Convert the repeating decimal 0.77 to a rational number in simplest form

Answer 1

To convert the repeating decimal \(0.77\) (where "7" repeats) to a rational number, we can set up an equation.

Let: \[ x = 0.77777\ldots \]

Next, we will multiply both sides of the equation by \(10\) to move the decimal point: \[ 10x = 7.77777\ldots \]

Now we can subtract the first equation from this new equation: \[ 10x - x = 7.77777\ldots - 0.77777\ldots \]

This simplifies to: \[ 9x = 7 \]

Now, we can solve for \(x\) by dividing both sides by \(9\): \[ x = \frac{7}{9} \]

Therefore, the repeating decimal \(0.77\) as a rational number in simplest form is: \[ \boxed{\frac{7}{9}} \]