Question

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form:

(vi) $\frac{19}{3125}$

Answer 1

Solution:

Step 1: Showing that the given rational number is a terminating decimal:

A rational number having the factors in the denominator as a power of $2$ or $5$ or a product of powers of $2$ and $5$ is a terminating decimal.

Given rational number is $\frac{19}{3125}$.

In the given rational number, the denominator has a factor as the power of $5$ , that is, $3125=5^5$. So, it is a terminating decimal.

Step 2: Expressing in decimal form:

$$\begin{gathered} \frac{19}{3125}=\frac{19}{5^5} \\ \Rightarrow \frac{19}{3125}=\frac{10000}{3125}\times \frac{19}{10000}[\mathrm{Multiplying}\mathrm{and}\mathrm{dividing}\mathrm{by}10000] \\ \Rightarrow \frac{19}{3125}=3.2\times 0.0019 \\ \Rightarrow \frac{19}{3125}=0.00608 \end{gathered}$$

Final answer: Hence, the rational number $\frac{19}{3125}$is a terminating decimal and its decimal form is $0.00608$