Rational Numbers in Ascending Order
We will learn how to arrange the rational numbers in ascending order.
General method to arrange from smallest to largest rational numbers (increasing):
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 smaller numerator is smaller.
Solved examples on rational numbers in ascending order:
1. Arrange the rational numbers \(\frac{-7}{10}\), \(\frac{5}{-8}\) and \(\frac{2}{-3}\) in ascending order:
Solution:
We first write the given rational numbers so that their denominators are positive.
We have,
\(\frac{5}{-8}\) = \(\frac{5 × (-1)}{(-8) × (-1)}\) = \(\frac{-5}{8}\) and \(\frac{2}{-3}\) = \(\frac{2 × (-1)}{(-3) × (-1)}\) = \(\frac{-2}{3}\)
Thus, the given rational numbers with positive denominators are
\(\frac{-7}{10}\), \(\frac{-5}{8}\), \(\frac{-2}{3}\)
Now, LCM of the denominators 10, 8 and 3 is 2 × 2 × 2 × 3 × 5 = 120
We now write the numerators so that they have a common denominator 120 as follows:
\(\frac{-7}{10}\) = \(\frac{(-7) × 12}{10 × 12}\) = \(\frac{-84}{120}\),
\(\frac{-5}{8}\) = \(\frac{(-5) × 15}{8 × 15}\) = \(\frac{-75}{120}\) and
\(\frac{-2}{3}\) = \(\frac{(-2) × 40}{3 × 40}\) = \(\frac{-80}{120}\).
Comparing the numerators of these numbers, we get,
- 84 < -80 < -75
Therefore, \(\frac{-84}{120}\) < \(\frac{-80}{120}\) < \(\frac{-75}{120}\) ⇒ \(\frac{-7}{10}\) < \(\frac{-2}{3}\) < \(\frac{-5}{8}\) ⇒ \(\frac{-7}{10}\) < \(\frac{2}{-3}\) < \(\frac{5}{-8}\)
Hence, the given numbers when arranged in ascending order are:
\(\frac{-7}{10}\), \(\frac{2}{-3}\), \(\frac{5}{-8}\)
2. Arrange the rational numbers \(\frac{5}{8}\), \(\frac{5}{-6}\), \(\frac{7}{-4}\) and \(\frac{3}{5}\) in ascending order.
Solution:
First we write each one of the given rational numbers with positive denominator.
Clearly, denominators of \(\frac{5}{8}\) and \(\frac{3}{5}\) are positive.
The denominators of \(\frac{5}{-6}\) and \(\frac{7}{-4}\) are negative.
So, we express \(\frac{5}{-6}\) and \(\frac{7}{-4}\) with positive denominator as follows:
\(\frac{5}{-6}\) = \(\frac{5 × (-1)}{(-6) × (-1)}\) = \(\frac{-5}{6}\) and \(\frac{7}{-4}\) = \(\frac{7 × (-1)}{(-4) × (-1)}\) = \(\frac{-7}{4}\)
Thus, the given rational numbers with positive denominators are
\(\frac{5}{8}\), \(\frac{-5}{6}\), \(\frac{-7}{4}\) and \(\frac{3}{5}\)
Now, LCM of the denominators 8, 6, 4 and 5 is 2 × 2 × 2 × 3 × 5 = 120
Now we convert each of the rational numbers to their equivalent rational number with common denominator 120 as follows:
\(\frac{5}{8}\) = \(\frac{5 × 15}{8 × 15}\), [Multiplying the numerator and denominator by 120 ÷ 8 = 15]
⇒ \(\frac{5}{8}\) = \(\frac{75}{120}\)
\(\frac{-5}{6}\) = \(\frac{(-5) × 20}{6 × 20}\), [Multiplying the numerator and denominator by 120 ÷ 6 = 20]
⇒ \(\frac{-5}{6}\) = \(\frac{-100}{120}\)
\(\frac{-7}{4}\) = \(\frac{(-7) × 30}{4 × 30}\), [Multiplying the numerator and denominator by 120 ÷ 4 = 30]
⇒ \(\frac{-7}{4}\) = \(\frac{-210}{120}\) and
\(\frac{3}{5}\) = \(\frac{3 × 24}{5 × 24}\), [Multiplying the numerator and denominator by 120 ÷ 5 = 24]
⇒ \(\frac{3}{5}\) = \(\frac{72}{120}\)
Comparing the numerators of these numbers, we get,
-210 < -100 < 72 < 75
Therefore, \(\frac{-210}{120}\) < \(\frac{-100}{120}\) < \(\frac{72}{120}\) < \(\frac{75}{120}\) ⇒ \(\frac{-7}{4}\) < \(\frac{-5}{6}\) < \(\frac{3}{5}\) < 5/8 ⇒ \(\frac{7}{-4}\) < \(\frac{5}{-6}\) < \(\frac{3}{5}\) < \(\frac{5}{8}\)
Hence, the given numbers when arranged in ascending order are:
\(\frac{7}{-4}\), \(\frac{5}{-6}\), \(\frac{3}{5}\), \(\frac{5}{8}\).
● 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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