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