\( +\frac{2}{3}+1 \frac{2}{3}=\frac{61}{3} \quad \) 6. \( 6 \frac{5}{12}+4 \frac{11}{12} \)

To solve \( 6 \frac{5}{12} + 4 \frac{11}{12} \), first, convert the mixed numbers into improper fractions. For \( 6 \frac{5}{12} \): \[ 6 \times 12 + 5 = 72 + 5 = 77 \] So, \( 6 \frac{5}{12} = \frac{77}{12} \). For \( 4 \frac{11}{12} \): \[ 4 \times 12 + 11 = 48 + 11 = 59 \] So, \( 4 \frac{11}{12} = \frac{59}{12} \). Now, add the two fractions: \[ \frac{77}{12} + \frac{59}{12} = \frac{77 + 59}{12} = \frac{136}{12} \]. This simplifies to \( \frac{34}{3} \) or \( 11 \frac{1}{3} \). Mathematics has been essential throughout history, shaping civilizations and solving problems from ancient times. Ancient Egyptians used basic arithmetic and geometry for tasks like land surveying and budgeting, while the Greeks developed foundational concepts, like the Pythagorean theorem, which still influences modern mathematics. When performing operations with mixed numbers and improper fractions, common pitfalls include forgetting to convert completely to improper fractions or struggling with finding a common denominator for addition. To avoid mistakes, always simplify your fractions at the end and double-check that you’ve combined numerators correctly.