Ask a New Question
Menu

Convert 0.09¯¯¯¯¯ to a rational number in simplest form. Show your work.

1 answer

To convert the repeating decimal \( 0.09\overline{9} \) into a rational number, we can follow these steps.

Let \( x = 0.09\overline{9} \). This means that \( x = 0.099999...\).

To isolate the repeating part, we can multiply \( x \) by a power of 10 that moves the decimal point to the right, placing the repeating part as the leading part.

  1. Multiply by 10: \[ 10x = 0.99999... \]

  2. Now, create another equation by multiplying \( x \) by 100: \[ 100x = 9.99999... \]

Now, we have two equations:

  • \( 10x = 0.99999... \) (1)
  • \( 100x = 9.99999... \) (2)

Next, we can subtract equation (1) from equation (2).

  1. Subtract the two equations: \[ 100x - 10x = 9.99999... - 0.99999... \] \[ 90x = 9 \]

  2. Solve for \( x \): \[ x = \frac{9}{90} \]

  3. Simplify the fraction: \[ x = \frac{1}{10} \]

Thus, the rational number in simplest form that represents \( 0.09\overline{9} \) is \(\frac{1}{10}\).