How do you convert 0.23 (3 repeating) as a fraction?
The Correct Answer and Explanation is :
To convert the repeating decimal (0.\overline{23}) (where “23” repeats infinitely) into a fraction, follow these steps:
Step 1: Define the repeating decimal
Let (x = 0.\overline{23}), where the “23” repeats forever. This means:
[
x = 0.2323232323\ldots
]
Step 2: Multiply both sides by 100
Since the repeating block is two digits long (“23”), multiply both sides of the equation by 100 to shift the decimal point two places to the right:
[
100x = 23.2323232323\ldots
]
Step 3: Subtract the original equation from the new equation
Now, subtract the original equation (x = 0.\overline{23}) from the new equation (100x = 23.\overline{23}):
[
100x – x = 23.\overline{23} – 0.\overline{23}
]
This simplifies to:
[
99x = 23
]
Step 4: Solve for (x)
Now, solve for (x) by dividing both sides by 99:
[
x = \frac{23}{99}
]
Final Answer:
Thus, the repeating decimal (0.\overline{23}) as a fraction is:
[
x = \frac{23}{99}
]
Explanation of the process:
- Why multiplying by 100 works: When dealing with repeating decimals, multiplying by a power of 10 (in this case, 100) shifts the decimal point so that the repeating part aligns with the original decimal. This allows you to subtract the two equations and eliminate the repeating decimal portion.
- The subtraction step: Subtracting the equations allows us to get rid of the infinite repeating decimal part, leaving just the non-repeating portion, which is easier to manipulate algebraically.
- The result: The final fraction, ( \frac{23}{99} ), is in its simplest form because 23 is a prime number and does not divide evenly into 99. Thus, (0.\overline{23} = \frac{23}{99}).
This method can be applied to any repeating decimal to convert it into a fraction.