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

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

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

Addition of Rational Numbers

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

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers

8th Grade Math Practice 

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