Rewrite 3.2 as a simplified fraction.
Understand the Problem
The question is asking to rewrite the repeating decimal 3.2 as a simplified fraction. The focus is on converting the decimal to its fractional form and simplifying it if possible.
Answer
The simplified fraction is $\frac{29}{9}$.
Answer for screen readers
The simplified fraction is $\frac{29}{9}$.
Steps to Solve
-
Identify the decimal structure
The decimal $3.2$ has a repeating portion. We can represent it as $3.\overline{2}$, or $3.222\ldots$. -
Set up an equation
Let $x = 3.\overline{2}$. So, we have:
$$ x = 3.2222... $$ -
Multiply to remove the decimal
Multiply both sides of the equation by $10$ to move the decimal point:
$$ 10x = 32.2222... $$ -
Subtract the original equation
Now subtract the original equation $x = 3.2222...$ from $10x = 32.2222...$:
$$ 10x - x = 32.222... - 3.222... $$
This simplifies to:
$$ 9x = 29 $$ -
Solve for x
Now, divide both sides by $9$:
$$ x = \frac{29}{9} $$ -
Final Result
Thus, $3.\overline{2}$ as a simplified fraction is:
$$ \frac{29}{9} $$
The simplified fraction is $\frac{29}{9}$.
More Information
The fraction $\frac{29}{9}$ is an improper fraction and represents the repeating decimal $3.\overline{2}$. Improper fractions can often be converted to mixed numbers, which in this case would be $3 \frac{2}{9}$.
Tips
- Confusing repeating and non-repeating decimals: Make sure to accurately identify which digits repeat.
- Improper subtraction: When subtracting the two equations, confirm that the decimal parts align correctly.
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