Equivalent Rational Numbers

We will learn how to find the equivalent rational numbers by multiplication and division.

Equivalent rational numbers by multiplication:

If $\frac{a}{b}$ is a rational number and m is a non-zero integer then $\frac{a × m}{b × m}$ is a rational number equivalent to $\frac{a}{b}$.

For example, rational numbers $\frac{12}{15}$, $\frac{20}{25}$, $\frac{-28}{-35}$, $\frac{-48}{-60}$ are equivalent to the rational number $\frac{4}{5}$.

We know that if we multiply the numerator and denominator of a fraction by the same positive integer, the value of the fraction does not change.

For example, the fractions $\frac{3}{7}$ and $\frac{21}{49}$ are equal because the numerator and the denominator of $\frac{21}{49}$ can be obtained by multiplying each of the numerator and denominator of $\frac{3}{7}$ by 7.

Also, $\frac{-3}{4}$ = $\frac{-3 × (-1)}{4 × (-1)}$ = $\frac{3}{-4}$, $\frac{-3}{4}$ = $\frac{-3 × 2}{4 × 2}$ = $\frac{-6}{8}$, $\frac{-3}{4}$ = $\frac{-3 × (-2)}{4 × (-2)}$ = $\frac{6}{-8}$ and so on …….

Therefore, $\frac{-3}{4}$ = $\frac{-3 × (-1)}{4 × (-1)}$ = $\frac{-3 × 2}{4 × 2}$ = $\frac{(-3) × (-2)}{4 × (-2)}$ and so on …….

Note: If the denominator of a rational number is a negative integer, then by using the above property, we can make it positive by multiplying its numerator and denominator by -1.

For example, $\frac{5}{-7}$ = $\frac{5 × (-1)}{(-7) × (-1)}$ = $\frac{-5}{7}$

Equivalent rational numbers by division:

If $\frac{a}{b}$ is a rational number and m is a common divisor of a and b, then $\frac{a ÷ m}{b ÷ m}$ is a rational number equivalent to $\frac{a}{b}$.

For example, rational numbers $\frac{-48}{-60}$, $\frac{-28}{-35}$, $\frac{20}{25}$, $\frac{12}{15}$ are equivalent to the rational number $\frac{4}{5}$.

We know that if we divide the numerator and denominator of a fraction by a common divisor, then the value of the fraction does not change.

For example, $\frac{48}{64}$ = $\frac{48 ÷ 16}{64 ÷ 16}$ = $\frac{3}{4}$

Similarly, we have
$\frac{-75}{100}$ = $\frac{(-75) ÷ 5}{100 ÷ 5}$ = $\frac{-15}{20}$ = $\frac{(-15) ÷ 5}{20 ÷ 5}$ = $\frac{-3}{4}$, and $\frac{42}{-56}$ = $\frac{42 ÷ 2}{(-56 ) ÷ 2}$ = $\frac{21}{-28}$ = $\frac{21 ÷ (-7)}{(-28) ÷ (-7)}$ = $\frac{-3}{4}$

Solved examples:

1. Find the two rational numbers equivalent to $\frac{3}{7}$.

Solution:

$\frac{3}{7}$ = $\frac{3 × 4}{7 × 4}$ = $\frac{12}{28}$ and

$\frac{3}{7}$ = $\frac{3 × 11}{7 × 11}$ = $\frac{33}{77}$

Therefore, the two rational numbers equivalent to $\frac{3}{7}$ are $\frac{12}{28}$ and $\frac{33}{77}$

2. Determine the smallest equivalent rational number of $\frac{210}{462}$.

Solution:

$\frac{210}{462}$ = $\frac{210 ÷ 2}{462 ÷ 2}$ = $\frac{105}{231}$ = $\frac{105 ÷ 3}{231 ÷ 3}$ = $\frac{35}{77}$ = $\frac{35 ÷ 7}{77 ÷ 7}$ = $\frac{5}{11}$

Therefore, the least equivalent rational number of $\frac{210}{462}$ is $\frac{5}{11}$

3. Write each of the following rational numbers with positive denominator:

$\frac{3}{-7}$, $\frac{11}{-28}$, $\frac{-19}{-13}$

Solution:

In order to express a rational number with positive denominator, we multiply its numerator and denominator by -1.

Therefore,

$\frac{3}{-7}$ = $\frac{3 × (-1)}{(-7) × (-1)}$ = $\frac{-3}{7}$,

$\frac{11}{-28}$ = $\frac{11 × (-1)}{(-28) × (-1)}$ = $\frac{-11}{28}$,

and $\frac{-19}{-13}$ = $\frac{(-19) × (-1)}{(-13) × (-1)}$ = $\frac{19}{13}$

4. Express $\frac{-3}{7}$ as a rational number with numerator:

(i) -15; (ii) 21

Solution: