Question

Write the value of $2.\overline {134} $ in the form of a simple fraction.

Answer 1

Hint: First, assign the number to the variable. Then, multiply it by 1000 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 answer:
Given: - The recurring decimal is $2.\overline {134} $.
Let the repeating number be X. Then,
$X = 2.\overline {134} $ …… (1)
Now, multiply both sides by 1000 to get one set of the repeating numbers to the left side of the decimal.
$1000X = 2134.\overline {134} $ …… (2)
Now, subtract equation (1) from the equation (2) to remove the recurring number,
$999X = 2132$
Divide both sides of the equation by 999, we get,
$\dfrac{{999X}}{{999}} = \dfrac{{2132}}{{999}}$
Cancel out the common factor from the left side of the equation,
$X = \dfrac{{2132}}{{999}}$

Hence, the value of $2.\overline {134} $ in the form of a simple fraction is $\dfrac{{2132}}{{999}}$.

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 to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527.