Math worksheet for evaluating expressions with rational numbers.
Worksheet titled "Operations on Rational Numbers-WS-8" with 16 math problems involving fractions, decimals, and operations like addition, subtraction, multiplication, division, and exponentiation.
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Step-by-step solution for: 301 Moved Permanently
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Show Answer Key & Explanations
Step-by-step solution for: 301 Moved Permanently
1) $\frac{2}{5} \times \frac{-1}{2} = \frac{-2}{10} = \frac{-1}{5}$, then $\frac{-1}{5} \div 2 = \frac{-1}{10}$. Next, $\frac{-11}{9} \div \frac{-3}{2} = \frac{-11}{9} \times \frac{2}{-3} = \frac{22}{27}$. Finally, $\frac{-1}{10} - \frac{22}{27} = \frac{-27}{270} - \frac{220}{270} = \frac{-247}{270}$.
2) First, $\frac{-16}{9} + \frac{1}{5} = \frac{-80}{45} + \frac{9}{45} = \frac{-71}{45}$. Squaring gives $\left(\frac{-71}{45}\right)^2 = \frac{5041}{2025}$. Then, $\frac{11}{8} - \frac{1}{10} = \frac{55}{40} - \frac{4}{40} = \frac{51}{40}$. Subtracting: $\frac{5041}{2025} - \frac{51}{40} = \frac{40328}{16200} - \frac{20655}{16200} = \frac{19673}{16200}$.
3) $\frac{-8}{5} \div \frac{-4}{3} = \frac{-8}{5} \times \frac{3}{-4} = \frac{24}{20} = \frac{6}{5}$. Then, $\frac{6}{5} \times 1 = \frac{6}{5}$, and $\frac{6}{5} \div \frac{1}{7} = \frac{6}{5} \times 7 = \frac{42}{5}$. Finally, $\frac{42}{5} - \frac{4}{3} = \frac{126}{15} - \frac{20}{15} = \frac{106}{15}$.
4) Combine all terms: $\frac{-11}{10} - 2 + \frac{9}{10} - \frac{6}{5} - \frac{1}{2} = \left(\frac{-11}{10} + \frac{9}{10}\right) + (-2) + \left(-\frac{6}{5}\right) + \left(-\frac{1}{2}\right) = \frac{-2}{10} - 2 - \frac{6}{5} - \frac{1}{2} = \frac{-1}{5} - 2 - \frac{6}{5} - \frac{1}{2} = \left(\frac{-1}{5} - \frac{6}{5}\right) - 2 - \frac{1}{2} = \frac{-7}{5} - 2 - \frac{1}{2} = \frac{-14}{10} - \frac{20}{10} - \frac{5}{10} = \frac{-39}{10}$.
5) Inside the parentheses: $\frac{-9}{7} + 2 \div \frac{1}{2} - \frac{-4}{7} = \frac{-9}{7} + 4 + \frac{4}{7} = \left(\frac{-9}{7} + \frac{4}{7}\right) + 4 = \frac{-5}{7} + 4 = \frac{-5}{7} + \frac{28}{7} = \frac{23}{7}$. Then, $\frac{-6}{5} \times \frac{23}{7} = \frac{-138}{35}$.
6) First, $\frac{-1}{2} - 9 = \frac{-1}{2} - \frac{18}{2} = \frac{-19}{2}$. Next, $\frac{-7}{4} - \frac{7}{8} = \frac{-14}{8} - \frac{7}{8} = \frac{-21}{8}$. Then, $\frac{-3}{2} \times \frac{-19}{2} \times \frac{-21}{8} = \frac{(-3) \times (-19) \times (-21)}{2 \times 2 \times 8} = \frac{-1197}{32}$.
7) First, $\frac{13}{10} + \frac{7}{5} = \frac{13}{10} + \frac{14}{10} = \frac{27}{10}$. Then, $\frac{-6}{7} \times -10 \times \frac{27}{10} = \frac{-6}{7} \times -27 = \frac{162}{7}$. Finally, $\frac{162}{7} \div \frac{-8}{5} = \frac{162}{7} \times \frac{5}{-8} = \frac{810}{-56} = \frac{-405}{28}$.
