Question

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

(iv) $\frac{171}{800}$

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 $\frac{171}{800}$.

In the given rational number, the denominator has a factor as the product of powers of $2$ and $5$, that is $800=2^5\times 5^2$. So, it is a terminating decimal.

Step 2: Expressing in decimal form:

$$\begin{gathered} \frac{171}{800}=\frac{171}{2^5\times 5^2} \\ \Rightarrow \frac{171}{800}=\frac{171}{2^3\times 2^2\times 5^2} \\ \Rightarrow \frac{171}{800}=\frac{171}{8\times {(2\times 5)}^2} \\ \Rightarrow \frac{171}{800}=\frac{171}{8\times 100} \\ \Rightarrow \frac{171}{800}=\frac{171}{8}\times \frac{1}{100} \\ \Rightarrow \frac{171}{800}=\frac{21.375}{100} \\ \Rightarrow \frac{171}{800}=0.21375 \end{gathered}$$

Final answer : Hence, the rational number $\frac{171}{800}$ is a terminating decimal and its decimal form is $0.21375$