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

To find the fraction equivalent to the decimal \( 0.4\overline{35} \), which represents \( 0.4353535\ldots \), follow these steps: ### Step 1: Understand the Decimal The decimal \( 0.4\overline{35} \) can be expressed as: \[ x = 0.4353535\ldots \] Here, the non-repeating part is "4", and the repeating part is "35". ### Step 2: Isolate the Repeating Part Let’s separate the non-repeating and repeating parts: \[ x = 0.4 + 0.0353535\ldots \] Let \( y = 0.0353535\ldots \). ### Step 3: Convert the Repeating Decimal to a Fraction To convert \( y \) into a fraction: \[ y = 0.\overline{35} \] Let’s set up an equation for \( y \): \[ 100y = 3.5353535\ldots \] \[ y = 0.0353535\ldots \] Subtract the second equation from the first: \[ 100y - y = 3.5353535\ldots - 0.0353535\ldots \] \[ 99y = 3.5 \] \[ y = \frac{3.5}{99} = \frac{7}{198} \] ### Step 4: Combine the Parts Now, substitute \( y \) back into the expression for \( x \): \[ x = 0.4 + \frac{7}{198} \] Convert \( 0.4 \) to a fraction with denominator 198: \[ 0.4 = \frac{4}{10} = \frac{396}{990} \] So: \[ x = \frac{396}{990} + \frac{35}{990} = \frac{431}{990} \] ### Step 5: Simplify the Fraction Check if \( \frac{431}{990} \) can be simplified. Since 431 is a prime number and does not divide 990, the fraction is already in its simplest form. ### Final Answer \[ 0.4\overline{35} = \frac{431}{990} \] **Answer:** \(\boxed{\dfrac{431}{990}}\)