Question

Convert given decimal \[1.27\] in \[\dfrac{p}{q}\] form.

Answer 1

Hint: Here, we need to convert \[1.27\] in \[\dfrac{p}{q}\] form. We will solve this question by forming two equations and subtracting them. We will assume the value of to form the first equation. Then, we will multiply the first equation by 100 to form the second equation. Now we will subtract the two equations to remove the decimal part. After this we will be left with only whole numbers in the equation. Then, we will divide the numbers to get the \[\dfrac{p}{q}\] form.

Complete step-by-step answer:
We will first form two equations and will subtract them to remove the numbers in the decimal places.
Let \[x = 1.27272727\].
Now, we can observe that 27 repeats in the decimal expansion.
Since 27 is two digits, we will multiply the expression by 100 to find the value of \[100x\].
Multiplying both sides of the expression by 100, we get
\[100x = 127.272727\]
We will extend the decimal places such that the number of decimal places in the value of \[x\] and \[100x\] is the same.
The number of decimal places in \[x = 1.27272727\] is 8.
Thus, extending the decimal places in \[100x = 127.272727\] from 6 to 8, we get
\[100x = 127.27272727\]
Next, we need to subtract the two equations such that numbers in the decimal places are 0.
Subtracting the equation \[x = 1.27272727\] from the equation \[100x = 127.27272727\], we get
\[100x - x = 127.27272727 - 1.27272727\]
Subtracting the terms in the equation, we get
\[\begin{array}{l} \Rightarrow 99x = 126.00000000\\ \Rightarrow 99x = 126\end{array}\]
The decimal places are removed from the expression.
Now, we will divide both sides by 99 to get the value of \[x\] in the \[\dfrac{p}{q}\] form.
Dividing both sides by 99, we get
\[\begin{array}{l} \Rightarrow \dfrac{{99x}}{{99}} = \dfrac{{126}}{{99}}\\ \Rightarrow x = \dfrac{{126}}{{99}}\end{array}\]
\[\therefore\] The \[\dfrac{p}{q}\] form of the number \[1.27\] is \[\dfrac{{126}}{{99}}\].

Note: We should remember this method to convert a decimal into the \[\dfrac{p}{q}\] form. We might make a mistake in this question by converting the decimal \[1.27\] into the fraction \[\dfrac{{127}}{{100}}\] 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.