Question

The number 0.121212.. in the form p/q is equal to
A. \[\dfrac{4}{{11}}\]
B. \[\dfrac{2}{{11}}\]
C. \[\dfrac{{\text{4}}}{{{\text{33}}}}\]
D. \[\dfrac{2}{{{\text{33}}}}\]

Answer 1

Hint: We need to convert a non terminating and recurring decimal number into the form p/q. First, we assume the decimal to be a variable. Then we multiply the variable with a suitable power of 10. Then we subtract the variable from its multiple to cancel all the decimals. We get the required fraction when we solve for the variable.

Complete step by step Answer:

We need to convert the decimal number to its rational form.
Let us assume \[{\text{x = 0}}{\text{.121212}}...\] …1
Then, \[{\text{100x = 12}}{\text{.1212}}...\] …2
Equation (2) – equation (1) gives,
\[{\text{100x - x = 12}}{\text{.1212}}...{\text{ - 0}}{\text{.121212}}...\]
\[ \Rightarrow {\text{99x = 12}}\]
After solving for x, we get,
\[{\text{x = }}\dfrac{{{\text{12}}}}{{{\text{99}}}}{\text{ = }}\dfrac{{\text{4}}}{{{\text{33}}}}\]
Therefore, the required rational number is \[\dfrac{{\text{4}}}{{{\text{33}}}}\]
So, the correct answer is option C.

Note: The number in the form p/q where p and q are integers and p and have no common factor other than 1 are called rational numbers. In this problem, we multiplied the variable with 100 as 2 decimal places are recurring. If only one decimal place is recurring, we will be multiplying the variable with 10. We must simplify the ratio we get such that the p and q we got must have only 1 as its common factor. The numbers which cannot be written in the form p/q are known as irrational numbers. They are non-terminating and non-recurring decimal numbers. Rational numbers and irrational numbers together form the real number system. The real numbers are the numbers that can be represented on a number line.