What is 0.58 repeating as a fraction?
The Correct Answer and Explanation is:
To convert 0.5858‾0.58\overline{58}0.5858 (where 58 repeats) into a fraction, follow these steps:
Step 1: Let x=0.585858…x = 0.585858\ldotsx=0.585858…
Step 2: Multiply both sides of the equation by 100
Since the repeating block has two digits, we multiply by 100 to shift the decimal point two places to the right:100x=58.585858…100x = 58.585858\ldots100x=58.585858…
Step 3: Subtract the original x=0.585858…x = 0.585858\ldotsx=0.585858… from this new equation:
100x−x=58.585858…−0.585858…100x – x = 58.585858\ldots – 0.585858\ldots100x−x=58.585858…−0.585858…99x=5899x = 5899x=58
Step 4: Solve for xxx
x=5899x = \frac{58}{99}x=9958
Final Answer:
5899\boxed{\frac{58}{99}}9958
Explanation
Repeating decimals can be expressed as fractions using algebra. In this case, we are given a repeating decimal 0.585858…0.585858\ldots0.585858… where the digits “58” repeat endlessly. The goal is to express this value as a fraction of two integers.
We start by letting xxx represent the repeating decimal, so x=0.585858…x = 0.585858\ldotsx=0.585858…. To eliminate the repeating portion, we multiply xxx by 100 because the repeating segment “58” has two digits. This multiplication moves the decimal point two places to the right and gives 100x=58.585858…100x = 58.585858\ldots100x=58.585858…. Notice how the decimal part of 100x100x100x is the same as the decimal part of xxx, which allows us to eliminate the repeating digits through subtraction.
Subtracting xxx from 100x100x100x removes the repeating part:100x−x=58.585858…−0.585858…=58100x – x = 58.585858\ldots – 0.585858\ldots = 58100x−x=58.585858…−0.585858…=58
This results in 99x=5899x = 5899x=58, and by solving for xxx, we find x=5899x = \frac{58}{99}x=9958. This fraction is already in its simplest form because 58 and 99 do not have any common factors besides 1.
Therefore, the repeating decimal 0.585858…0.585858\ldots0.585858… is exactly equal to the fraction 5899\frac{58}{99}9958. This method works for any repeating decimal and allows you to convert it into a precise fractional representation.
