# 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 \(\begin{align}\frac{1}{2}\end{align}\)

## 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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