- The box-filling problems are solved by calculating the numerator differences: 7.1: \(\boxed{6/25}\), 7.2: \(\boxed{6/12}\), 7.3: \(\boxed{5/15}\), 7.4: \(\boxed{4/8}\), 7.5: \(\boxed{13/25}\), 7.6: \(\boxed{4/10}\).
- The arithmetic operations are performed by calculating the combined numerators: A.1: \(\boxed{1/3}\), A.2: \(\boxed{9/40}\), A.3: \(\boxed{15/17}\), A.4: \(\boxed{1/10}\), A.5: \(\boxed{5\frac{3}{8}}\), A.6: \(\boxed{6\frac{1}{5}}\).
Explanation
- Problem Analysis and Strategy
The problems involve performing arithmetic operations with fractions. To subtract fractions with the same denominator, we subtract the numerators while keeping the common denominator unchanged. For mixed numbers, we subtract the whole number parts and the fractional parts separately, or convert to improper fractions if needed. For expressions involving a whole number and a fraction, we represent the whole number as a fraction with the same denominator.
- Solving the Box-Filling Problems (7.1-7.6)
- \(\frac{14}{25} - \frac{8}{25} = \frac{14-8}{25} = \frac{6}{25}\)
- \(\frac{7}{12} - \frac{1}{12} = \frac{7-1}{12} = \frac{6}{12}\)
- \(\frac{12}{15} - \frac{7}{15} = \frac{12-7}{15} = \frac{5}{15}\)
- \(\frac{7}{8} - \frac{3}{8} = \frac{7-3}{8} = \frac{4}{8}\)
- \(\frac{17}{25} - \frac{4}{25} = \frac{17-4}{25} = \frac{13}{25}\)
- \(\frac{7}{10} - \frac{3}{10} = \frac{7-3}{10} = \frac{4}{10}\)
- Solving the Arithmetic Operations (A.1-A.6)
A.1. \(\frac{5}{9} - \frac{2}{9} = \frac{3}{9} = \frac{1}{3}\)
A.2. \(\frac{5}{40} + \frac{6}{40} - \frac{2}{40} = \frac{5+6-2}{40} = \frac{9}{40}\)
A.3. \(\frac{19}{17} + \frac{2}{17} - \frac{6}{17} = \frac{19+2-6}{17} = \frac{15}{17}\)
A.4. \(1 - \frac{9}{10} = \frac{10}{10} - \frac{9}{10} = \frac{1}{10}\)
A.5. \(9\frac{4}{8} - 4\frac{1}{8} = (9-4) + (\frac{4}{8} - \frac{1}{8}) = 5 + \frac{3}{8} = 5\frac{3}{8}\)
A.6. \(15\frac{7}{5} - 10\frac{1}{5} = (15-10) + (\frac{7}{5} - \frac{1}{5}) = 5 + \frac{6}{5} = 5 + 1\frac{1}{5} = 6\frac{1}{5}\)
Examples
Understanding how to perform arithmetic operations with fractions is essential for everyday tasks, such as measuring ingredients for cooking. For instance, if a recipe calls for \(1\frac{3}{4}\) cups of flour and you only need half a batch, you would divide the fractions accordingly. Similarly, when woodworking, subtracting fractional measurements is common to ensure pieces fit perfectly. Mastery of these skills ensures accuracy in both professional and domestic projects.