To convert the repeating decimal 0.35555555555 to a fraction, you can use the following method:
Let's denote the repeating part by "overlining" it. So 0.35555555555 can be denoted as 0.3(5), where (5) means that the digit 5 is repeating indefinitely.
Now, let's assign the value x to this number:
x = 0.3(5)
To eliminate the repeating part, we need to find a number that, when multiplied by x, shifts the decimal point in such a way that it aligns with the repeating part. Since the 5 repeats after one digit, we multiply by 10:
10x = 3.555555555...
Now let's subtract the original x from this 10x. This subtraction will align the repeating 5s and allow us to eliminate them:
10x - x = 3.555555555... - 0.35555555555
9x = 3.2
Now we simply solve for x:
x = 3.2 / 9
To convert this mixed number into an improper fraction, multiply the whole number (3) by the denominator (9) and add the numerator (2):
x = (3 * 9 + 2) / 9
x = (27 + 2) / 9
x = 29 / 9
Therefore, the fraction representation of the repeating decimal 0.35555555555 is 29/9. This is the fraction in its simplest form, as 29 and 9 have no common factors other than 1.