301 Moved Permanently - Free Printable

Problem: Evaluate each expression involving rational numbers.

We will solve each expression step by step, following the order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

---

#### 1. $\left(\frac{2}{5} \times \frac{-1}{2}\right) \div 2 - \frac{-11}{9} \div \frac{-3}{2}$

1. Simplify inside the parentheses:
$$
\frac{2}{5} \times \frac{-1}{2} = \frac{2 \times -1}{5 \times 2} = \frac{-2}{10} = \frac{-1}{5}
$$

2. Divide by 2:
$$
\frac{-1}{5} \div 2 = \frac{-1}{5} \times \frac{1}{2} = \frac{-1}{10}
$$

3. Simplify the second term:
$$
\frac{-11}{9} \div \frac{-3}{2} = \frac{-11}{9} \times \frac{2}{-3} = \frac{-11 \times 2}{9 \times -3} = \frac{-22}{-27} = \frac{22}{27}
$$

4. Combine the results:
$$
\frac{-1}{10} - \frac{22}{27}
$$
Find a common denominator (270):
$$
\frac{-1}{10} = \frac{-27}{270}, \quad \frac{22}{27} = \frac{220}{270}
$$
$$
\frac{-27}{270} - \frac{220}{270} = \frac{-27 - 220}{270} = \frac{-247}{270}
$$

Answer:
$$
\boxed{\frac{-247}{270}}
$$

---

#### 2. $\left(\frac{-16}{9} + \frac{1}{5}\right)^2 - \left(\frac{11}{8} - \frac{-1}{10}\right)$

1. Simplify inside the first parentheses:
$$
\frac{-16}{9} + \frac{1}{5} = \frac{-16 \times 5 + 1 \times 9}{45} = \frac{-80 + 9}{45} = \frac{-71}{45}
$$

2. Square the result:
$$
\left(\frac{-71}{45}\right)^2 = \frac{(-71)^2}{45^2} = \frac{5041}{2025}
$$

3. Simplify inside the second parentheses:
$$
\frac{11}{8} - \frac{-1}{10} = \frac{11}{8} + \frac{1}{10} = \frac{11 \times 10 + 1 \times 8}{80} = \frac{110 + 8}{80} = \frac{118}{80} = \frac{59}{40}
$$

4. Subtract the results:
$$
\frac{5041}{2025} - \frac{59}{40}
$$
Find a common denominator (8100):
$$
\frac{5041}{2025} = \frac{5041 \times 4}{8100} = \frac{20164}{8100}, \quad \frac{59}{40} = \frac{59 \times 202.5}{8100} = \frac{11805}{8100}
$$
$$
\frac{20164}{8100} - \frac{11805}{8100} = \frac{20164 - 11805}{8100} = \frac{8359}{8100}
$$

Answer:
$$
\boxed{\frac{8359}{8100}}
$$

---

#### 3. $\frac{-8}{5} \div \frac{-4}{3} \times 1 \div \frac{1}{7} - \frac{4}{3}$

1. Simplify step by step:
$$
\frac{-8}{5} \div \frac{-4}{3} = \frac{-8}{5} \times \frac{3}{-4} = \frac{-8 \times 3}{5 \times -4} = \frac{-24}{-20} = \frac{6}{5}
$$
$$
\frac{6}{5} \times 1 = \frac{6}{5}
$$
$$
\frac{6}{5} \div \frac{1}{7} = \frac{6}{5} \times 7 = \frac{6 \times 7}{5} = \frac{42}{5}
$$

2. Subtract $\frac{4}{3}$:
$$
\frac{42}{5} - \frac{4}{3}
$$
Find a common denominator (15):
$$
\frac{42}{5} = \frac{42 \times 3}{15} = \frac{126}{15}, \quad \frac{4}{3} = \frac{4 \times 5}{15} = \frac{20}{15}
$$
$$
\frac{126}{15} - \frac{20}{15} = \frac{126 - 20}{15} = \frac{106}{15}
$$

Answer:
$$
\boxed{\frac{106}{15}}
$$

---

#### 4. $\frac{-11}{10} - 2 + \frac{9}{10} - \frac{6}{5} - \frac{1}{2}$

1. Combine like terms:
$$
\frac{-11}{10} + \frac{9}{10} = \frac{-11 + 9}{10} = \frac{-2}{10} = \frac{-1}{5}
$$

2. Convert all terms to have a common denominator (10):
$$
-2 = \frac{-20}{10}, \quad \frac{6}{5} = \frac{12}{10}, \quad \frac{1}{2} = \frac{5}{10}
$$

