(7-5 1/4)-(1 4/9+4 5/9-5 1/2)

To solve the expression \((7 - 5 \frac{1}{4}) - (1 \frac{4}{9} + 4 \frac{5}{9} - 5 \frac{1}{2})\), we first need to convert all mixed numbers to improper fractions for easier calculation. 1. **Convert mixed numbers to improper fractions:** - \(7 = \frac{7 \times 1}{1} = \frac{7}{1}\) - \(5 \frac{1}{4} = 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}\) - \(1 \frac{4}{9} = 1 + \frac{4}{9} = \frac{9}{9} + \frac{4}{9} = \frac{13}{9}\) - \(4 \frac{5}{9} = 4 + \frac{5}{9} = \frac{36}{9} + \frac{5}{9} = \frac{41}{9}\) - \(5 \frac{1}{2} = 5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2}\) 2. **Substitute the improper fractions back into the expression:** \[ (7 - 5 \frac{1}{4}) = \left(\frac{7}{1} - \frac{21}{4}\right) \] We need a common denominator to subtract these fractions. The least common multiple of 1 and 4 is 4. \[ = \left(\frac{7 \times 4}{4} - \frac{21}{4}\right) = \left(\frac{28}{4} - \frac{21}{4}\right) = \frac{7}{4} \] 3. **Now calculate \((1 \frac{4}{9} + 4 \frac{5}{9} - 5 \frac{1}{2})\):** \[ = \left(\frac{13}{9} + \frac{41}{9} - \frac{11}{2}\right) \] First, combine \(\frac{13}{9}\) and \(\frac{41}{9}\): \[ = \frac{54}{9} = 6 \] Next, we need to convert \(6\) to a fraction with a common denominator with \(-\frac{11}{2}\): \[ 6 = \frac{6 \times 2}{2} = \frac{12}{2} \] Now combine: \[ = \left(\frac{12}{2} - \frac{11}{2}\right) = \frac{1}{2} \] 4. **Combine everything back into the original expression:** \[ \frac{7}{4} - \frac{1}{2} \] To subtract these fractions, convert \(\frac{1}{2}\) to have a denominator of 4: \[ \frac{1}{2} = \frac{2}{4} \] Now, perform the subtraction: \[ = \left(\frac{7}{4} - \frac{2}{4}\right) = \frac{5}{4} \] 5. **Final Result:** \[ \frac{5}{4} = 1 \frac{1}{4} \] Therefore, the result of the expression \((7 - 5 \frac{1}{4}) - (1 \frac{4}{9} + 4 \frac{5}{9} - 5 \frac{1}{2})\) is \(\frac{5}{4}\) or \(1 \frac{1}{4}\).