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
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
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