Convert 0.27... to a rational number in simplest form.
To convert a repeating decimal to a rational number, we can multiply the decimal by a power of 10 to "shift" the repeating part to the left of the decimal point.
Let's use the variable x to represent the repeating decimal:
x = 0.27...
To shift the repeating part, we multiply both sides of the equation by 100 (since there are 2 digits in the repeating part):
100x = 27.27...
Next, we subtract the original equation from the shifted equation to eliminate the repeating part:
100x - x = 27.27... - 0.27...
Simplifying the equation:
99x = 27
Divide both sides by 99:
x = 27/99
To simplify further, we can divide both the numerator and denominator by their greatest common divisor (GCD), which is 9:
x = (27/9)/(99/9)
x = 3/11
Therefore, 0.27... as a rational number in simplest form is 3/11.