Write the recurring decimal 0.26 as a fraction.
The Correct Answer and Explanation is :
To convert the recurring decimal ( 0.\overline{26} ) (where the digits 26 repeat indefinitely) into a fraction, follow these steps:
Step 1: Define the decimal as a variable
Let:
[
x = 0.\overline{26}
]
This means ( x = 0.26262626\ldots ).
Step 2: Eliminate the repeating decimal
To eliminate the repeating part, we multiply both sides of the equation by 100 (since the repeating block consists of two digits, “26”):
[
100x = 26.26262626\ldots
]
Now, subtract the original equation (( x = 0.26262626\ldots )) from this new equation:
[
100x – x = 26.26262626\ldots – 0.26262626\ldots
]
This simplifies to:
[
99x = 26
]
Step 3: Solve for ( x )
Now, solve for ( x ) by dividing both sides by 99:
[
x = \frac{26}{99}
]
Step 4: Simplify the fraction
The fraction ( \frac{26}{99} ) is already in its simplest form because 26 and 99 have no common factors other than 1. So, the fraction representation of ( 0.\overline{26} ) is:
[
x = \frac{26}{99}
]
Explanation:
This process of converting a repeating decimal into a fraction involves recognizing the periodic pattern, multiplying by a power of 10 to shift the decimal point, and then subtracting the original equation from the shifted equation to eliminate the repeating part. The result is an equation that can be solved algebraically.
The fraction ( \frac{26}{99} ) is the simplest form because the numerator (26) and the denominator (99) share no common factors.
Final Answer:
The recurring decimal ( 0.\overline{26} ) as a fraction is ( \frac{26}{99} ).