express 0.4$$ \overline{21} $$ in a form of a/b

To express $ 0.4\overline{21} $ as a fraction in the form of $ \frac{a}{b} $, let's break down the process.

Given:
\[
x = 0.4\overline{21}
\]
This decimal has a repeating part of "21" that starts after the initial "4".

### Step 1: Represent the Decimal in Terms of $ x $
Let:
\[
x = 0.4212121212\ldots
\]

### Step 2: Eliminate the Repeating Decimal
1. Multiply $ x $ by 100 to shift the decimal point to the right by two places, so that the repeating part "21" lines up.
   \[
   100x = 42.1212121212\ldots
   \]
   
2. Now we can see that $ x = 0.4212121212\ldots $ and $ 100x = 42.1212121212\ldots $. 

### Step 3: Set Up an Equation to Solve for $ x $
Subtract the original equation from the new one:
\[
100x - x = 42.1212121212\ldots - 0.4212121212\ldots
\]
This simplifies to:
\[
99x = 41.7
\]

### Step 4: Solve for $ x $
Now, divide both sides by 99 to isolate $ x $:
\[
x = \frac{41.7}{99}
\]

### Step 5: Convert 41.7 to a Fraction
Rewrite 41.7 as a fraction:
\[
41.7 = \frac{417}{10}
\]
So:
\[
x = \frac{417}{10} \times \frac{1}{99} = \frac{417}{990}
\]

### Step 6: Simplify the Fraction
Now, simplify $\frac{417}{990}$ by finding the greatest common divisor (GCD) of 417 and 990, which is 3:
\[
x = \frac{417 \div 3}{990 \div 3} = \frac{139}{330}
\]

Thus,
\[
0.4\overline{21} = \frac{139}{330}
\]

Would you like further details on this solution?

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Here are 5 questions for further exploration:
1. How do we find the GCD of two numbers?
2. Why do repeating decimals convert neatly into fractions?
3. What if a decimal repeats only partially, e.g., $ 0.4\overline{2}1$?
4. How can we confirm our fraction form is correct by reconverting it to a decimal?
5. What methods exist for converting more complex repeating decimals?

**Tip:** When converting repeating decimals, multiplying by powers of 10 is a key step to line up the repeating parts.

Learn how to express the repeating decimal 0.4$$ \overline{21} $$ as a fraction in the form a/b. The solution involves algebraic manipulation, using GCD for simplification, and understanding how repeating decimals can be converted to fractions. Suitable for Grades 8-10 students.

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