Convert the decimal $0.\overline{5}$ to a fraction.
Answer (2)
Let x = 0. 5 .
Multiply by 10: 10 x = 5. 5 .
Subtract the equations: 9 x = 5 .
Solve for x : x = 9 5 .
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0. 5 into a fraction. This means the digit 5 repeats infinitely after the decimal point.
Setting up the Equation Let x = 0. 5 . This means x = 0.5555...
Multiplying by 10 To eliminate the repeating decimal, we multiply both sides of the equation by 10: 10 x = 5.5555...
Subtracting the Equations Now, we subtract the original equation from the new equation: 10 x − x = 5.5555... − 0.5555... This simplifies to: 9 x = 5
Solving for x Finally, we solve for x by dividing both sides by 9: x = 9 5 Thus, the repeating decimal 0. 5 is equal to the fraction 9 5 .
Examples
Repeating decimals can be converted to fractions, which is useful in various real-life scenarios. For example, if you are calculating probabilities and one of the probabilities is expressed as a repeating decimal, you would need to convert it to a fraction to perform accurate calculations. Another example is in financial calculations, where interest rates or growth rates might be expressed as repeating decimals, and converting them to fractions allows for precise computations.
The decimal 0. 5 can be converted to a fraction by letting x = 0. 5 , multiplying by 10, and solving the resulting equation. This results in the fraction 9 5 . Thus, 0. 5 is equal to 9 5 .
;
