Evaluate: (3/4 + 1 5/7) / (4/7 of 2 1/3) / ((1 3/7 - 5/8) * 2/3)

Question

Evaluate: (3/4 + 1 5/7) / (4/7 of 2 1/3) / ((1 3/7 - 5/8) * 2/3)

Asked on March 26, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-26T05:33:33.674Z

To evaluate the expression, we follow the order of operations (BODMAS/PEMDAS): Brackets, Orders (powers/roots), Division/Multiplication (from left to right), Addition/Subtraction (from left to right). "Of" means multiplication. First, convert all mixed numbers to improper fractions: $1\frac{5}{7} = \frac{1 \times 7 + 5}{7} = \frac{12}{7}$ $2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$ $1\frac{3}{7} = \frac{1 \times 7 + 3}{7} = \frac{10}{7}$ $2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3}$ Substitute these into the expression: $$ \frac{\frac{3}{4} + \frac{12}{7} \div \frac{4}{7} \text{ of } \frac{7}{3}}{\left(\frac{10}{7} - \frac{5}{8}\right) \times \frac{8}{3}} $$ Step 1: Evaluate the numerator. The numerator is $\frac{3}{4} + \frac{12}{7} \div \frac{4}{7} \text{ of } \frac{7}{3}$. First, calculate the "of" part: $$ \frac{4}{7} \text{ of } \frac{7}{3} = \frac{4}{7} \times \frac{7}{3} = \frac{4 \times 7}{7 \times 3} = \frac{4}{3} $$ Now the numerator becomes: $$ \frac{3}{4} + \frac{12}{7} \div \frac{4}{3} $$ Next, perform the division: $$ \frac{12}{7} \div \frac{4}{3} = \frac{12}{7} \times \frac{3}{4} = \frac{12 \times 3}{7 \times 4} = \frac{3 \times 4 \times 3}{7 \times 4} = \frac{9}{7} $$ Now the numerator becomes: $$ \frac{3}{4} + \frac{9}{7} $$ Perform the addition by finding a common denominator (28): $$ \frac{3}{4} + \frac{9}{7} = \frac{3 \times 7}{4 \times 7} + \frac{9 \times 4}{7 \times 4} = \frac{21}{28} + \frac{36}{28} = \frac{21 + 36}{28} = \frac{57}{28} $$ So, the numerator is $\frac{57}{28}$. Step 2: Evaluate the denominator. The denominator is $\left(\frac{10}{7} - \frac{5}{8}\right) \times \frac{8}{3}$. First, calculate the expression inside the brackets by finding a common denominator (56): $$ \frac{10}{7} - \frac{5}{8} = \frac{10 \times 8}{7 \times 8} - \frac{5 \times 7}{8 \times 7} = \frac{80}{56} - \frac{35}{56} = \frac{80 - 35}{56} = \frac{45}{56} $$ Now the denominator becomes: $$ \frac{45}{56} \times \frac{8}{3} $$ Perform the multiplication: $$ \frac{45}{56} \times \frac{8}{3} = \frac{45 \times 8}{56 \times 3} = \frac{(15 \times 3) \times 8}{(7 \times 8) \times 3} = \frac{15}{7} $$ So, the denominator is $\frac{15}{7}$. Step 3: Divide the numerator by the denominator. $$ \frac{\frac{57}{28}}{\frac{15}{7}} = \frac{57}{28} \div \frac{15}{7} $$ To divide by a fraction, multiply by its reciprocal: $$ \frac{57}{28} \times \frac{7}{15} $$ Simplify before multiplying: $$ \frac{57}{4 \times 7} \times \frac{7}{15} = \frac{57}{4 \times 15} $$ Both 57 and 15 are divisible by 3: $57 \div 3 = 19$ and $15 \div 3 = 5$. $$ \frac{19}{4 \times 5} = \frac{19}{20} $$ The final answer is $\boxed{\frac{19}{20}}$.

Accepted Answer · 2026-03-26T05:33:33.674Z

To evaluate the expression, we follow the order of operations (BODMAS/PEMDAS): Brackets, Orders (powers/roots), Division/Multiplication (from left to right), Addition/Subtraction (from left to right). "Of" means multiplication. First, convert all mixed numbers to improper fractions: $1\frac{5}{7} = \frac{1 \times 7 + 5}{7} = \frac{12}{7}$ $2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$ $1\frac{3}{7} = \frac{1 \times 7 + 3}{7} = \frac{10}{7}$ $2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3}$ Substitute these into the expression: $$ \frac{\frac{3}{4} + \frac{12}{7} \div \frac{4}{7} \text{ of } \frac{7}{3}}{\left(\frac{10}{7} - \frac{5}{8}\right) \times \frac{8}{3}} $$ Step 1: Evaluate the numerator. The numerator is $\frac{3}{4} + \frac{12}{7} \div \frac{4}{7} \text{ of } \frac{7}{3}$. First, calculate the "of" part: $$ \frac{4}{7} \text{ of } \frac{7}{3} = \frac{4}{7} \times \frac{7}{3} = \frac{4 \times 7}{7 \times 3} = \frac{4}{3} $$ Now the numerator becomes: $$ \frac{3}{4} + \frac{12}{7} \div \frac{4}{3} $$ Next, perform the division: $$ \frac{12}{7} \div \frac{4}{3} = \frac{12}{7} \times \frac{3}{4} = \frac{12 \times 3}{7 \times 4} = \frac{3 \times 4 \times 3}{7 \times 4} = \frac{9}{7} $$ Now the numerator becomes: $$ \frac{3}{4} + \frac{9}{7} $$ Perform the addition by finding a common denominator (28): $$ \frac{3}{4} + \frac{9}{7} = \frac{3 \times 7}{4 \times 7} + \frac{9 \times 4}{7 \times 4} = \frac{21}{28} + \frac{36}{28} = \frac{21 + 36}{28} = \frac{57}{28} $$ So, the numerator is $\frac{57}{28}$. Step 2: Evaluate the denominator. The denominator is $\left(\frac{10}{7} - \frac{5}{8}\right) \times \frac{8}{3}$. First, calculate the expression inside the brackets by finding a common denominator (56): $$ \frac{10}{7} - \frac{5}{8} = \frac{10 \times 8}{7 \times 8} - \frac{5 \times 7}{8 \times 7} = \frac{80}{56} - \frac{35}{56} = \frac{80 - 35}{56} = \frac{45}{56} $$ Now the denominator becomes: $$ \frac{45}{56} \times \frac{8}{3} $$ Perform the multiplication: $$ \frac{45}{56} \times \frac{8}{3} = \frac{45 \times 8}{56 \times 3} = \frac{(15 \times 3) \times 8}{(7 \times 8) \times 3} = \frac{15}{7} $$ So, the denominator is $\frac{15}{7}$. Step 3: Divide the numerator by the denominator. $$ \frac{\frac{57}{28}}{\frac{15}{7}} = \frac{57}{28} \div \frac{15}{7} $$ To divide by a fraction, multiply by its reciprocal: $$ \frac{57}{28} \times \frac{7}{15} $$ Simplify before multiplying: $$ \frac{57}{4 \times 7} \times \frac{7}{15} = \frac{57}{4 \times 15} $$ Both 57 and 15 are divisible by 3: $57 \div 3 = 19$ and $15 \div 3 = 5$. $$ \frac{19}{4 \times 5} = \frac{19}{20} $$ The final answer is $\boxed{\frac{19}{20}}$.