4) × 4}{9 × 4}\) = \(\frac{-16}{36}\) and, \(\frac{-5}{12}\) = \(\frac{(-5) × 3}{12 × 3}\) = \(\frac{-15}{36}\)

Therefore, -15 > -16 ⇒ \(\frac{-15}{36}\) > \(\frac{-16}{36}\) ⇒ \(\frac{-5}{12}\) > \(\frac{-4}{9}\) ⇒ \(\frac{5}{-12}\) > \(\frac{-4}{9}\).

● **Rational Numbers**

Introduction of Rational Numbers

Is Every Rational Number a Natural Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational Number

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

**8th Grade Math Practice****From Comparison of Rational Numbers to HOME PAGE**

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● **Rational Numbers - Worksheets**

Worksheet on Equivalent Rational Numbers

Worksheet on Lowest form of a Rational Number

Worksheet on Standard form of a Rational Number

Worksheet on Equality of Rational Numbers

Worksheet on Comparison of Rational Numbers

Worksheet on Representation of Rational Number on a Number Line

Worksheet on Adding Rational Numbers

Worksheet on Properties of Addition of Rational Numbers

Worksheet on Subtracting Rational Numbers

Worksheet on Addition and
Subtraction of Rational Number

Worksheet on Rational Expressions Involving Sum and Difference

Worksheet on Multiplication of Rational Number

Worksheet on Properties of Multiplication of Rational Numbers

Worksheet on Division of Rational Numbers

Worksheet on Properties of Division of Rational Numbers

Worksheet on Finding Rational Numbers between Two Rational Numbers

Worksheet on Word Problems on Rational Numbers

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