What is the fraction of 0.72 repeating
The Correct Answer and Explanation is :
To convert the repeating decimal (0.72\overline{72}) (where “72” repeats indefinitely) into a fraction, we can follow these steps:
- Set up the equation: Let (x = 0.72\overline{72}). [
x = 0.72727272\ldots
] - Multiply by a power of 10: Since the repeating part has two digits, we multiply both sides of the equation by 100 to move the decimal point two places to the right. [
100x = 72.727272\ldots
] - Set up another equation: Now we have two equations:
- (x = 0.727272\ldots)
- (100x = 72.727272\ldots)
- Subtract the first equation from the second: [
100x – x = 72.727272\ldots – 0.727272\ldots
]
[
99x = 72
] - Solve for (x): [
x = \frac{72}{99}
] - Simplify the fraction: We can simplify (\frac{72}{99}) by finding the greatest common divisor (GCD) of 72 and 99. The GCD is 9. [
\frac{72 \div 9}{99 \div 9} = \frac{8}{11}
]
Therefore, the repeating decimal (0.72\overline{72}) as a fraction is (\frac{8}{11}).
Explanation:
Repeating decimals can be tricky, but converting them to fractions is a systematic process. The method of using variables to represent the repeating decimal allows us to establish a relationship between the decimal and its fractional form. By multiplying by a power of ten corresponding to the length of the repeating part, we create a new equation that aligns with the original equation, making it possible to eliminate the repeating portion through subtraction.
In this case, (0.72\overline{72}) signifies that “72” continues indefinitely, prompting us to adjust our equations accordingly. The resulting equation (99x = 72) highlights the relationship between the decimal and the integer values, leading us to the simple fraction form. Finally, simplifying the fraction by its GCD not only provides the correct answer but also demonstrates the importance of reduction in making fractions easier to work with and understand.