Question

Express the following decimal in the form of $\dfrac{p}{q}$:
                     \[0.4\overline{7}\]

Answer 1

Hint: In the above type of question we will assume a variable to the given decimal and we will multiply it by such a factor so that after multiplying decimal place digits become equal to that before multiplying.

Complete step-by-step answer:

Let \[x=0.47777................(1)\]
Multiplying both sides of equation (1) by 10, we get
\[10x=4.77777................(2)\]

Again multiplying it by 10 , we get
\[100x=47.777777...............(3)\]

Now subtracting (2) from (3), we get
\[\begin{align}
  & 90x=43 \\
 & x=\dfrac{43}{90} \\
\end{align}\]

Therefore, the given decimal can be expressed in the form of $\dfrac{p}{q}$ as \[\dfrac{43}{90}\] .

Note: If the decimal number is a non terminating non recurring decimal number , you can not convert it into $\dfrac{p}{q}$ form, since it is an irrational number.

If the decimal is a terminating decimal, we will first find the number of decimal places in the given decimal number. Take 1 annexed with as many zeroes as the number of decimal places given in the decimal. Multiply and divide the given decimal with this number. Reduce the above rational number to simplest form.

Non terminating decimals are those which keep on continuing after the decimal point. They don't come to an end or if they do it after a long interval.
The rational number for which the long division terminates after a finite number of steps is known as the terminating decimal.