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:  

(i) In order to -3 as a rational number with  numerator -15, we first find a number  which when multiplied by -3 gives -15.

Clearly, such number is (-15) ÷ (-3) = 5
Multiplying the numerator and denominator of $\frac{-3}{7}$ by 5, we have

$\frac{-3}{7}$ = $\frac{(-3) × 5}{7 × 5}$ = $\frac{-15}{35}$

Thus, the required rational number is $\frac{-15}{35}$.

(ii) In order to express $\frac{-3}{7}$ as a rational number with numerator 21, we first find a number which when multiplied with -3 gives 21. 

Clearly, such a number is 21 ÷ (-3) = -7 

Multiplying the numerator and denominator of $\frac{-3}{7}$ by (-7), we have

$\frac{-3}{7}$ = $\frac{(-3) × (-7)}{7 × (-7)}$ = $\frac{21}{-49}$

These are the above examples on equivalent rational numbers.

● 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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