Convert the decimal $0 . \overline{27}$ to a fraction.
Answer (2)
Let x = 0. 27 .
Multiply by 100: 100 x = 27. 27 .
Subtract the original equation: 99 x = 27 .
Solve for x and simplify: x = 99 27 = 11 3 .
The decimal 0. 27 converted to a fraction is 11 3 .
Explanation
Understanding the Problem We are given the repeating decimal 0. 27 and we want to convert it to a fraction. This means the decimal representation is 0.27272727... where the digits 27 repeat indefinitely.
Setting up the Equation Let x = 0. 27 . To eliminate the repeating part, we multiply x by 100, since the repeating block has length 2. This gives us 100 x = 27. 27 = 27.272727...
Eliminating the Repeating Part Now, we subtract the original equation from the multiplied equation:
100 x − x = 27. 27 − 0. 27
Simplifying the Equation This simplifies to:
99 x = 27
Solving for x Now, we solve for x by dividing both sides by 99:
x = 99 27
Simplifying the Fraction Finally, we simplify the fraction by finding the greatest common divisor (GCD) of 27 and 99. The GCD of 27 and 99 is 9. So, we divide both the numerator and the denominator by 9:
x = 99 ÷ 9 27 ÷ 9 = 11 3
Final Answer Therefore, the decimal 0. 27 is equal to the fraction 11 3 .
Examples
Repeating decimals are commonly encountered when dealing with fractions in everyday situations, such as dividing quantities or calculating proportions. For example, if you want to divide a pizza into 11 equal slices, each slice would represent 11 1 of the pizza. Converting this fraction to a decimal gives you 0. 09 , which is a repeating decimal. Understanding how to convert repeating decimals to fractions allows you to work with exact values instead of approximations, ensuring accuracy in calculations involving proportions and divisions.
The decimal 0. 27 can be converted to a fraction by letting x = 0. 27 and multiplying by 100 to eliminate the repetition. This results in 99 x = 27 , leading to x = 99 27 = 11 3 . Therefore, the final result is 11 3 .
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