Write the decimal 0.27 repeated as a fraction in simplest form

Write the decimal 0.27 repeated as a fraction in simplest form

The Correct Answer and Explanation is:

To express (0.\overline{27}) (where 27 repeats indefinitely) as a fraction in its simplest form, we follow a systematic approach. This is a repeating decimal, and the goal is to convert it into a fraction.

Step 1: Define the Repeating Decimal

Let ( x = 0.\overline{27} ). This means that ( x = 0.27272727\ldots ), with the digits 27 repeating infinitely.

Step 2: Eliminate the Repeating Decimal

To get rid of the repeating part, we multiply both sides of the equation by 100. This shifts the decimal point two places to the right.

[
100x = 27.272727\ldots
]

Step 3: Set Up the Subtraction

Now, subtract the original equation from the new equation:

[
100x = 27.272727\ldots
]
[
x = 0.272727\ldots
]

Subtracting these two equations:

[
100x – x = 27.272727\ldots – 0.272727\ldots
]
[
99x = 27
]

Step 4: Solve for (x)

Now, solve for (x) by dividing both sides of the equation by 99:

[
x = \frac{27}{99}
]

Step 5: Simplify the Fraction

To simplify the fraction, we find the greatest common divisor (GCD) of 27 and 99. The GCD of 27 and 99 is 9. Divide both the numerator and the denominator by 9:

[
\frac{27}{99} = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}
]

Thus, the decimal (0.\overline{27}) as a fraction in simplest form is:

[
x = \frac{3}{11}
]

Conclusion

Therefore, (0.\overline{27} = \frac{3}{11}). This fraction is in its simplest form because the numerator and denominator have no common factors other than 1. This method can be applied to any repeating decimal by following the same steps, multiplying by a power of 10 to shift the decimal point and solving the resulting equation.