What is 0.58 repeating as a fraction?
The Correct Answer and Explanation is:
To express 0.58‾0.\overline{58}0.58 (which means 0.585858… repeating) as a fraction, we can follow a step-by-step process to convert the repeating decimal into a rational number. Here’s the full explanation:
Step 1: Let x=0.58‾x = 0.\overline{58}x=0.58
We define the repeating decimal as xxx:x=0.585858…x = 0.585858…x=0.585858…
The “58” is the repeating part, and we want to convert this repeating decimal into a fraction.
Step 2: Multiply both sides of the equation by 100
Since the decimal repeats every two digits, we multiply both sides of the equation by 100 to shift the decimal point two places to the right:100x=58.585858…100x = 58.585858…100x=58.585858…
Step 3: Subtract the original equation from the new equation
Now subtract the original equation x=0.585858…x = 0.585858…x=0.585858… from the new equation 100x=58.585858…100x = 58.585858…100x=58.585858…:100x−x=58.585858…−0.585858…100x – x = 58.585858… – 0.585858…100x−x=58.585858…−0.585858…
This simplifies to:99x=5899x = 5899x=58
Step 4: Solve for xxx
To find xxx, divide both sides of the equation by 99:x=5899x = \frac{58}{99}x=9958
Thus, the repeating decimal 0.58‾0.\overline{58}0.58 is equal to the fraction 5899\frac{58}{99}9958.
Step 5: Check if the fraction can be simplified
We need to check if the fraction 5899\frac{58}{99}9958 can be simplified. To do this, we find the greatest common divisor (GCD) of 58 and 99. The prime factorization of 58 is 2×292 \times 292×29, and the prime factorization of 99 is 32×113^2 \times 1132×11. Since there are no common factors between 58 and 99, the GCD is 1, meaning the fraction is already in its simplest form.
Final Answer:
The repeating decimal 0.58‾0.\overline{58}0.58 is equal to the fraction 5899\frac{58}{99}9958.