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

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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:** $$\boxed{\dfrac{431}{990}}$$
The fraction equivalent to $$ 0.4\overline{35} $$ is $$ \frac{431}{990} $$.

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

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