What is 0.14 repeating as a fraction
The correct answer and explanation is:
To convert 0.14‾0.\overline{14} (where the “14” repeats indefinitely) into a fraction, follow these steps:
- Let x=0.14‾x = 0.\overline{14}. This means x=0.14141414…x = 0.14141414\ldots, where the digits “14” repeat infinitely.
- Multiply both sides by 100. This step shifts the decimal point two places to the right: 100x=14.14141414…100x = 14.14141414\ldots
- Subtract the original equation from the new equation. Now subtract x=0.14141414…x = 0.14141414\ldots from 100x=14.14141414…100x = 14.14141414\ldots: 100x−x=14.14141414…−0.14141414…100x – x = 14.14141414\ldots – 0.14141414\ldots This simplifies to: 99x=1499x = 14
- Solve for xx. To find xx, divide both sides of the equation by 99: x=1499x = \frac{14}{99}
Thus, 0.14‾0.\overline{14} as a fraction is 1499\frac{14}{99}.
Why does this work?
The reason this method works is because multiplying by a power of 10 shifts the repeating decimal, which makes it easier to subtract and isolate the repeating part. Subtracting removes the repeating decimals, allowing you to solve for the unknown fraction.
Simplifying the fraction
In this case, 1499\frac{14}{99} is already in its simplest form because 14 and 99 have no common factors other than 1. Thus, the fraction remains 1499\frac{14}{99}.
Conclusion
So, the repeating decimal 0.14‾0.\overline{14} is equivalent to the fraction 1499\frac{14}{99}. This method of subtracting the equations after shifting the decimal is a reliable way to convert repeating decimals into fractions.