What is 0.27 repeated converted to a rational number in the simplest form
To convert 0.27 repeated to a rational number in simplest form, we can represent it as an infinite geometric series. Let x = 0.27 repeated.
x = 0.272727...
100x = 27.272727...
Now, subtract the two equations:
(100x - x) = (27.272727...) - (0.272727...)
99x = 27
To simplify, divide both sides by 99:
x = 27/99
The rational number in simplest form is 27/99, which can be further reduced by dividing both numerator and denominator by their greatest common divisor:
x = 3/11
Therefore, 0.27 repeated as a rational number in simplest form is 3/11.
To convert 0.27 with the repeating decimal 27 to a rational number in its simplest form, you need to follow these steps:
Step 1: Let x be the repeating decimal number 0.27.
x = 0.272727...
Step 2: Multiply both sides of the equation by 100 (for a two-digit repeating decimal) to remove the repeating part from the decimal.
100x = 27.272727...
Step 3: Subtract the original equation from the result in Step 2 to eliminate the repeating part.
100x - x = 27.272727... - 0.272727...
99x = 27
Step 4: Solve for x by dividing both sides of the equation by 99.
x = 27/99
Step 5: Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator, which is 9.
x = 3/11
Hence, 0.27 repeating as a rational number in its simplest form is 3/11.