How do you write 0.05 repeating as a fraction
The correct Answer and Explanation is:
To convert the repeating decimal 0.055‾0.05\overline{5}0.055 (which is 0.055555…0.055555…0.055555…) into a fraction, follow these steps:
- Set Up the Equation: Let x=0.055‾x = 0.05\overline{5}x=0.055.
- Eliminate the Repeating Part: Multiply both sides by 10 to shift the decimal point:10x=0.55‾10x = 0.5\overline{5}10x=0.55
- Align the Repeating Decimals: Now, multiply the original equation by 100 to shift two decimal places:100x=5.55‾100x = 5.5\overline{5}100x=5.55
- Subtract the Equations: Now, subtract the first equation from the second:100x−10x=5.55‾−0.55‾100x – 10x = 5.5\overline{5} – 0.5\overline{5}100x−10x=5.55−0.55Simplifying this gives:90x=590x = 590x=5
- Solve for xxx: Divide both sides by 90:x=590x = \frac{5}{90}x=905
- Simplify the Fraction: The fraction 590\frac{5}{90}905 can be simplified. The greatest common divisor (GCD) of 5 and 90 is 5:x=5÷590÷5=118x = \frac{5 \div 5}{90 \div 5} = \frac{1}{18}x=90÷55÷5=181
Thus, 0.055‾0.05\overline{5}0.055 as a fraction is 118\frac{1}{18}181.
Explanation
Repeating decimals can be converted into fractions by setting up equations that eliminate the repeating part through multiplication. The key steps involve creating equations that clearly outline the relationship between the decimal and its fractional equivalent.
In this case, we started by defining the repeating decimal as xxx. By multiplying by 10 and then by 100, we aligned the decimal places so that the repeating parts could be eliminated through subtraction. This subtraction is crucial as it leads to a simple equation involving only xxx.
After isolating xxx, we divided to solve for the fraction, and then simplified it to its lowest terms. Understanding the relationship between decimals and fractions is essential in mathematics, especially for operations that require precise values. Converting 0.055‾0.05\overline{5}0.055 into 118\frac{1}{18}181 allows for easier manipulation in various mathematical contexts, such as addition, subtraction, and more complex calculations involving rational numbers.