Question

Convert \[1.473\] in \[\dfrac{p}{q}\] form, where $p \in Z$ and \[q \in N\].

Answer 1

Hint:
Here, we need to convert \[1.473\] in \[\dfrac{p}{q}\] form. We will assume the decimal number given to us to be \[x\]. We will then multiply the expression by 1000, and then subtract the two equations such that the numbers in the decimal places will be 0. Then, we will simplify the equation to find the value of \[x\], and hence, get the value of \[1.473\] in \[\dfrac{p}{q}\] form.

Complete step by step solution:
We will solve this question by forming two equations and subtracting them. The purpose of this is to remove the numbers in the decimal places.
Let \[x = 1.473473473\].
Now, we can observe that 473 repeats in the decimal expansion.
Since 473 is three digits, we will multiply the expression by 1000 to find the value of \[1000x\].
Multiplying both sides of the expression by 1000, we get
\[1000x = 1473.473473\]
We will extend the decimal places such that the number of decimal places in the value of \[x\] and \[1000x\] is the same.
The number of decimal places in \[x = 1.27272727\] is 9.
Thus, extending the decimal places in \[1000x = 1473.473473\] from 6 to 9, we get
\[1000x = 1473.473473473\]
Next, we need to subtract the two equations such that numbers in the decimal places are 0.
Subtracting the equation \[x = 1.473473473\] from the equation \[1000x = 1473.473473473\], we get
\[1000x - x = 1473.473473473 - 1.473473473\]
Subtracting the terms in the equation, we get
\[\begin{array}{l} \Rightarrow 999x = 1472.000000000\\ \Rightarrow 999x = 1472\end{array}\]
The decimal places are removed from the expression.
Now, we will divide both sides by 999 to get the value of \[x\] in the \[\dfrac{p}{q}\] form.
Dividing both sides by 999, we get
$ \Rightarrow \dfrac{999x}{999} = \dfrac{1472}{999} \\
\Rightarrow x= \dfrac{1472}{999}$

$\therefore$ The number \[1.473\] is converted into the \[\dfrac{p}{q}\] form, \[\dfrac{{1472}}{{999}}\].

Note:
You should remember this method to convert a decimal into the \[\dfrac{p}{q}\] form. A common mistake in this question is to convert the decimal \[1.473\] into the fraction \[\dfrac{{1473}}{{100}}\] and leave the answer at that. Whenever we need to convert a decimal to \[\dfrac{p}{q}\] form, it is usually a question involving a decimal with repeating expansion, usually denoted by a bar sign over the repeating digits.