Simplifying Arithmetic Expression with Mixed Fractions

To find the value of the expression $5\frac{7}{9} \div 4\frac{1}{3} \times 6\frac{9}{14} + \frac{7}{9} \times 18$, we follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Converting Mixed Fractions to Improper Fractions

• $5\frac{7}{9} = \frac{(5 \times 9) + 7}{9} = \frac{45 + 7}{9} = \frac{52}{9}$
• $4\frac{1}{3} = \frac{(4 \times 3) + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3}$
• $6\frac{9}{14} = \frac{(6 \times 14) + 9}{14} = \frac{84 + 9}{14} = \frac{93}{14}$

Step-by-Step Calculation

1. Division: $ 5\frac{7}{9} \div 4\frac{1}{3} = \frac{52}{9} \div \frac{13}{3} $ $ = \frac{52}{9} \times \frac{3}{13} $ $ = \frac{52 \div 13}{9 \div 3} \times \frac{3 \div 3}{13 \div 13} $ $ = \frac{4}{3} \times \frac{1}{1} = \frac{4}{3} $
2. Multiplication (First Part): $ \frac{4}{3} \times 6\frac{9}{14} = \frac{4}{3} \times \frac{93}{14} $ $ = \frac{4 \div 2}{3 \div 3} \times \frac{93 \div 3}{14 \div 2} $ $ = \frac{2}{1} \times \frac{31}{7} = \frac{62}{7} $
3. Multiplication (Second Part): $ \frac{7}{9} \times 18 = \frac{7}{9} \times \frac{18}{1} $ $ = \frac{7}{1} \times \frac{18 \div 9}{9 \div 9} $ $ = 7 \times 2 = 14 $
4. Addition: $ \frac{62}{7} + 14 $ Find a common denominator: $ = \frac{62}{7} + \frac{14 \times 7}{7} $ $ = \frac{62}{7} + \frac{98}{7} $ $ = \frac{62 + 98}{7} = \frac{160}{7} $
5. Convert Result to Mixed Fraction: $ \frac{160}{7} = 22 \text{ remainder } 6 $ $ = 22\frac{6}{7} $

Final Result

The value of the expression is $22\frac{6}{7}$.