# Ex.1.3 Q6 Number System Solution - NCERT Maths Class 9

## Question

Look at the several examples of rational numbers in the form \(\begin{align}\frac{p}{q}\end{align}\) \(\begin{align}(q \ne 0)\end{align}\) where ** \(p\)** and

*are integers with no common factors other than*

**\(q\)****\(1\)**and having terminating decimal representation (expansions). Can you guess what property

*must satisfy?*

**\(q\)**

## Text Solution

**Steps:**

We shall look at some examples of rational numbers in the form of \(\begin{align}\frac{{\rm{p}}}{{\rm{q}}}\end{align}\) \(\begin{align}(q \ne 0)\end{align}\) where decimal representations are terminating.

\(\begin{align}\frac{2}{5} &= 0.4 \qquad \qquad\frac{3}{{100}} = 0.03\\\\\frac{{27}}{{16}} & = 1.6875 \qquad \quad \frac{{33}}{{50}} = 0.66\end{align}\)

We observed that the denominators of above rational numbers are in the form of \(\begin{align}{2^a} \times {5^b}\end{align}\) Where, * \(a\) *and

*are whole numbers.*

**\(b\)**Hence if * \(q\)* is in the form \(\begin{align}{2^a} \times {5^b}\end{align}\) then \(\begin{align}\frac{{\rm{p}}}{{\rm{q}}}\end{align}\) is a terminating decimal.