8) $\left(\frac{-5}{3}\right)^2 = \frac{25}{9}$. Then, $\frac{-2}{3} - 2 = \frac{-2}{3} - \frac{6}{3} = \frac{-8}{3}$. So, $\frac{25}{9} \times \frac{-8}{3} = \frac{-200}{27}$. Finally, $\frac{-200}{27} \div \frac{1}{2} = \frac{-200}{27} \times 2 = \frac{-400}{27}$.
9) $\left(\frac{-4}{3}\right)^2 = \frac{16}{9}$. Then, $\frac{-2}{3} - \frac{5}{3} = \frac{-7}{3}$. So, $\frac{16}{9} \times \frac{-7}{3} = \frac{-112}{27}$. Finally, $\frac{-112}{27} \div \frac{-3}{2} = \frac{-112}{27} \times \frac{2}{-3} = \frac{224}{81}$.
10) Inside the parentheses: $\frac{10}{9} - \frac{9}{7} = \frac{70}{63} - \frac{81}{63} = \frac{-11}{63}$. Then, $\frac{5}{7} - \frac{-11}{63} = \frac{45}{63} + \frac{11}{63} = \frac{56}{63} = \frac{8}{9}$. The denominator is $1 - (-1) = 2$. So, $\frac{8}{9} \div 2 = \frac{8}{9} \times \frac{1}{2} = \frac{4}{9}$.
11) Numerator: $\frac{-3}{2} \times \frac{-11}{7} \times -1 = \frac{33}{14} \times -1 = \frac{-33}{14}$. Denominator: $\frac{-7}{4} - 2 = \frac{-7}{4} - \frac{8}{4} = \frac{-15}{4}$. So, $\frac{-33}{14} \div \frac{-15}{4} = \frac{-33}{14} \times \frac{4}{-15} = \frac{132}{210} = \frac{22}{35}$.
12) Inside the parentheses: $\frac{1}{4} - \frac{3}{2} \times \frac{2}{3} - 2 = \frac{1}{4} - 1 - 2 = \frac{1}{4} - 3 = \frac{1}{4} - \frac{12}{4} = \frac{-11}{4}$. Then, $\frac{-11}{4} \div \frac{7}{4} = \frac{-11}{4} \times \frac{4}{7} = \frac{-11}{7}$.
13) Numerator: $-6 \times \frac{-4}{3} \times \frac{-13}{9} = 8 \times \frac{-13}{9} = \frac{-104}{9}$. Denominator: $\frac{2}{5} - \frac{4}{3} = \frac{6}{15} - \frac{20}{15} = \frac{-14}{15}$. So, $\frac{-104}{9} \div \frac{-14}{15} = \frac{-104}{9} \times \frac{15}{-14} = \frac{1560}{126} = \frac{260}{21}$.
14) First, inside the innermost parentheses: $\frac{-5}{3} - \frac{7}{4} = \frac{-20}{12} - \frac{21}{12} = \frac{-41}{12}$. Then, $\frac{-3}{2} \times \frac{-41}{12} \times 2 = \frac{-3}{2} \times \frac{-82}{12} = \frac{246}{24} = \frac{41}{4}$. Finally, $\frac{-11}{9} \div \frac{41}{4} = \frac{-11}{9} \times \frac{4}{41} = \frac{-44}{369}$.
15) First, $\frac{1}{5} \times \frac{3}{10} = \frac{3}{50}$. Then, combine: $-1 + \frac{-5}{7} + 2 + \frac{3}{50} = (-1 + 2) + \left(\frac{-5}{7} + \frac{3}{50}\right) = 1 + \left(\frac{-250}{350} + \frac{21}{350}\right) = 1 + \frac{-229}{350} = \frac{350}{350} - \frac{229}{350} = \frac{121}{350}$.
