Question

Express \[0.12\overline{3}\] in the form of \[\dfrac{p}{q},q\ne 0\] and p, q are integers.

Answer 1

Hint: In this problem, we have to find the fraction for the given decimal. We can first assume x to the given decimal. We can then multiply the both to 10 and 100, we will get two equations. We can then subtract the both equations to get the value of x, that is the exact fraction of the given decimal number.

Complete step by step answer:
We know that the given decimal number to be converted into its fraction is \[0.12\overline{3}\] , where 3 is repeated, so we can write as 0.12333….
We can assume the decimal number to x, we get
\[\Rightarrow x=0.123333...\] ….. (1)
Now we can multiply the number 10 on both sides were in right-hand side the decimal point moves one point to the right, we get
\[\Rightarrow 100x=12.33...\] ….. (2)
We can now subtract the equation (1) and (2), we get
\[\Rightarrow 99x=12.21\]
We can now divide the number 90 on both the sides, we get
\[\Rightarrow x=\dfrac{12.21}{99}\]
\[\Rightarrow x=\dfrac{1221}{9900}\]
Therefore, the fractional form of the decimal 0.123 (3 being repeated) is \[\dfrac{407}{3300}\].

Note: Students make mistakes while reading the question properly, here ‘3 is being repeated’ is an important point to be noted. We should also know that if we multiply a decimal number by 10 and 100, we should move the decimal point 1 and 2 points to the right respectively.