Question

Find ten rational numbers between \[\dfrac{{ - 2}}{5}\] and \[\dfrac{1}{2}\]
A) \[\dfrac{{ - 7}}{{20}},\dfrac{{11}}{{20}},\dfrac{{ - 5}}{{20}},\dfrac{{ - 4}}{{20}},\dfrac{{ - 3}}{{20}},\dfrac{{ - 2}}{{20}},\dfrac{{ - 1}}{{20}},0,\dfrac{1}{{20}}\] and \[\dfrac{2}{{20}}\]
B) \[\dfrac{{ - 7}}{{20}},\dfrac{{ - 6}}{{20}},\dfrac{{ - 5}}{{20}},\dfrac{{ - 4}}{{20}},\dfrac{{ - 3}}{{20}},\dfrac{{ - 2}}{{20}},\dfrac{{ - 1}}{{20}},0,\dfrac{1}{{20}}\] and \[\dfrac{2}{{20}}\]
C) \[\dfrac{{ - 7}}{{20}},\dfrac{{ - 6}}{{20}},\dfrac{{ - 5}}{{20}},\dfrac{{ - 4}}{{20}},\dfrac{{ - 3}}{{20}},\dfrac{{ - 2}}{{20}},\dfrac{{ - 1}}{{20}},0,\dfrac{1}{{20}}\] and \[\dfrac{{13}}{{20}}\]
D) \[\dfrac{{ - 7}}{{20}},\dfrac{{ - 6}}{{20}},\dfrac{{ - 5}}{{20}},\dfrac{{ - 4}}{{20}},\dfrac{{ - 3}}{{20}},\dfrac{{ - 2}}{{20}},\dfrac{{ - 1}}{{20}},0,\dfrac{{15}}{{20}}\] and \[\dfrac{2}{{20}}\]

Answer 1

Hint: Here, we will make the denominator of both the given fraction same by multiplying and dividing them with a suitable number. Then we will write all the rational numbers that come between the two numbers. We will compare the numbers with the given options to get the required answer.

Complete step-by-step answer:
The two rational numbers given are \[\dfrac{{ - 2}}{5}\] and \[\dfrac{1}{2}\]
As we have the option of the question with denominator 20 so we will convert the denominator of both the term to 20.
First, we will multiply and divide the first number by 4.
\[\dfrac{{ - 2}}{5} = - \dfrac{2}{5} \times \dfrac{4}{4} = - \dfrac{8}{{20}}\]……………….\[\left( 1 \right)\]
Now, we will multiply and divide the second number with 10. Therefore, we get
\[\dfrac{1}{2} = \dfrac{1}{2} \times \dfrac{{10}}{{10}} = \dfrac{{10}}{{20}}\]………………….…..\[\left( 2 \right)\]
From equation \[\left( 1 \right)\] and \[\left( 2 \right)\] we get the two numbers as \[ - \dfrac{8}{{20}}\] and \[\dfrac{{10}}{{20}}\].
So the rational numbers between the above two numbers are:
\[ - \dfrac{7}{{20}}, - \dfrac{6}{{20}}, - \dfrac{5}{{20}}, - \dfrac{4}{{20}}, - \dfrac{3}{{20}}, - \dfrac{2}{{20}}, - \dfrac{1}{{20}},0,\dfrac{1}{{20}},\dfrac{2}{{20}}\]

Hence, option (B) is correct.

Note:
Rational numbers are those that can be expressed in the form of \[\dfrac{p}{q}\] where \[q \ne 0\]. If we find the decimal expansion of a rational number it either terminates after a finite number or the digit starts to repeat them over and over again. If it is not a rational number that means it is an irrational number. The set of all rational numbers together with addition and multiplication operations forms a field. The set of a rational number is countable but the set of irrational numbers is uncountable and as real number is a union of rational and irrational numbers so it is also uncountable.