Convert 0.8181818181 ... to a rational number. Let x be the rational number you're looking for.
x= 0.8181818181 ...
Since there are two digits in the repeating cycle, multiply both sides of the equation by 10^2=100. 100x=81.81818181 ...
To eliminate the repeating decimal, we can subtract x from 100x:
100x - x = 81.81818181 ... - 0.81818181 ...
Simplifying both sides:
99x = 81
Now we can solve for x by dividing both sides by 99:
x = 81/99
We can simplify this fraction by dividing the numerator and denominator by their greatest common divisor, which is 9:
x = (81/9) / (99/9)
x = 9/11
Therefore, the rational number equivalent of 0.8181818181 ... is 9/11.
Next, subtract the original equation from the new equation to eliminate the repeating part:
100x - x = 81.81818181 ... - 0.8181818181 ...
This simplifies to:
99x = 81
To find the value of x, divide both sides of the equation by 99:
x = 81/99
Simplifying the fraction:
x = 9/11
Therefore, the rational number equivalent to 0.8181818181 ... is 9/11.