Question

Write the following rational numbers in their decimal form and also state which are terminating.
(a)$\dfrac{3}{8}$
(b)$\dfrac{229}{400}$
(c)$4\dfrac{1}{5}$
(d)$\dfrac{2}{11}$
(e)$\dfrac{8}{125}$

Answer 1

Hint: Find the decimal representation of the given fractions by dividing the numerator by denominator of the fraction. Any decimal representation will be of terminating type, if there are fixed digits after the decimal and termed as non-terminating if there is no end of digits after the decimal.

Complete step-by-step answer:
As we know for converting any fraction to decimal form we have to divide the given fractions. And if numbers after decimals are fixed, then they are said to be terminating, otherwise non-terminating.
(a)$\dfrac{3}{8}$
let us divide 3 by 8 as following
$8\overset{0.375}{\overline{\left){\begin{align}
  & 30 \\
 & \underline{24} \\
 & 60 \\
 & \underline{56} \\
 & 40 \\
 & \underline{40} \\
 & \underline{00} \\
\end{align}}\right.}}$
Hence, the decimal representation of $\dfrac{3}{8}\to 0.375$ and it is terminating after the decimal.

(b)$\dfrac{229}{400}$
Let us divide 229 by 400 by following way:
$400\overset{0.5725}{\overline{\left){\begin{align}
  & 2290 \\
 & \underline{2000} \\
 & 2900 \\
 & \underline{2800} \\
 & 1000 \\
 & \underline{0800} \\
 & 2000 \\
 & \underline{2000} \\
 & \underline{\text{ 0 }} \\
\end{align}}\right.}}$
Hence, the decimal representation of $\dfrac{229}{400}\to 0.5725$and it is terminating.

(c)$4\dfrac{1}{5}$
Let us convert this mixed fraction as simple fraction:
$4\dfrac{1}{5}=\dfrac{21}{5}$
$5\overset{4.2}{\overline{\left){\begin{align}
  & 21 \\
 & \underline{20} \\
 & 10 \\
 & \underline{10} \\
 & \underline{00} \\
\end{align}}\right.}}$
Hence, the decimal representation of $\dfrac{21}{5}\to 4.2$ and so, it is terminating.

(d)$\dfrac{2}{11}$
Let us divide 2 by 11 as:
$11\overset{0.1818181}{\overline{\left){\begin{align}
  & 20 \\
 & \underline{11} \\
 & 90 \\
 & \underline{88} \\
 & 20 \\
 & \underline{11} \\
 & 90 \\
 & \underline{88} \\
 & 20 \\
 & \underline{11} \\
 & 9.0 \\
 & \underline{88} \\
 & 20 \\
 & \underline{11} \\
 & \underline{\text{ 9}} \\
\end{align}}\right.}}$
Hence, the decimal representation of $\dfrac{20}{11}\to 0.18181818.......,0.\overline{18}$ .
And as the digits after the decimal is not fixed,
So, $\dfrac{2}{11}$ is non-terminating.

(e)$\dfrac{8}{125}$
Let us convert the given fraction to decimal by dividing 8 by 125 as:
$125\overset{0.064}{\overline{\left){\begin{align}
  & 800 \\
 & \underline{750} \\
 & 500 \\
 & \underline{500} \\
 & \underline{\text{ 0 }} \\
\end{align}}\right.}}$
Hence, the decimal representation of $\dfrac{8}{125}\to 0.064$ and so, it is terminating.

Note: Be clear with the terminology of terminating and non-terminating.
Here, we need to know proper division rules for decimals. As one may get confused with the division of $\dfrac{8}{125}$ in a way that we have to put one extra zero after decimal. So, be careful with all these fractions.