To solve the problem [tex]\(4 \frac{4}{9} - 3 \frac{1}{9}\)[/tex], we will follow a series of detailed steps.
### Step 1: Convert mixed numbers to improper fractions
First, let's convert each mixed number to an improper fraction:
- For [tex]\(4 \frac{4}{9}\)[/tex]:
- Multiply the whole number (4) by the denominator (9): [tex]\(4 \times 9 = 36\)[/tex]
- Add the numerator (4) to the result: [tex]\(36 + 4 = 40\)[/tex]
- Place the result over the original denominator: [tex]\(\frac{40}{9}\)[/tex]
- For [tex]\(3 \frac{1}{9}\)[/tex]:
- Multiply the whole number (3) by the denominator (9): [tex]\(3 \times 9 = 27\)[/tex]
- Add the numerator (1) to the result: [tex]\(27 + 1 = 28\)[/tex]
- Place the result over the original denominator: [tex]\(\frac{28}{9}\)[/tex]
### Step 2: Subtract the improper fractions
Now, subtract the improper fractions [tex]\(\frac{40}{9}\)[/tex] and [tex]\(\frac{28}{9}\)[/tex]:
- Since the denominators are the same, subtract the numerators:
[tex]\[
\frac{40}{9} - \frac{28}{9} = \frac{40 - 28}{9} = \frac{12}{9}
\][/tex]
### Step 3: Simplify the resulting fraction
Next, simplify the fraction [tex]\(\frac{12}{9}\)[/tex]:
- Find the greatest common divisor (GCD) of 12 and 9, which is 3.
- Divide both the numerator and the denominator by the GCD:
[tex]\[
\frac{12 \div 3}{9 \div 3} = \frac{4}{3}
\][/tex]
### Step 4: Convert to a mixed number (if necessary)
Finally, convert [tex]\(\frac{4}{3}\)[/tex] to a mixed number:
- Divide the numerator (4) by the denominator (3) to get the whole number part and the remainder:
- The quotient is 1 (whole number part)
- The remainder is 1, forming the fraction part [tex]\(\frac{1}{3}\)[/tex]
- Thus, [tex]\(\frac{4}{3}\)[/tex] can be written as [tex]\(1 \frac{1}{3}\)[/tex]
### Final Answer
So, [tex]\(4 \frac{4}{9} - 3 \frac{1}{9} = 1 \frac{1}{3}\)[/tex].
