1. Stating the problems: a. Evaluate 7 \(\frac{3}{5}\) - 8 \(\frac{2}{7}\) b. Evaluate \(\frac{4}{7}\) + 2 \(\frac{7}{14}\) c. Evaluate 2 \(\frac{1}{8}\) + \(\frac{11}{3}\) d. Evaluate 12 \(\frac{3}{8}\) - 8 \(\frac{2}{9}\) e. Evaluate 8 - 6 \(\frac{4}{5}\) f. Evaluate 9 \(\frac{7}{8}\) \(\times\) \(\frac{4}{5}\) 2. Simplifying each mixed number to improper fractions: a. 7 \(\frac{3}{5}\) = \(\frac{7 \times 5 + 3}{5} = \frac{35 + 3}{5} = \frac{38}{5}\) 8 \(\frac{2}{7}\) = \(\frac{8 \times 7 + 2}{7} = \frac{56 + 2}{7} = \frac{58}{7}\) b. 2 \(\frac{7}{14}\) = 2 \(\frac{1}{2}\) since \(\frac{7}{14} = \frac{1}{2}\), converted to improper fraction: 2 \(\frac{1}{2}\) = \(\frac{2 \times 2 + 1}{2} = \frac{5}{2}\) c. 2 \(\frac{1}{8}\) = \(\frac{2 \times 8 + 1}{8} = \frac{17}{8}\) \(\frac{11}{3}\) is already an improper fraction d. 12 \(\frac{3}{8}\) = \(\frac{12 \times 8 + 3}{8} = \frac{99}{8}\) 8 \(\frac{2}{9}\) = \(\frac{8 \times 9 + 2}{9} = \frac{74}{9}\) e. 6 \(\frac{4}{5}\) = \(\frac{6 \times 5 + 4}{5} = \frac{34}{5}\) f. 9 \(\frac{7}{8}\) = \(\frac{9 \times 8 + 7}{8} = \frac{79}{8}\) 3. Calculating each expression: a. \(\frac{38}{5} - \frac{58}{7} = \frac{38 \times 7}{35} - \frac{58 \times 5}{35} = \frac{266}{35} - \frac{290}{35} = \frac{266 - 290}{35} = \frac{-24}{35}\) b. \(\frac{4}{7} + \frac{5}{2} = \frac{4 \times 2}{14} + \frac{5 \times 7}{14} = \frac{8}{14} + \frac{35}{14} = \frac{43}{14}\) c. \(\frac{17}{8} + \frac{11}{3} = \frac{17 \times 3}{24} + \frac{11 \times 8}{24} = \frac{51}{24} + \frac{88}{24} = \frac{139}{24}\) d. \(\frac{99}{8} - \frac{74}{9} = \frac{99 \times 9}{72} - \frac{74 \times 8}{72} = \frac{891}{72} - \frac{592}{72} = \frac{299}{72}\) e. \(8 - \frac{34}{5} = \frac{8 \times 5}{5} - \frac{34}{5} = \frac{40}{5} - \frac{34}{5} = \frac{6}{5}\) f. \(\frac{79}{8} \times \frac{4}{5} = \frac{79 \times 4}{8 \times 5} = \frac{316}{40} = \frac{79}{10}\) 4. Next, solving the custom operations: a. Red circle operation \(a \blacklozenge b = 5b - 3a\) a) Calculate \(6 \blacklozenge 4 = 5 \times 4 - 3 \times 6 = 20 - 18 = 2\) b) Calculate \(4 \blacklozenge 6 = 5 \times 6 - 3 \times 4 = 30 - 12 = 18\) b. Blue circle operation is multiplication of a sequence accordingly. Given: \(2 \odot 3 = 2 \times 3 \times 4 = 24\) \(3 \odot 4 = 3 \times 4 \times 5 \times 6 = 360\) \(5 \odot 2 = 5 \times 6 = 30\) Pattern: For \(a \odot b\), multiply the integers from \(a\) to \(a + b - 1\). a) Calculate \(6 \odot 3 = 6 \times 7 \times 8 = 336\) b) Calculate \(5 \odot 4 = 5 \times 6 \times 7 \times 8 = 1680\) Final answers: a. \(-\frac{24}{35}\) b. \(\frac{43}{14}\) c. \(\frac{139}{24}\) d. \(\frac{299}{72}\) e. \(\frac{6}{5}\) f. \(\frac{79}{10}\) 5 \blacklozenge 6 = 18, 6 \blacklozenge 4 = 2 6 \odot 3 = 336, 5 \odot 4 = 1680