What is 0.36 repeating expressed as a fraction in simplest form?
The Correct Answer and Explanation is :
To express ( 0.\overline{36} ) (which means 0.363636…, where “36” repeats indefinitely) as a fraction, follow these steps:
Step 1: Let ( x = 0.\overline{36} )
We will set up an equation where ( x ) represents the repeating decimal:
[
x = 0.\overline{36}
]
Step 2: Eliminate the repeating decimal
To eliminate the repeating part, multiply both sides of the equation by 100 (since “36” consists of two digits, multiplying by 100 shifts the decimal point two places to the right):
[
100x = 36.\overline{36}
]
Step 3: Subtract the original equation from the new equation
Now, subtract the original equation ( x = 0.\overline{36} ) from the equation ( 100x = 36.\overline{36} ):
[
100x – x = 36.\overline{36} – 0.\overline{36}
]
Simplifying both sides:
[
99x = 36
]
Step 4: Solve for ( x )
To solve for ( x ), divide both sides by 99:
[
x = \frac{36}{99}
]
Step 5: Simplify the fraction
To simplify ( \frac{36}{99} ), find the greatest common divisor (GCD) of 36 and 99. The GCD of 36 and 99 is 9. Now, divide both the numerator and the denominator by 9:
[
\frac{36}{99} = \frac{36 \div 9}{99 \div 9} = \frac{4}{11}
]
Thus, ( 0.\overline{36} = \frac{4}{11} ).
Conclusion
The repeating decimal ( 0.\overline{36} ) is equivalent to the fraction ( \frac{4}{11} ).
Explanation
This process works because by subtracting the equation where the repeating decimal is shifted, we can eliminate the repeating part. The result is a simple fraction, and simplifying that fraction makes it easier to work with and understand. This method of converting repeating decimals into fractions is a standard technique in algebra.