3. Combine all terms:
$$
\frac{-1}{5} - \frac{20}{10} + \frac{9}{10} - \frac{12}{10} - \frac{5}{10} = \frac{-2}{10} - \frac{20}{10} + \frac{9}{10} - \frac{12}{10} - \frac{5}{10}
$$
$$
= \frac{-2 - 20 + 9 - 12 - 5}{10} = \frac{-30}{10} = -3
$$

Answer:
$$
\boxed{-3}
$$

---

#### 5. $\frac{-6}{5} \left(\frac{-9}{7} + 2 \div \frac{1}{2} - \frac{-4}{7}\right)$

1. Simplify inside the parentheses:
$$
2 \div \frac{1}{2} = 2 \times 2 = 4
$$
$$
\frac{-9}{7} + 4 - \frac{-4}{7} = \frac{-9}{7} + 4 + \frac{4}{7}
$$
Combine fractions:
$$
\frac{-9}{7} + \frac{4}{7} = \frac{-9 + 4}{7} = \frac{-5}{7}
$$
Add 4:
$$
\frac{-5}{7} + 4 = \frac{-5}{7} + \frac{28}{7} = \frac{-5 + 28}{7} = \frac{23}{7}
$$

2. Multiply by $\frac{-6}{5}$:
$$
\frac{-6}{5} \times \frac{23}{7} = \frac{-6 \times 23}{5 \times 7} = \frac{-138}{35}
$$

Answer:
$$
\boxed{\frac{-138}{35}}
$$

---

#### 6. $\frac{-3}{2} \times \left(\frac{-1}{2} - 9\right)\left(\frac{-7}{4} - \frac{7}{8}\right)$

1. Simplify inside the first set of parentheses:
$$
\frac{-1}{2} - 9 = \frac{-1}{2} - \frac{18}{2} = \frac{-1 - 18}{2} = \frac{-19}{2}
$$

2. Simplify inside the second set of parentheses:
$$
\frac{-7}{4} - \frac{7}{8} = \frac{-7 \times 2}{8} - \frac{7}{8} = \frac{-14}{8} - \frac{7}{8} = \frac{-14 - 7}{8} = \frac{-21}{8}
$$

3. Multiply the results:
$$
\frac{-3}{2} \times \frac{-19}{2} \times \frac{-21}{8}
$$
$$
\frac{-3 \times -19 \times -21}{2 \times 2 \times 8} = \frac{-1197}{32}
$$

Answer:
$$
\boxed{\frac{-1197}{32}}
$$

---

#### 7. $\frac{-6}{7} \times -10 \times \left(\frac{13}{10} + \frac{7}{5}\right) \div \frac{-8}{5}$

1. Simplify inside the parentheses:
$$
\frac{13}{10} + \frac{7}{5} = \frac{13}{10} + \frac{14}{10} = \frac{13 + 14}{10} = \frac{27}{10}
$$

2. Multiply:
$$
\frac{-6}{7} \times -10 \times \frac{27}{10} = \frac{-6 \times -10 \times 27}{7 \times 10} = \frac{1620}{70} = \frac{162}{7}
$$

3. Divide by $\frac{-8}{5}$:
$$
\frac{162}{7} \div \frac{-8}{5} = \frac{162}{7} \times \frac{5}{-8} = \frac{162 \times 5}{7 \times -8} = \frac{810}{-56} = \frac{-405}{28}
$$

Answer:
$$
\boxed{\frac{-405}{28}}
$$

---

#### 8. $\left(\frac{-5}{3}\right)^2 \times \left(\frac{-2}{3} - 2\right) \div \frac{1}{2}$

1. Simplify the square:
$$
\left(\frac{-5}{3}\right)^2 = \frac{(-5)^2}{3^2} = \frac{25}{9}
$$

2. Simplify inside the parentheses:
$$
\frac{-2}{3} - 2 = \frac{-2}{3} - \frac{6}{3} = \frac{-2 - 6}{3} = \frac{-8}{3}
$$

