Question

Which of the following pairs represent the same rational number? (This question has multiple correct options)
A.\[\dfrac{{ - 7}}{{21}}\] and \[\dfrac{3}{9}\]
B.\[\dfrac{{ - 16}}{{20}}\] and \[\dfrac{{20}}{{ - 25}}\]
C.\[\dfrac{{ - 2}}{{ - 3}}\] and \[\dfrac{2}{3}\]
D.\[\dfrac{{ - 3}}{5}\] and \[\dfrac{{ - 12}}{{20}}\]
E.\[\dfrac{8}{{ - 5}}\] and \[\dfrac{{ - 24}}{{15}}\]

Answer 1

Hint First we will consider all the options individually and try to simplify both the numerator and denominator to compare the values. If both the values are the same then the pair represents the same rational number.

Complete step-by-step answer:
We know that the rational numbers are those numbers which can be written in the form of \[\dfrac{p}{q}\], where \[p\] is numerator, \[q\] is denominator, \[q \ne 0\] and both are integers.
First, we will consider option A.
Dividing the numerator and denominator by 7 in number \[\dfrac{{ - 7}}{{21}}\], we get
\[\dfrac{{ - 7 \div 7}}{{21 \div 7}} = \dfrac{{ - 1}}{3}\]
Dividing the numerator and denominator by 3 in number \[\dfrac{3}{9}\], we get
\[\dfrac{{3 \div 3}}{{9 \div 3}} = \dfrac{1}{3}\]
Since the numbers \[\dfrac{{ - 1}}{3}\] and \[\dfrac{1}{3}\] have same denominators and different numerators, on comparing both the values comes out to be different.
Considering option B,
Dividing the numerator and denominator by 4 in number \[\dfrac{{ - 16}}{{20}}\], we get
\[\dfrac{{ - 16 \div 4}}{{20 \div 4}} = \dfrac{{ - 4}}{5}\]
Dividing the numerator and denominator by \[ - 5\] in number \[\dfrac{{20}}{{ - 25}}\], we get
\[\dfrac{{20 \div \left( { - 5} \right)}}{{ - 25 \div \left( { - 5} \right)}} = \dfrac{{ - 4}}{5}\]
Since the numbers \[\dfrac{{ - 4}}{5}\] and \[\dfrac{{ - 4}}{5}\] have same denominators and numerators, on comparing both the values comes out to be same.
Thus, option B is correct.
Considering option C,
Dividing the numerator and denominator by \[ - 1\] in number \[\dfrac{{ - 2}}{{ - 3}}\], we get
\[\dfrac{{ - 2 \div - 1}}{{ - 3 \div - 1}} = \dfrac{2}{3}\]
Since the numbers \[\dfrac{2}{3}\] and \[\dfrac{2}{3}\] have same denominators and same numerators, comparing both the values comes out to be the same.
Thus, option C is also correct.
We will now consider option D.
Dividing the numerator and denominator by 4 in number \[\dfrac{{ - 12}}{{20}}\], we get

\[\dfrac{{ - 12 \div 4}}{{20 \div 4}} = \dfrac{{ - 3}}{5}\]
Since the numbers \[\dfrac{{ - 3}}{5}\] and \[\dfrac{{ - 3}}{5}\] have same denominators and numerators, on comparing both the values comes out to be same.
Thus, option D is also correct.
Considering option E,
Dividing the numerator and denominator by \[ - 3\] in number \[\dfrac{{ - 24}}{{15}}\], we get
\[\dfrac{{ - 24 \div \left( { - 3} \right)}}{{15 \div \left( { - 3} \right)}} = \dfrac{8}{{ - 5}}\]
Since the numbers \[\dfrac{8}{{ - 5}}\] and \[\dfrac{8}{{ - 5}}\] have same denominators and numerators, on comparing both the values comes out to be same.
Thus, option E is also correct.
Hence, options B, C, D and E are correct.

Note In solving these types of questions, students must be careful while dealing with the sign of the given rational number. Students should also know that the sign could be transferred from denominator to numerator and numerator to denominator by multiplying both the numerator and denominator with \[ - 1\]. The key step is the simplification of the rational numbers to the same denominator and numerators for comparison of numbers.