16) First, $\frac{5}{6} \times 2 = \frac{10}{6} = \frac{5}{3}$. Then, $\frac{-8}{7} \div \frac{5}{4} = \frac{-8}{7} \times \frac{4}{5} = \frac{-32}{35}$. So, $\frac{5}{3} + \frac{3}{7} - \frac{-32}{35} = \frac{5}{3} + \frac{3}{7} + \frac{32}{35}$. Find a common denominator (105): $\frac{175}{105} + \frac{45}{105} + \frac{96}{105} = \frac{316}{105}$.
2) First, $\frac{-16}{9} + \frac{1}{5} = \frac{-80}{45} + \frac{9}{45} = \frac{-71}{45}$. Squaring gives $\left(\frac{-71}{45}\right)^2 = \frac{5041}{2025}$. Then, $\frac{11}{8} - \frac{1}{10} = \frac{55}{40} - \frac{4}{40} = \frac{51}{40}$. Subtracting: $\frac{5041}{2025} - \frac{51}{40} = \frac{40328}{16200} - \frac{20655}{16200} = \frac{19673}{16200}$.
3) $\frac{-8}{5} \div \frac{-4}{3} = \frac{-8}{5} \times \frac{3}{-4} = \frac{24}{20} = \frac{6}{5}$. Then, $\frac{6}{5} \times 1 = \frac{6}{5}$, and $\frac{6}{5} \div \frac{1}{7} = \frac{6}{5} \times 7 = \frac{42}{5}$. Finally, $\frac{42}{5} - \frac{4}{3} = \frac{126}{15} - \frac{20}{15} = \frac{106}{15}$.
4) Combine all terms: $\frac{-11}{10} - 2 + \frac{9}{10} - \frac{6}{5} - \frac{1}{2} = \left(\frac{-11}{10} + \frac{9}{10}\right) + (-2) + \left(-\frac{6}{5}\right) + \left(-\frac{1}{2}\right) = \frac{-2}{10} - 2 - \frac{6}{5} - \frac{1}{2} = \frac{-1}{5} - 2 - \frac{6}{5} - \frac{1}{2} = \left(\frac{-1}{5} - \frac{6}{5}\right) - 2 - \frac{1}{2} = \frac{-7}{5} - 2 - \frac{1}{2} = \frac{-14}{10} - \frac{20}{10} - \frac{5}{10} = \frac{-39}{10}$.
5) Inside the parentheses: $\frac{-9}{7} + 2 \div \frac{1}{2} - \frac{-4}{7} = \frac{-9}{7} + 4 + \frac{4}{7} = \left(\frac{-9}{7} + \frac{4}{7}\right) + 4 = \frac{-5}{7} + 4 = \frac{-5}{7} + \frac{28}{7} = \frac{23}{7}$. Then, $\frac{-6}{5} \times \frac{23}{7} = \frac{-138}{35}$.
6) First, $\frac{-1}{2} - 9 = \frac{-1}{2} - \frac{18}{2} = \frac{-19}{2}$. Next, $\frac{-7}{4} - \frac{7}{8} = \frac{-14}{8} - \frac{7}{8} = \frac{-21}{8}$. Then, $\frac{-3}{2} \times \frac{-19}{2} \times \frac{-21}{8} = \frac{(-3) \times (-19) \times (-21)}{2 \times 2 \times 8} = \frac{-1197}{32}$.
7) First, $\frac{13}{10} + \frac{7}{5} = \frac{13}{10} + \frac{14}{10} = \frac{27}{10}$. Then, $\frac{-6}{7} \times -10 \times \frac{27}{10} = \frac{-6}{7} \times -27 = \frac{162}{7}$. Finally, $\frac{162}{7} \div \frac{-8}{5} = \frac{162}{7} \times \frac{5}{-8} = \frac{810}{-56} = \frac{-405}{28}$.
8) $\left(\frac{-5}{3}\right)^2 = \frac{25}{9}$. Then, $\frac{-2}{3} - 2 = \frac{-2}{3} - \frac{6}{3} = \frac{-8}{3}$. So, $\frac{25}{9} \times \frac{-8}{3} = \frac{-200}{27}$. Finally, $\frac{-200}{27} \div \frac{1}{2} = \frac{-200}{27} \times 2 = \frac{-400}{27}$.
