About Rational Numbers with Number Line Representation

$\frac{1}{2},\frac{6}{7},\frac{9}{10},\frac{3}{10}$

$3.45=\frac{345}{100}$

$0.2=\frac{1}{5}$

$11.575=\frac{11575}{1000}$

$\frac{1}{9}=\mathrm{0.11111......}$

$\frac{1}{3}=\mathrm{0.3333......}$

$\frac{2}{3}=\mathrm{0.66666......}$

$\frac{3}{7},\frac{4}{5},\frac{6}{8}$

$\frac{-5}{7},\frac{-4}{3},\frac{-1}{2}$

Convert $\frac{12}{20}$ into its standard form
HCF of numerator 12 and denominator 20 is 4
Divide both 12 and 20 by 4
$\frac{12÷4}{20÷4}=\frac{3}{5}$

Convert $\frac{14}{-18}$ into its standard form
HCF of numerator 14 and denominator 18 is 2
Divide both 14 and - 18 by - 2
$\frac{14÷\left(-2\right)}{-18÷\left(-2\right)}=\frac{-7}{9}$

$\frac{4}{6}$ is equivalent to $\frac{2}{3}$
because, $\frac{2}{3}\times \frac{2}{2}=\frac{4}{6}$

Which is greater $\frac{2}{3}$ or $\frac{-4}{5}$ ?
Since, both rational numbers have opposite signs, so the positive rational will be greater than the negative rational number
∴ $\frac{2}{3}>\frac{-4}{5}$

Which is greater $\frac{2}{7}$ or $\frac{4}{5}$ ?
Since, both rational numbers have same signs, so find the LCM of denominators 7 and 5, which will be 35
Now, multiply $\frac{2}{7}$ and $\frac{4}{5}$ with the number so that denominators of both become equal to LCM 35.
$\frac{2}{7}\times \frac{5}{5}=\frac{10}{35}$
$\frac{4}{5}\times \frac{7}{7}=\frac{28}{35}$
Compare the numerators 28 and 35
28 > 10
∴ $\frac{28}{35}>\frac{10}{35}$
or $\frac{4}{5}>\frac{2}{7}$

Which is smaller $\frac{-3}{4}$ or $\frac{-5}{9}$?
Both rational numbers have same signs, so find the LCM of denominators 4 and 9, which will be 36
Now, multiply $\frac{-3}{4}$ and $\frac{-5}{9}$ with the number so that denominators of both become equal to LCM 36.
$\frac{-3}{4}\times \frac{9}{9}=\frac{-27}{36}$
$\frac{-5}{9}\times \frac{4}{4}=\frac{-20}{36}$
Compare the numerators - 27 and - 20
- 27 < - 20
∴ $\frac{-27}{36}<\frac{-20}{36}$
or $\frac{-3}{4}<\frac{-5}{9}$

Example 1: Represent $\frac{3}{7}$ on the number line.
Step 1
Draw a number line with positive and negative numbers on the right and left hand sides of 0 respectively.
Step 2
Here, denominator is 7, ∴ n = 7.
So, divide 0 and 1 on the number line into 7 equal points and mark each point as
$\frac{0}{7},\frac{1}{7},\frac{2}{7},\frac{3}{7}\frac{4}{7},\frac{5}{7},\frac{6}{7},\frac{7}{7}$
Length of each part between two adjacent points is $\frac{1}{7}$ .
Now, start from 0 and mark the point $\frac{3}{7}$ on the number line with A.
So, $\frac{3}{7}$ is represented on the number line with point A

Example 2: Represent $\frac{7}{4}$ on the number line.
Step 1
Draw the number line.
Step 2
Here, the denominator is 4, so n = 4.
Divide 0 and 1 into 4 equal points
Because the value of denominator is less than value of numerator, so do the same from 1 to 2, i.e. divide them into 4 equal parts also.
Length of each part between two adjacent points is $\frac{1}{4}$.
Mark the point $\frac{7}{4}$ on the number line with A.
So, $\frac{7}{4}$ is represented on the number line with point A.

Example 3: Represent $\frac{2}{7}$ on the number line.
To represent $\frac{2}{7}$ on the number line, we divide 0 to 1 into 7 equal parts.