Express the repeating decimal as the ratio of two integers. 0.\overline{5} = 0.5555… The ratio of two integers is (Type an integer or a simplified fraction
The Correct Answer and Explanation is:
To express the repeating decimal 0.5‾=0.5555…0.\overline{5} = 0.5555\ldots0.5=0.5555… as a ratio of two integers, we follow a standard algebraic method:
Step-by-step solution:
Letx=0.5‾x = 0.\overline{5}x=0.5
This means:x=0.5555…x = 0.5555\ldotsx=0.5555…
Now multiply both sides of the equation by 10 to move the decimal point one place to the right:10x=5.5555…10x = 5.5555\ldots10x=5.5555…
Now subtract the original equation (x=0.5555…x = 0.5555\ldotsx=0.5555…) from this new equation:10x−x=5.5555…−0.5555…10x – x = 5.5555\ldots – 0.5555\ldots10x−x=5.5555…−0.5555…9x=59x = 59x=5
Now solve for xxx:x=59x = \frac{5}{9}x=95
Final Answer:
59\boxed{\frac{5}{9}}95
Textbook-style Explanation
Repeating decimals are decimal numbers in which a digit or group of digits repeat infinitely. A common problem in algebra is converting a repeating decimal into a rational number, which means expressing it as a ratio of two integers (a fraction in simplest form). In this case, we are given the repeating decimal 0.5‾0.\overline{5}0.5, which is shorthand notation for the infinite decimal 0.5555…0.5555\ldots0.5555…, where the digit 5 repeats forever.
To convert this repeating decimal to a fraction, we use a simple algebraic trick. First, we assign a variable to the repeating decimal. Let x=0.5555…x = 0.5555\ldotsx=0.5555…. To eliminate the repeating part, we multiply both sides of the equation by 10. This moves the decimal one place to the right and gives us 10x=5.5555…10x = 5.5555\ldots10x=5.5555….
Now, notice that both xxx and 10x10x10x have the same repeating decimal part 0.5555…0.5555\ldots0.5555…. If we subtract the original equation from the new one, the repeating parts cancel out:10x−x=5.5555…−0.5555…⇒9x=510x – x = 5.5555\ldots – 0.5555\ldots \Rightarrow 9x = 510x−x=5.5555…−0.5555…⇒9x=5
Solving this simple equation for xxx, we divide both sides by 9:x=59x = \frac{5}{9}x=95
Thus, 0.5‾=590.\overline{5} = \frac{5}{9}0.5=95. This shows that the repeating decimal is a rational number. All repeating decimals can be expressed in this way using algebra, and this method provides a reliable way to convert them to exact fractions.