9) $\left(\frac{-4}{3}\right)^2 = \frac{16}{9}$. Then, $\frac{-2}{3} - \frac{5}{3} = \frac{-7}{3}$. So, $\frac{16}{9} \times \frac{-7}{3} = \frac{-112}{27}$. Finally, $\frac{-112}{27} \div \frac{-3}{2} = \frac{-112}{27} \times \frac{2}{-3} = \frac{224}{81}$.
10) Inside the parentheses: $\frac{10}{9} - \frac{9}{7} = \frac{70}{63} - \frac{81}{63} = \frac{-11}{63}$. Then, $\frac{5}{7} - \frac{-11}{63} = \frac{45}{63} + \frac{11}{63} = \frac{56}{63} = \frac{8}{9}$. The denominator is $1 - (-1) = 2$. So, $\frac{8}{9} \div 2 = \frac{8}{9} \times \frac{1}{2} = \frac{4}{9}$.
11) Numerator: $\frac{-3}{2} \times \frac{-11}{7} \times -1 = \frac{33}{14} \times -1 = \frac{-33}{14}$. Denominator: $\frac{-7}{4} - 2 = \frac{-7}{4} - \frac{8}{4} = \frac{-15}{4}$. So, $\frac{-33}{14} \div \frac{-15}{4} = \frac{-33}{14} \times \frac{4}{-15} = \frac{132}{210} = \frac{22}{35}$.
12) Inside the parentheses: $\frac{1}{4} - \frac{3}{2} \times \frac{2}{3} - 2 = \frac{1}{4} - 1 - 2 = \frac{1}{4} - 3 = \frac{1}{4} - \frac{12}{4} = \frac{-11}{4}$. Then, $\frac{-11}{4} \div \frac{7}{4} = \frac{-11}{4} \times \frac{4}{7} = \frac{-11}{7}$.
13) Numerator: $-6 \times \frac{-4}{3} \times \frac{-13}{9} = 8 \times \frac{-13}{9} = \frac{-104}{9}$. Denominator: $\frac{2}{5} - \frac{4}{3} = \frac{6}{15} - \frac{20}{15} = \frac{-14}{15}$. So, $\frac{-104}{9} \div \frac{-14}{15} = \frac{-104}{9} \times \frac{15}{-14} = \frac{1560}{126} = \frac{260}{21}$.
14) First, inside the innermost parentheses: $\frac{-5}{3} - \frac{7}{4} = \frac{-20}{12} - \frac{21}{12} = \frac{-41}{12}$. Then, $\frac{-3}{2} \times \frac{-41}{12} \times 2 = \frac{-3}{2} \times \frac{-82}{12} = \frac{246}{24} = \frac{41}{4}$. Finally, $\frac{-11}{9} \div \frac{41}{4} = \frac{-11}{9} \times \frac{4}{41} = \frac{-44}{369}$.
15) First, $\frac{1}{5} \times \frac{3}{10} = \frac{3}{50}$. Then, combine: $-1 + \frac{-5}{7} + 2 + \frac{3}{50} = (-1 + 2) + \left(\frac{-5}{7} + \frac{3}{50}\right) = 1 + \left(\frac{-250}{350} + \frac{21}{350}\right) = 1 + \frac{-229}{350} = \frac{350}{350} - \frac{229}{350} = \frac{121}{350}$.
16) First, $\frac{5}{6} \times 2 = \frac{10}{6} = \frac{5}{3}$. Then, $\frac{-8}{7} \div \frac{5}{4} = \frac{-8}{7} \times \frac{4}{5} = \frac{-32}{35}$. So, $\frac{5}{3} + \frac{3}{7} - \frac{-32}{35} = \frac{5}{3} + \frac{3}{7} + \frac{32}{35}$. Find a common denominator (105): $\frac{175}{105} + \frac{45}{105} + \frac{96}{105} = \frac{316}{105}$.
Parent Tip: Review the logic above to help your child master the concept of operations with rational numbers worksheet.