3. Multiply:
$$
\frac{25}{9} \times \frac{-8}{3} = \frac{25 \times -8}{9 \times 3} = \frac{-200}{27}
$$

4. Divide by $\frac{1}{2}$:
$$
\frac{-200}{27} \div \frac{1}{2} = \frac{-200}{27} \times 2 = \frac{-400}{27}
$$

Answer:
$$
\boxed{\frac{-400}{27}}
$$

---

#### 9. $\left(\frac{-4}{3}\right)^2 \times \left(\frac{-2}{3} - \frac{5}{3}\right) \div \frac{-3}{2}$

1. Simplify the square:
$$
\left(\frac{-4}{3}\right)^2 = \frac{(-4)^2}{3^2} = \frac{16}{9}
$$

2. Simplify inside the parentheses:
$$
\frac{-2}{3} - \frac{5}{3} = \frac{-2 - 5}{3} = \frac{-7}{3}
$$

3. Multiply:
$$
\frac{16}{9} \times \frac{-7}{3} = \frac{16 \times -7}{9 \times 3} = \frac{-112}{27}
$$

4. Divide by $\frac{-3}{2}$:
$$
\frac{-112}{27} \div \frac{-3}{2} = \frac{-112}{27} \times \frac{2}{-3} = \frac{-112 \times 2}{27 \times -3} = \frac{-224}{-81} = \frac{224}{81}
$$

Answer:
$$
\boxed{\frac{224}{81}}
$$

---

#### 10. $\left(\frac{5}{7} - \left(\frac{10}{9} - \frac{9}{7}\right)\right) \div (1 - (-1))$

1. Simplify inside the inner parentheses:
$$
\frac{10}{9} - \frac{9}{7} = \frac{10 \times 7}{63} - \frac{9 \times 9}{63} = \frac{70}{63} - \frac{81}{63} = \frac{70 - 81}{63} = \frac{-11}{63}
$$

2. Simplify the outer parentheses:
$$
\frac{5}{7} - \left(\frac{-11}{63}\right) = \frac{5}{7} + \frac{11}{63} = \frac{5 \times 9}{63} + \frac{11}{63} = \frac{45}{63} + \frac{11}{63} = \frac{45 + 11}{63} = \frac{56}{63} = \frac{8}{9}
$$

3. Simplify the divisor:
$$
1 - (-1) = 1 + 1 = 2
$$

4. Divide:
$$
\frac{8}{9} \div 2 = \frac{8}{9} \times \frac{1}{2} = \frac{8 \times 1}{9 \times 2} = \frac{8}{18} = \frac{4}{9}
$$

Answer:
$$
\boxed{\frac{4}{9}}
$$

---

#### 11. $\left(\frac{-3}{2} \times \frac{-11}{7} \times -1\right) \div \left(\frac{-7}{4} - 2\right)$

1. Simplify inside the first set of parentheses:
$$
\frac{-3}{2} \times \frac{-11}{7} \times -1 = \frac{-3 \times -11 \times -1}{2 \times 7} = \frac{-33}{14}
$$

2. Simplify inside the second set of parentheses:
$$
\frac{-7}{4} - 2 = \frac{-7}{4} - \frac{8}{4} = \frac{-7 - 8}{4} = \frac{-15}{4}
$$

3. Divide:
$$
\frac{-33}{14} \div \frac{-15}{4} = \frac{-33}{14} \times \frac{4}{-15} = \frac{-33 \times 4}{14 \times -15} = \frac{-132}{-210} = \frac{132}{210} = \frac{22}{35}
$$

Answer:
$$
\boxed{\frac{22}{35}}
$$

---

#### 12. $\left(\frac{-1}{4} - \frac{3}{2} \times \frac{2}{3} - 2\right) \div \frac{7}{4}$

1. Simplify inside the parentheses:
$$
\frac{3}{2} \times \frac{2}{3} = 1
$$
$$
\frac{-1}{4} - 1 - 2 = \frac{-1}{4} - \frac{4}{4} - \frac{8}{4} = \frac{-1 - 4 - 8}{4} = \frac{-13}{4}
$$

2. Divide by $\frac{7}{4}$:
$$
\frac{-13}{4} \div \frac{7}{4} = \frac{-13}{4} \times \frac{4}{7} = \frac{-13 \times 4}{4 \times 7} = \frac{-13}{7}
$$

Answer:
$$
\boxed{\frac{-13}{7}}
$$

---

#### 13. $\left(-6 \times \frac{-4}{3} \times \frac{-13}{9}\right) \div \left(\frac{2}{5} - \frac{4}{3}\right)$

1. Simplify inside the first set of parentheses:
$$
-6 \times \frac{-4}{3} \times \frac{-13}{9} = \frac{-6 \times -4 \times -13}{3 \times 9} = \frac{-312}{27} = \frac{-104}{9}
$$

2. Simplify inside the second set of parentheses:
$$
\frac{2}{5} - \frac{4}{3} = \frac{2 \times 3}{15} - \frac{4 \times 5}{15} = \frac{6}{15} - \frac{20}{15} = \frac{6 - 20}{15} = \frac{-14}{15}
$$

