# Ex.1.2 Q4 Rational Numbers Solution - NCERT Maths Class 8

## Question

Find ten rational numbers between \(\begin{align}\frac{{ - 2}}{5}\end{align}\) 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:**

\(\begin{align}\frac{{ - 2}}{5} = \frac{{ - 2 \times 2}}{{5 \times 2}} = \frac{{ - 4}}{{10}}\end{align}\) [multiplying both numerator and denominator by \(2\)]

\(\begin{align}\frac{1}{2} = \frac{{1 \times 5}}{{2 \times 5}} = \frac{5}{{10}}\end{align}\) [multiplying both numerator and denominator by \(5\)]

[Multiplying both numerators and denominators by the same number]

Now, there are \(9\) rational numbers between \(\begin{align}\frac{{ - 4}}{{10}}\end{align}\) and \(\begin{align}\frac{5}{{10}}\end{align},\) but we need \(10\) numbers. So, we should again multiply both numerator and denominator by \(2\) in the two rational numbers \(\begin{align}\frac{{ - 4}}{{10}}\end{align}\) and \(\begin{align}\frac{5}{{10}}\end{align}\)

\(\begin{align}\frac{{ - 4 \times 2}}{{10 \times 2}} = \frac{{ - 8}}{{20}}\end{align}\)and \(\begin{align}\frac{5}{{10}} \times \frac{2}{2} = \frac{{10}}{{20}}\end{align}\)

The ten rational numbers between \(\begin{align}\frac{{ - 2}}{5}\end{align}\) and \(\begin{align}\frac{1}{2}\end{align}\) which can be taken as.

\[\begin{align}&\frac{- 7}{20},\;\;\frac{- 6}{20},\;\;\frac{- 5}{20}, \;\;\frac{- 4}{20},\;\;\frac{- 3}{20},\;\; \\ & \frac{- 2}{20},\;\;\frac{1}{20},\;\;0,\;\;\frac{1}{20} \;\; \text{and} \;\; \frac{2}{20}\end{align}\]

[There are many more such rational numbers.]