Question

How do you convert 0.243(43 repeating) to a fraction ?

Answer 1

Hint: The number given in the question is a repeating decimal number. We can convert the repeating decimal number to fraction by taking the number x and then doing some arithmetic operation to cancel out all the repeating terms. Then we can convert it into a fraction.

Complete step-by-step answer:
The given number is 0.24343… where 43 is repeating
Let’s take the number equal to x
So we can write x = 0.24343… eq1
We can multiply 1000 both sides and subtract one from another such that the repeating term will cancel out.
Multiplying 1000 with LHS and RHS we get
$\Rightarrow 100x=243.4343....$ eq2
Now if we subtract eq1 from eq2 all repeating numbers will be cancelled out
$\Rightarrow 999x=243$
Now we can see that there are no repeating terms present. We can divide both LHS and RHS by 999
$\Rightarrow x={\displaystyle \frac{243}{999}}$
243 and 999 have common factor 27 so we can reduce the fraction
$\Rightarrow x={\displaystyle \frac{9}{37}}$
So the fraction form of 0.24343… is ${\displaystyle \frac{9}{37}}$

Note: Always remember that all infinitely long decimal numbers are not irrational , only
non repetitive infinitely long numbers are irrational numbers. We can convert all repetitive numbers to fractions. While converting to fractions, multiply the number with power of 10 such that when we subtract the number from the result all the repetitive terms get cancelled out.
When we write irrational numbers such as e we will not find specific parts repeating in the decimal form.