Question

Express \[0.\overline {75} \] as a rational number?

Answer 1

Hint: The given question needs to be solved by assuming a variable “x” and then equating the given number with the variable and then solving further, after removing decimal and solving for the variable.

Complete step-by-step solution:
The given question is to write a decimal number into rational number, let first assume a variable “x” which is a rational number, on solving we get:
\[ \Rightarrow x = 0.757575... \to eq1\]
Now multiplying by 100 on both the sides of equation we get:
\[ \Rightarrow 100x = 75.7575... \to eq2\]
Now subtracting both the equations we get:
\[
   \Rightarrow 100x - x = 75.7575... - 0.7575... \\
   \Rightarrow 99x = 75 \\
   \Rightarrow x = \dfrac{{75}}{{99}} = \dfrac{{25}}{{33}} \\
 \]
Here we find the rational number, of the given decimal number.
Additional Information: To solve the given question here we first assume a variable and then multiply by hundred because the decimal place on which bar was given was two, hence we multiply by 100 and then solve further.

Note: To solve this kind of question we need to solve for the decimal places, hence to solve for the hundred decimal places we make two equations, and then solve both equations by subtracting both the equations and get the answer.