How do you represent a repeating decimal as a fraction?

To represent a repeating decimal as a fraction, setting a variable ( x ) equal to the decimal is an effective method. This process typically involves the following steps:

1. Assign the repeating decimal to a variable, for instance, let ( x = 0.777\ldots ) for the decimal 0.777...

2. Multiply both sides of the equation by a power of 10 that moves the decimal point to the right past the repeating part. In this case, multiplying by 10 gives ( 10x = 7.777\ldots ).

3. This allows you to set up a system of equations. Subtract the first equation from the second: ( 10x - x = 7.777\ldots - 0.777\ldots ), leading to ( 9x = 7 ).

4. Finally, solve for ( x ) by dividing both sides by 9, resulting in ( x = \frac{7}{9} ).

By setting ( x ) equal to the decimal and manipulating the equations, you can effectively derive the fraction representation of the repeating decimal. This method is systematic and can be repeated for any repeating decimal, making it a powerful