Question

How to convert 6.1252525… in fraction.

Answer 1

Hint: Now to solve the given equation we will first write x = 6.1252525.. Now we will multiply the given equation by 10 to form a new equation. Now again we will multiply the obtained equation by 100 and subtract the previous equation with this obtained equation. Now we divide the whole equation by coefficient of x and hence we will get x in fraction form.

Complete step-by-step solution:
Now in general to convert decimals into fraction we first write the x = given decimal. Now we multiply the equation with ${{10}^{n}}$ such that recurring starts after the decimal point. Now we will count the number of recurring digits. Let us say there are p recurring digits then we multiply the hole equation by ${{10}^{p}}$ . Now subtract the two obtained equations such that we get an equation in the form of $ax=b$ . Now divide the equation of the form ax = b by a. Hence we get $x=\dfrac{b}{a}$ . Hence we have converted x into decimal form.
Now first consider the given number 6.1252525…
Let us say x = 6.1252525…
Multiplying the whole equation by 10 we get the recurring numbers after decimal point,
$\Rightarrow 10x=61.252525....................\left( 1 \right)$
Now we will multiplying the above equation by 100 since there are 2 recurring digits,
$\Rightarrow 1000x=6125.252525............\left( 2 \right)$
Now subtracting equation (1) from equation (2) we get,
$\Rightarrow 1000x-10x=6125.252525....-61.252525...$
$\Rightarrow 990x=6064$
Now dividing the whole equation by 90 we get,
$\Rightarrow x=\dfrac{6064}{990}$
But we know that x is just 6.1252525…
Hence we can write $6.1252525...=\dfrac{6064}{990}$. Hence 6.1252525… in fraction is $\dfrac{6064}{990} =6\dfrac{22}{249}$.

Note: Note that every rational number can be expressed in the form of $\dfrac{p}{q}$ where p and q are integers. If the number of digits after decimal are finite then just divide and multiply the number by ${{10}^{n}}$ where n denotes number of digits after decimal. If the digits after decimal are recurring infinite then we use the above method. If the digits after decimal are infinite and non-recurring then the number is irrational and cannot be denoted as a fraction.