What is 0.63 repeating expressed as a fraction in simplest form?
The Correct Answer and Explanation is:
To express 0.63 repeating as a fraction, follow these steps:
Step 1: Let x=0.63‾x = 0.\overline{63}x=0.63
This means the decimal is repeating every two digits (63). So, we can set up the equation: x=0.6363636363…x = 0.6363636363\ldotsx=0.6363636363…
Step 2: Eliminate the repeating decimal
Multiply both sides of the equation by 100. This shifts the decimal point two places to the right: 100x=63.6363636363…100x = 63.6363636363\ldots100x=63.6363636363…
Now we have two equations:
- x=0.6363636363…x = 0.6363636363\ldotsx=0.6363636363…
- 100x=63.6363636363…100x = 63.6363636363\ldots100x=63.6363636363…
Step 3: Subtract the first equation from the second
Now subtract x=0.6363636363…x = 0.6363636363\ldotsx=0.6363636363… from 100x=63.6363636363…100x = 63.6363636363\ldots100x=63.6363636363…: 100x−x=63.6363636363…−0.6363636363…100x – x = 63.6363636363\ldots – 0.6363636363\ldots100x−x=63.6363636363…−0.6363636363…
Simplifying both sides: 99x=6399x = 6399x=63
Step 4: Solve for xxx
Now, divide both sides by 99 to isolate xxx: x=6399x = \frac{63}{99}x=9963
Step 5: Simplify the fraction
We can simplify 6399\frac{63}{99}9963 by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 63 and 99 is 9. So, divide both by 9: 63÷999÷9=711\frac{63 \div 9}{99 \div 9} = \frac{7}{11}99÷963÷9=117
Thus, the decimal 0.63‾0.\overline{63}0.63 is equivalent to the fraction 711\frac{7}{11}117.
Final Answer:
The repeating decimal 0.63‾0.\overline{63}0.63 is expressed as the fraction 711\frac{7}{11}117.