3. Divide:
$$
\frac{-104}{9} \div \frac{-14}{15} = \frac{-104}{9} \times \frac{15}{-14} = \frac{-104 \times 15}{9 \times -14} = \frac{-1560}{-126} = \frac{1560}{126} = \frac{260}{21}
$$

Answer:
$$
\boxed{\frac{260}{21}}
$$

---

#### 14. $\frac{-11}{9} \div \left(\frac{-3}{2} \left(\frac{-5}{3} - \frac{7}{4}\right) \times 2\right)$

1. Simplify inside the innermost parentheses:
$$
\frac{-5}{3} - \frac{7}{4} = \frac{-5 \times 4}{12} - \frac{7 \times 3}{12} = \frac{-20}{12} - \frac{21}{12} = \frac{-20 - 21}{12} = \frac{-41}{12}
$$

2. Multiply by $\frac{-3}{2}$:
$$
\frac{-3}{2} \times \frac{-41}{12} = \frac{-3 \times -41}{2 \times 12} = \frac{123}{24} = \frac{41}{8}
$$

3. Multiply by 2:
$$
\frac{41}{8} \times 2 = \frac{41 \times 2}{8} = \frac{82}{8} = \frac{41}{4}
$$

4. Divide:
$$
\frac{-11}{9} \div \frac{41}{4} = \frac{-11}{9} \times \frac{4}{41} = \frac{-11 \times 4}{9 \times 41} = \frac{-44}{369}
$$

Answer:
$$
\boxed{\frac{-44}{369}}
$$

---

#### 15. $-1 + \frac{-5}{7} + 2 + \frac{1}{5} \times \frac{3}{10}$

1. Simplify the multiplication:
$$
\frac{1}{5} \times \frac{3}{10} = \frac{1 \times 3}{5 \times 10} = \frac{3}{50}
$$

2. Combine all terms:
$$
-1 + \frac{-5}{7} + 2 + \frac{3}{50}
$$
Convert all terms to have a common denominator (350):
$$
-1 = \frac{-350}{350}, \quad \frac{-5}{7} = \frac{-250}{350}, \quad 2 = \frac{700}{350}, \quad \frac{3}{50} = \frac{21}{350}
$$
$$
\frac{-350}{350} + \frac{-250}{350} + \frac{700}{350} + \frac{21}{350} = \frac{-350 - 250 + 700 + 21}{350} = \frac{121}{350}
$$

Answer:
$$
\boxed{\frac{121}{350}}
$$

---

#### 16. $\frac{5}{6} \times 2 + \frac{3}{7} - \frac{-8}{7} \div \frac{5}{4}$

1. Simplify the multiplication:
$$
\frac{5}{6} \times 2 = \frac{5 \times 2}{6} = \frac{10}{6} = \frac{5}{3}
$$

2. Simplify the division:
$$
\frac{-8}{7} \div \frac{5}{4} = \frac{-8}{7} \times \frac{4}{5} = \frac{-8 \times 4}{7 \times 5} = \frac{-32}{35}
$$

3. Combine all terms:
$$
\frac{5}{3} + \frac{3}{7} - \frac{-32}{35} = \frac{5}{3} + \frac{3}{7} + \frac{32}{35}
$$
Find a common denominator (105):
$$
\frac{5}{3} = \frac{5 \times 35}{105} = \frac{175}{105}, \quad \frac{3}{7} = \frac{3 \times 15}{105} = \frac{45}{105}, \quad \frac{32}{35} = \frac{32 \times 3}{105} = \frac{96}{105}
$$
$$
\frac{175}{105} + \frac{45}{105} + \frac{96}{105} = \frac{175 + 45 + 96}{105} = \frac{316}{105}
$$

Answer:
$$
\boxed{\frac{316}{105}}
$$

---

Final Answers:

1. $\boxed{\frac{-247}{270}}$
2. $\boxed{\frac{8359}{8100}}$
3. $\boxed{\frac{106}{15}}$
4. $\boxed{-3}$
5. $\boxed{\frac{-138}{35}}$
6. $\boxed{\frac{-1197}{32}}$
7. $\boxed{\frac{-405}{28}}$
8. $\boxed{\frac{-400}{27}}$
9. $\boxed{\frac{224}{81}}$
10. $\boxed{\frac{4}{9}}$
11. $\boxed{\frac{22}{35}}$
12. $\boxed{\frac{-13}{7}}$
13. $\boxed{\frac{260}{21}}$
14. $\boxed{\frac{-44}{369}}$
15. $\boxed{\frac{121}{350}}$
16. $\boxed{\frac{316}{105}}$