In this chapter we will learn about the concept of equivalent rational number & the method to identify them.
To understand the chapter, you should first have basic knowledge of rational number. Click the red link to understand the basics.
What are equivalent rational numbers ?
The rational numbers are said to be equivalent when on simplification they have same values.
For example;
Consider the two rational number \mathtt{\ \frac{1}{2} \ \&\ \frac{3}{6}} .
Simplify both the fraction to decimal value.
\mathtt{\frac{1}{2}} ⟹ 0.5
\mathtt{\frac{3}{6}} ⟹ 0.5
You can see that both the above numbers have same decimal value. Hence they are equivalent rational numbers.
Conclusion
Rational numbers are equivalent when they have same values.
Producing equivalent rational numbers
From a given rational number, you can generate multiple equivalent numbers by;
(a) Multiplying same number in numerator & denominator
(b) Dividing same number in numerator & denominator
Let us understand both the process with the help of example.
Generating equivalent rational number by multiplication
Let \mathtt{\frac{a}{b}} is the given rational number.
Multiply numerator & denominator by integer p.
\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{p\times a}{p\times b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{p.a}{p.b}}
Here the newly generated rational number \mathtt{\frac{p.a}{p.b}} and \mathtt{\frac{a}{b}} are the equivalent fraction.
I hope you understood the concept. Let us see some examples for better clarity.
Example 01
Find two equivalent rational number of \mathtt{\frac{7}{5}}
Solution
(i) First equivalent rational number.
Multiply numerator & denominator by 3.
\mathtt{\Longrightarrow \frac{7}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 3}{5\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{21}{15}}
We got the fraction 21 / 15.
(ii) Second equivalent rational number
Multiply numerator and denominator by 4
\mathtt{\Longrightarrow \frac{7}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 4}{5\times 4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{28}{20}}
We got the fraction 28 / 20.
Hence, \mathtt{\frac{7}{5} ,\ \frac{21}{15} \ \&\frac{28}{20}} are all equivalent rational number since on simplification they will produce same value.
Example 02
Fine two equivalent rational number of \mathtt{\frac{-3}{10}}
Solution
(a) First equivalent rational number
Multiply numerator and denominator by -3
\mathtt{\Longrightarrow \frac{-3}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-3\times -3}{10\times -3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9}{-30}}\\\ \\ \mathtt{\Longrightarrow \frac{-9}{30}}
Hence, -9/30 is the first equivalent rational number.
(b) Second equivalent rational number
Multiply numerator and denominator by 2.
\mathtt{\Longrightarrow \frac{-3}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-3\times 2}{10\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-6}{20}}
Hence, -6/20 is the second equivalent fraction.
Hence, \mathtt{\frac{-3}{10} ,\ \frac{-9}{30} \ \&\frac{-6}{20}} are all equivalent rational number since they have same values.
Generating equivalent rational number by division
Let \mathtt{\frac{a}{b}} be the given rational number.
If we divide both numerator and denominator by integer m, we will get its equivalent number.
\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\div m}{b\div m}}
Let us solve some examples related to the concept.
Example 01
Find two equivalent rational number of \mathtt{\frac{6}{54}}
Solution
(a) First equivalent number.
Divide numerator & denominator by 3
\mathtt{\Longrightarrow \frac{6}{54}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\div 3}{54\div 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{18}}
Hence, 2/18 is the first equivalent rational number
(b) Second equivalent rational number
Divide numerator and denominator by 2
\mathtt{\Longrightarrow \frac{6}{54}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\div 2}{54\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{27}}
So, 3/27 is the second equivalent number.
Hence, \mathtt{\frac{6}{54} ,\ \frac{2}{18} \ \&\ \frac{\ 3}{27}} are the equivalent rational numbers since all of them have same values.
Example 02
Find equivalent rational number of \mathtt{\frac{28}{14}}
Solution
(a) First equivalent fraction
Divide numerator & denominator by 7
\mathtt{\Longrightarrow \frac{28}{14}}\\\ \\ \mathtt{\Longrightarrow \ \frac{28\div 7}{14\div 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{2}}
Hence, we got one equivalent rational number 4 / 2.
(b) Second equivalent number
Divide numerator and denominator by 2.
\mathtt{\Longrightarrow \frac{28}{14}}\\\ \\ \mathtt{\Longrightarrow \ \frac{28\div 2}{14\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{14}{7}}
So, 14 / 7 is the second equivalent rational number.
Hence, \mathtt{\frac{28}{14} ,\ \frac{4}{2} \ \&\ \frac{\ 14}{7}} are the three equivalent rational number since they have same values.