2.0345 with the 45 repeating as the simplest fraction possible

To express the decimal \( 2.0345454545\ldots \) (where 45 is repeating) as the simplest fraction, we can solve it step by step.

Let \( x = 2.0345454545\ldots \).

### Step 1: Isolate the repeating part.
Separate the integer part and the repeating decimal:
\[
x = 2 + 0.0345454545\ldots
\]
Let \( y = 0.0345454545\ldots \), so we now have \( x = 2 + y \).

### Step 2: Set up an equation for \( y \).
Let \( y = 0.0345454545\ldots \). We can represent this repeating decimal as a fraction by using algebra.

First, set:
\[
y = 0.0345454545\ldots
\]
Multiply both sides by 1000 to shift the decimal point three places to the right:
\[
1000y = 34.5454545\ldots
\]
Next, multiply by 100 to shift the repeating part:
\[
100y = 3.45454545\ldots
\]
Now, subtract the second equation from the first:
\[
(1000y - 100y) = (34.5454545\ldots - 3.45454545\ldots)
\]
This simplifies to:
\[
900y = 31.09
\]
Thus:
\[
y = \frac{31.09}{900}
\]

### Step 3: Simplify the fraction.
Now simplify this fraction by multiplying both the numerator and denominator by 100 to eliminate the decimal:
\[
y = \frac{3109}{90000}
\]
Simplify this fraction if possible.

Finally, add the integer part (2) back:
\[
x = 2 + \frac{3109}{90000}
\]

This is the simplest fractional representation of \( 2.0345454545\ldots \).

Would you like to go into further details, or have any questions? Here are 5 related questions you might want to consider:

1. How do you convert non-repeating decimals to fractions?
2. How do you handle repeating decimals of varying lengths?
3. What if the repeating part started further into the decimal?
4. How do we simplify fractions in general?
5. How does this process change for negative repeating decimals?

**Tip:** When dealing with repeating decimals, identifying the repeating sequence and setting up an equation can help you convert it to a fraction.

Learn how to convert the repeating decimal 2.0345454545... (where 45 is repeating) to the simplest fraction. Step-by-step solution provided, covering concepts of repeating decimals and fraction simplification. Suitable for Grades 9-12.

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