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