Questions: Convert the following repeating decimal to a fraction in simplest form. .4 7

Solution Steps

Let \( x = 0.4\overline{7} \). This means that \( x \) represents the decimal \( 0.477777...\), where the digit \( 7 \) repeats indefinitely.

To eliminate the repeating part, we can express \( x \) in terms of a fraction. We can write: \[ x = 0.4 + 0.0\overline{7} \] Let \( y = 0.0\overline{7} \). Then, we can express \( y \) as: \[ y = \frac{7}{90} \] Thus, we have: \[ x = 0.4 + \frac{7}{90} \]

Next, we convert \( 0.4 \) to a fraction: \[ 0.4 = \frac{4}{10} = \frac{2}{5} \] Now, we need a common denominator to add the fractions: \[ x = \frac{2}{5} + \frac{7}{90} \] The least common multiple of \( 5 \) and \( 90 \) is \( 90 \). Therefore, we convert \( \frac{2}{5} \) to have a denominator of \( 90 \): \[ \frac{2}{5} = \frac{36}{90} \] Now we can add the fractions: \[ x = \frac{36}{90} + \frac{7}{90} = \frac{43}{90} \]

Final Answer