How do you change an infinite repeating decimal to a fraction?
To change an infinite repeating decimal to a fraction, follow these steps:
Step 1: Let x be the repeating decimal.
Step 2: Multiply both sides of the equation by a power of 10 that corresponds to the number of repeating digits. This will eliminate the decimal point and shift all repeating digits to the left of the decimal point.
Step 3: Subtract the original equation from the equation obtained in Step 2 to eliminate the repeating part. This will result in a new equation with a whole number and a decimal part, but with no repeating decimals.
Step 4: Solve the equation obtained in Step 3 for x.
Step 5: Write the original repeating decimal as a fraction by setting up a proportion with x as the numerator and the corresponding number of nines in the denominator.
For example, let's convert the repeating decimal 0.7272... to a fraction:
Step 1: Let x = 0.7272...
Step 2: Multiply both sides by 100 to eliminate the repeating decimal:
100x = 72.7272...
Step 3: Subtract the original equation from the equation obtained in Step 2:
100x - x = 72.7272... - 0.7272...
99x = 72
Step 4: Solve for x:
x = 72/99
Step 5: Write the original repeating decimal as a fraction:
0.7272... = 72/99
Simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor, we get:
0.7272... = 8/11
Hence, the infinite repeating decimal 0.7272... can be expressed as the fraction 8/11.