Question 7 Find the fraction which is equivalent to \( 0.4 \overline{35} \)

To convert the repeating decimal \( 0.4\overline{35} \) into a fraction, let's break it down. The decimal can be expressed as \( 0.4 + 0.0\overline{35} \). For \( 0.0\overline{35} \), we can set \( x = 0.0\overline{35} \). This gives us: \[ 100x = 3.5\overline{35} \] \[ 100x - x = 3.5 \] \[ 99x = 3.5 \] \[ x = \frac{3.5}{99} = \frac{35}{990} = \frac{7}{198} \] Now, adding \( 0.4 \) or \( \frac{4}{10} = \frac{2}{5} \) we have: \[ \frac{2}{5} + \frac{7}{198} \] Now we need a common denominator, which is 990. Therefore, we can rewrite: \[ \frac{2}{5} = \frac{2 \times 198}{5 \times 198} = \frac{396}{990} \] Now combine the two fractions: \[ \frac{396}{990} + \frac{35}{990} = \frac{431}{990} \] So, the fraction equivalent to \( 0.4\overline{35} \) is \( \frac{431}{990} \). Fun Fact: The concept of repeating decimals has been around for centuries! Mathematicians way back in the 9th century were already playing with the ideas of fractions and decimals, though it took a long time for people to actually use decimals as we do today. If you're trying to avoid confusion with repeating decimals, a handy tip is to always identify the repeating part and get a clear plan for writing it as a fraction. Common mistakes include skipping the steps in combining like fractions or misidentifying the repeating part's length, so keep an eye out!