To solve the problem \(3 \frac{4}{7} - 1 \frac{5}{14}\), we first need to convert both mixed numbers into improper fractions.
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Convert \(3 \frac{4}{7}\) into an improper fraction: \[ 3 \frac{4}{7} = \frac{(3 \times 7) + 4}{7} = \frac{21 + 4}{7} = \frac{25}{7} \]
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Convert \(1 \frac{5}{14}\) into an improper fraction: \[ 1 \frac{5}{14} = \frac{(1 \times 14) + 5}{14} = \frac{14 + 5}{14} = \frac{19}{14} \]
Now, we have: \[ \frac{25}{7} - \frac{19}{14} \]
To subtract the fractions, we need a common denominator. The least common multiple of 7 and 14 is 14.
- Convert \(\frac{25}{7}\) to have a denominator of 14: \[ \frac{25}{7} = \frac{25 \times 2}{7 \times 2} = \frac{50}{14} \]
Now we can subtract the two fractions: \[ \frac{50}{14} - \frac{19}{14} = \frac{50 - 19}{14} = \frac{31}{14} \]
Finally, converting \(\frac{31}{14}\) back to a mixed number: \[ \frac{31}{14} = 2 \frac{3}{14} \]
Therefore, the final answer is: \[ \boxed{2 \frac{3}{14}} \]