Final answer:
The fractional equivalent of the repeating decimal 0.5 is 5/9.
Step-by-step explanation:
Here's a detailed explanation of how to find the fractional equivalent of the repeating decimal 0.5:
1. Let x be the unknown fraction:
We don't know what the fraction is yet, so we use the variable "x" to represent it. This allows us to work with the unknown value symbolically.
2. Set up two equations:
a. Multiply x by 10: We write the decimal 0.5 as "0.5555..." to emphasize the repeating pattern. Multiplying x by 10 shifts the decimal one place to the right:
10x = 5.5555...
b. Define the original decimal: We write down the original decimal value as another equation:
x = 0.5555...
3. Subtract the second equation from the first:
By subtracting equation 2 from equation 1, we eliminate the repeating digits and isolate x:
10x - x = 5.5555... - 0.5555...
9x = 5
4. Solve for x:
Divide both sides by 9 to isolate x and find the solution:
x = 5 / 9
Therefore, the fractional equivalent of the repeating decimal 0.5 is 5/9.