Question

Find ten rational numbers between \[\dfrac{-2}{5}\] and \[\dfrac{1}{2}\] .
a) \[\dfrac{-7}{20},\text{ }\dfrac{-11}{20},\text{ }\dfrac{-5}{20},\text{ }\dfrac{-4}{20},\text{ }\dfrac{-3}{20},\text{ }\dfrac{-2}{20},\text{ }\dfrac{-1}{20},\text{ }0,\text{ }\dfrac{1}{20}\text{ }and\text{ }\dfrac{2}{20}.\]
b) \[\dfrac{-7}{20},\text{ }\dfrac{-6}{20},\text{ }\dfrac{-5}{20},\text{ }\dfrac{-4}{20},\text{ }\dfrac{-3}{20},\text{ }\dfrac{-2}{20},\text{ }\dfrac{-1}{20},\text{ }0,\text{ }\dfrac{1}{20}\text{ }and\text{ }\dfrac{2}{20}.\]
c) \[\dfrac{-7}{20},\text{ }\dfrac{-6}{20},\text{ }\dfrac{-5}{20},\text{ }\dfrac{-4}{20},\text{ }\dfrac{-3}{20},\text{ }\dfrac{-2}{20},\text{ }\dfrac{-1}{20},\text{ }0,\text{ }\dfrac{1}{20}\text{ }and\text{ }\dfrac{13}{20}.\]
d) \[\dfrac{-7}{20},\text{ }\dfrac{-6}{20},\text{ }\dfrac{-5}{20},\text{ }\dfrac{-4}{20},\text{ }\dfrac{-3}{20},\text{ }\dfrac{-2}{20},\text{ }\dfrac{-1}{20},\text{ }0,\text{ }\dfrac{15}{20}\text{ }and\text{ }\dfrac{2}{20}.\]

Answer 1

Hint:Firstly, we will have to make the denominators of the two fractions the same by calculating the L.C.M. of their denominators, i.e. by calculating the L.C.M. of 5 and 2.Then, after calculating their L.C.M., we will make their denominators the same by multiplying the denominators with a suitable number so that the product so obtained is the L.C.M. of the denominators.Similarity we take the multiple of L.CM and proceed the same by multiplying the denominator and numerator then we get required rational numbers.

Complete step-by-step answer:
Before solving this question, let us know about Rational Numbers and Irrational Numbers.
RATIONAL NUMBERS: Rational numbers are the numbers that are represented in the form of \[\dfrac{p}{q}\] where q is not equal to zero (q ≠ 0). Any fraction with non-zero denominators is a rational number. Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms.
IRRATIONAL NUMBERS: Irrational numbers are the numbers that cannot be represented as a simple fraction.
Now, to solve this question, we will find ten rational numbers between the two fractions mentioned in the question.
We will find them by the following method:-
Firstly, we will have to make the denominators of the two fractions the same by calculating the L.C.M. of their denominators, i.e. by calculating the L.C.M. of 5 and 2.
Then, after calculating their L.C.M., we will make their denominators the same by multiplying the denominators with a suitable number so that the product so obtained is the L.C.M. of the denominators.
Remember that when we are multiplying any number by the denominator, we will have to multiply the same number with the numerator also, otherwise, the answer will come out to be wrong.

Let us now find ten rational numbers between \[\dfrac{-2}{5}\] and \[\dfrac{1}{2}\] by the method explained in the hint provided above.
First of all, we will make the denominators of \[\dfrac{-2}{5}\] and \[\dfrac{1}{2}\] same by calculating the L.C.M. of their denominators.
L.C.M. of 5 and 2 = 10
\[\dfrac{-2\times 2}{5\times 2}=\dfrac{-4}{10}\]
\[\dfrac{1\times 5}{5\times 2}=\dfrac{5}{10}\]
We can see \[\dfrac{-4}{10}\] and \[\dfrac{5}{10}\] do not have 10 rational numbers between them. Therefore, we will take a multiple of the L.C.M. of 5 and 2 and make that the denominator of the two rational numbers.
Let us take 20 as their denominators.
\[\dfrac{-4\times 2}{10\times 2}=\dfrac{-8}{20}\]
\[\dfrac{5\times 2}{10\times 2}=\dfrac{10}{20}\]
So, 10 rational numbers those are between \[\dfrac{-8}{20}\] and \[\dfrac{10}{20}\] are:-
\[\dfrac{-7}{20},\text{ }\dfrac{-6}{20},\text{ }\dfrac{-5}{20},\text{ }\dfrac{-4}{20},\text{ }\dfrac{-3}{20},\text{ }\dfrac{-2}{20},\text{ }\dfrac{-1}{20},\text{ }0,\text{ }\dfrac{1}{20}\text{ }and\text{ }\dfrac{2}{20}.\]
Therefore, the answer of this question is (b)
\[\dfrac{-7}{20},\text{ }\dfrac{-6}{20},\text{ }\dfrac{-5}{20},\text{ }\dfrac{-4}{20},\text{ }\dfrac{-3}{20},\text{ }\dfrac{-2}{20},\text{ }\dfrac{-1}{20},\text{ }0,\text{ }\dfrac{1}{20}\text{ }and\text{ }\dfrac{2}{20}.\]

Note:-Remember that when we are multiplying any number by the denominator, we will have to multiply the same number with the numerator also, otherwise, the answer will come out to be wrong.We can find infinity of rational numbers between given two rational numbers.