# Rational Numbers in Descending Order

We will learn how to arrange the rational numbers in descending order.

General method to arrange from largest to smallest rational numbers (decreasing):

Step 1: Express the given rational numbers with positive denominator.

Step 2: Take the least common multiple (L.C.M.) of these positive denominator.

Step 3: Express each rational number (obtained in step 1) with this least common multiple (LCM) as the common denominator.

Step 4: The number having the greater numerator is greater.

Solved examples on rational numbers in descending order:

1. Arrange the numbers $$\frac{-3}{5}$$, $$\frac{7}{-10}$$ and $$\frac{-5}{8}$$ in descending order.

Solution:

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

We have;

$$\frac{7}{-10}$$ = $$\frac{7 × (-1)}{(-10) × (-1)}$$ = $$\frac{-7}{10}$$.

Thus, the given number are $$\frac{-3}{5}$$, $$\frac{-7}{10}$$ and $$\frac{-5}{8}$$.

L.C.M. of 5, 10, 8 is 40.

Now, $$\frac{-3}{5}$$ = $$\frac{(-3) × 8}{5 × 8}$$ = $$\frac{-24}{40}$$;

$$\frac{-7}{10}$$ = $$\frac{(-7) × 4}{10 × 4}$$ = $$\frac{-28}{40}$$

and $$\frac{-5}{8}$$ = $$\frac{(-5) × 5}{8 × 5}$$
= $$\frac{-25}{40}$$

Clearly, $$\frac{-24}{40}$$ > $$\frac{-25}{40}$$ > $$\frac{-28}{40}$$

Thus, $$\frac{-3}{5}$$ > $$\frac{-5}{8}$$ > $$\frac{-7}{10}$$, i.e., $$\frac{-3}{5}$$ > $$\frac{-5}{8}$$ > $$\frac{7}{-10}$$

Hence, the given numbers when arranged in descending order are: $$\frac{-3}{5}$$, $$\frac{-5}{8}$$, $$\frac{7}{-10}$$.

2. Arrange the following rational numbers in descending order: $$\frac{4}{9}$$, $$\frac{-5}{6}$$, $$\frac{-7}{-12}$$, $$\frac{11}{-24}$$.

Solution:

First we express the given rational numbers in the form so that their denominators are positive.

We have,

$$\frac{-7}{-12}$$ = $$\frac{(-7) × (-1)}{(-12) × (-1)}$$, [Multiplying the numerator and denominator by -1]

$$\frac{-7}{-12}$$ = $$\frac{7}{12}$$

and $$\frac{11}{-24}$$ = $$\frac{11 × (-1)}{(-24) × (-1)}$$ = $$\frac{-11}{24}$$

Thus, given rational numbers are:

$$\frac{4}{9}$$, $$\frac{-5}{6}$$, $$\frac{7}{12}$$, $$\frac{-11}{24}$$

Now, we find the LCM of 9, 6, 12 and 24.

Required LCM = 2 × 2 × 2 × 3 × 3 = 72.

We now write the rational numbers so that they have a common denominator 72.

We have,

$$\frac{4}{9}$$ = $$\frac{4 × 8}{9 × 8}$$, [Multiplying the numerator and denominator by 72 ÷ 9 = 8]

$$\frac{4}{9}$$ = $$\frac{32}{72}$$

$$\frac{-5}{6}$$ = $$\frac{-5 × 12}{6 × 12}$$, [Multiplying the numerator and denominator by 72 ÷ 6 = 12]

$$\frac{-5}{6}$$ = $$\frac{-60}{72}$$

$$\frac{7}{12}$$ = $$\frac{7 × 6}{12 × 6}$$, [Multiplying the numerator and denominator by 72 ÷ 12 = 6]

$$\frac{7}{12}$$ = $$\frac{42}{72}$$

$$\frac{-11}{24}$$ = $$\frac{-11 × 3}{24 × 3}$$, [Multiplying the numerator and denominator by 72 ÷ 24 = 3]

$$\frac{-11}{24}$$ = $$\frac{-33}{72}$$

Arranging the numerators of these rational numbers in descending order, we have

42 > 32 > -33 > -60

⇒ $$\frac{42}{72}$$ > $$\frac{32}{72}$$ > $$\frac{-33}{72}$$ > $$\frac{-60}{72}$$ ⇒ $$\frac{-7}{-12}$$ > $$\frac{4}{9}$$ > $$\frac{11}{-24}$$ > $$\frac{-5}{6}$$

Hence, the given numbers when arranged in descending order are:

$$\frac{-7}{-12}$$, $$\frac{4}{9}$$, $$\frac{11}{-24}$$, $$\frac{-5}{6}$$.

Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

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

Representation of 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 Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

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

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

To Find Rational Numbers