To convert the repeating decimal 0.63 (where 63 repeats indefinitely) into a fraction, we can use a simple algebraic method.
Let x = 0.636363…
Since the decimal repeats every two digits, we can multiply both sides of the equation by 100 (which moves the decimal point two places to the right):
100x = 63.636363…
Next, we can set up a system of equations:
1. x = 0.636363…
2. 100x = 63.636363…
Now, we can subtract the first equation from the second:
100x – x = 63.636363… – 0.636363…
This simplifies to:
99x = 63
Now, divide both sides by 99:
x = 63/99
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 9:
x = (63 ÷ 9) / (99 ÷ 9) = 7/11
Thus, the decimal 0.636363… can be represented as the fraction 7/11.