To simplify the expression \( 2 \frac{2}{4} + 3 \frac{2}{3} - \frac{1}{3} \), we can first convert the mixed numbers to improper fractions and then perform the calculations.
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Convert \( 2 \frac{2}{4} \): \[ 2 \frac{2}{4} = 2 + \frac{2}{4} = \frac{8}{4} + \frac{2}{4} = \frac{10}{4} = \frac{5}{2} \]
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Convert \( 3 \frac{2}{3} \): \[ 3 \frac{2}{3} = 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3} \]
Now we can rewrite the expression using the improper fractions:
\[ \frac{5}{2} + \frac{11}{3} - \frac{1}{3} \]
Combine the fractions \( \frac{11}{3} - \frac{1}{3} \):
\[ \frac{11}{3} - \frac{1}{3} = \frac{10}{3} \]
Now we have:
\[ \frac{5}{2} + \frac{10}{3} \]
Next, we need a common denominator to add these two fractions. The least common denominator between 2 and 3 is 6.
Convert \( \frac{5}{2} \) to have a denominator of 6:
\[ \frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6} \]
Convert \( \frac{10}{3} \) to have a denominator of 6:
\[ \frac{10}{3} = \frac{10 \times 2}{3 \times 2} = \frac{20}{6} \]
Now we can add them:
\[ \frac{15}{6} + \frac{20}{6} = \frac{35}{6} \]
The final answer is:
\[ \frac{35}{6} \]
This can also be expressed as a mixed number:
\[ 5 \frac{5}{6} \]
So, the simplified expression is \( 5 \frac{5}{6} \) or \( \frac{35}{6} \).
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