# Ex.1.2 Q5 Rational Numbers Solution - NCERT Maths Class 8

## Question

Find five rational numbers between,

(i) \begin{align}\;\frac{2}{3}\end{align} and \begin{align}\frac{4}{5}\end{align}

(ii)\begin{align}\;\frac{{ - 3}}{2}\end{align} and \begin{align}\frac{5}{3}\end{align}

(iii) \begin{align}\frac{1}{4}\end{align} and \begin{align}\frac{1}{2}\end{align}

Video Solution
Rational Numbers
Ex 1.2 | Question 5

## Text Solution

What is known?

Rational numbers.

What is unknown?

The rational numbers between given rational numbers.

Reasoning:

We can find infinitely many rational numbers between any two given rational numbers by taking the mean of the two rational numbers. Another method: We can make the denominator same for the two given rational numbers.

Steps:

(i)   \begin{align}\frac{2}{3}\end{align} and \begin{align}\frac{4}{5}\end{align}

\begin{align}\frac{2}{3} = \frac{{2 \times 20}}{{3 \times 20}} = \frac{{40}}{{60}}\end{align} [multiplying both numerator and denominator by \begin{align}20\end{align}]

\begin{align}\frac{4}{5} = \frac{{4 \times 12}}{{5 \times 12}} = \frac{{48}}{{60}}\end{align} [multiplying both numerator and denominator by \begin{align}12\end{align}]

The five rational numbers between \begin{align}\frac{2}{3}\end{align} and \begin{align}\frac{4}{5}\end{align} that can be taken are:

\begin{align}\frac{{41}}{{60}},\;\;\frac{{42}}{{60}},\;\;\frac{{43}}{{60}},\;\;\frac{{44}}{{60}},\;\;\frac{{45}}{{60}}\end{align}

(ii)   \begin{align}\frac{{ - 3}}{2}\end{align} and \begin{align}\frac{5}{3}\end{align}

\begin{align}\frac{{ - 3}}{2} = \frac{{ - 3 \times 3}}{{2 \times 3}} = \frac{{ - 9}}{6}\end{align} [multiplying both numerator and denominator by \begin{align}3\end{align}]

\begin{align}\frac{5}{3} = \frac{{5 \times 2}}{{2 \times 3}} = \frac{{10}}{6}\end{align} [multiplying both numerator denominator by \begin{align}2\end{align}]

\begin{align}\therefore\end{align} The five rational numbers between \begin{align}\frac{{ - 3}}{2}\end{align} and \begin{align}\frac{5}{3}\end{align} that can be taken are:

\begin{align} - \frac{8}{6},\;\; - \frac{7}{6},\;\;-1,\;\;\frac{5}{6},\;\;\frac{4}{6}\end{align}

[There can be more such rational numbers]

(iii)   \begin{align}\frac{1}{4}\end{align} and \begin{align}\frac{1}{2}\end{align}

\begin{align}\frac{1}{4} = \frac{{1 \times 8}}{{4 \times 8}} = \frac{8}{{32}}\end{align} [multiplying both numerator and denominator by \begin{align}8\end{align}]

\begin{align}\frac{1}{2} = \frac{{1 \times 16}}{{2 \times 16}} = \frac{{16}}{{32}}\end{align} [multiplying both numerator and denominator by \begin{align}16\end{align}]

Thus, five rational numbers between \begin{align}\frac{1}{2}\end{align} and \begin{align}\frac{1}{4}\end{align} that can be taken are:

\begin{align}\frac{9}{{32}},\;\;\frac{{10}}{{32}},\;\;\frac{{11}}{{32}},\;\;\frac{{12}}{{32}},\;\;\frac{{13}}{{32}}\end{align}

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