Convert 0.82 repeating to a rational number set X equal to 0.82 repeating then multiply both sides of the equation by what number

Answer 1

Let \( X = 0.828282...\) (where the "82" repeats).

To convert this repeating decimal into a rational number, we can multiply both sides of the equation by 100 (since the repeating part has 2 digits).

So, \[ 100X = 82.828282... \]

Now, we have two equations:

1. \( X = 0.828282...\)
2. \( 100X = 82.828282...\)

Next, we can subtract the first equation from the second: \[ 100X - X = 82.828282... - 0.828282... \] \[ 99X = 82 \]

Now, solve for \( X \): \[ X = \frac{82}{99} \]

Thus, \( 0.828282... \) can be expressed as the rational number \( \frac{82}{99} \).