Question

How do you convert $0.\overline {15} $ (15 being repeated) to a fraction?

Answer 1

Hint: First, assign the number to the variable. Then, multiply it by 100 to move one set of the repeating number to the left side of the decimal. Now, subtract the original number from the multiplied number. After that, divide the difference between the numbers with the coefficient of the variable. Thus, the fraction obtained will be the desired result.

Complete step-by-step solution:
Given: - The recurring decimal is $0.\overline {15} $.
Let the repeating number be X. Then,
$ \Rightarrow X = 0.\overline {15} $............…… (1)
Now, multiply both sides by 100 to get one set of the repeating numbers to the left side of the decimal.
$ \Rightarrow 100X = 15.\overline {15} $.............…… (2)
Now, subtract equation (1) from the equation (2) to remove the recurring number,
$ \Rightarrow 99X = 15$
Divide both sides of the equation by 99, we get,
$ \Rightarrow \dfrac{{99X}}{{99}} = \dfrac{{15}}{{99}}$
Cancel out the common factor from the left side of the equation,
$\therefore X = \dfrac{5}{{33}}$

Hence, the value of $0.\overline {15} $ in the form of a simple fraction is $\dfrac{5}{{33}}$.

Note: A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result, are irrational numbers. $\pi $ is a non-terminating, non-repeating decimal. The number e (Euler’s Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527.
Fun Fact:- The ancient Greek mathematician Pythagoras believed that all numbers were rational, but one of his students Hippasus proved (using geometry, it is thought) that you could not write the square root of 2 as a fraction, and so it was irrational. But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!