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Problem: Multiplying Fractions Using Cross-Cancellation

The task is to solve the given multiplication problems involving fractions using cross-cancellation. Let's go through each problem step by step.

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#### 1) \( \frac{9}{10} \times \frac{2}{3} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 9, and the denominator of the second fraction is 3. They share a common factor of 3.
- The numerator of the second fraction is 2, and the denominator of the first fraction is 10. They do not share any common factors.

- Step 2: Perform cross-cancellation.
- Divide 9 by 3: \( 9 \div 3 = 3 \)
- Divide 3 by 3: \( 3 \div 3 = 1 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 3 \times 2 = 6 \)
- Denominator: \( 10 \times 1 = 10 \)

- Step 4: Simplify the fraction.
- \( \frac{6}{10} \) can be simplified by dividing both numerator and denominator by their greatest common divisor (GCD), which is 2.
- \( \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \)

- Final Answer: \( \boxed{\frac{3}{5}} \)

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#### 2) \( \frac{12}{8} \times \frac{18}{16} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 12, and the denominator of the second fraction is 16. They share a common factor of 4.
- The numerator of the second fraction is 18, and the denominator of the first fraction is 8. They do not share any common factors.

- Step 2: Perform cross-cancellation.
- Divide 12 by 4: \( 12 \div 4 = 3 \)
- Divide 16 by 4: \( 16 \div 4 = 4 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 3 \times 18 = 54 \)
- Denominator: \( 8 \times 4 = 32 \)

- Step 4: Simplify the fraction.
- \( \frac{54}{32} \) can be simplified by dividing both numerator and denominator by their GCD, which is 2.
- \( \frac{54 \div 2}{32 \div 2} = \frac{27}{16} \)
- Convert to a mixed number: \( \frac{27}{16} = 1 \frac{11}{16} \)

- Final Answer: \( \boxed{\frac{27}{16} \text{ or } 1 \frac{11}{16}} \)

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#### 3) \( \frac{33}{7} \times \frac{14}{21} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 33, and the denominator of the second fraction is 21. They do not share any common factors.
- The numerator of the second fraction is 14, and the denominator of the first fraction is 7. They share a common factor of 7.

- Step 2: Perform cross-cancellation.
- Divide 14 by 7: \( 14 \div 7 = 2 \)
- Divide 7 by 7: \( 7 \div 7 = 1 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 33 \times 2 = 66 \)
- Denominator: \( 1 \times 21 = 21 \)

- Step 4: Simplify the fraction.
- \( \frac{66}{21} \) can be simplified by dividing both numerator and denominator by their GCD, which is 3.
- \( \frac{66 \div 3}{21 \div 3} = \frac{22}{7} \)
- Convert to a mixed number: \( \frac{22}{7} = 3 \frac{1}{7} \)

- Final Answer: \( \boxed{\frac{22}{7} \text{ or } 3 \frac{1}{7}} \)

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#### 4) \( \frac{6}{18} \times \frac{9}{42} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 6, and the denominator of the second fraction is 42. They share a common factor of 6.
- The numerator of the second fraction is 9, and the denominator of the first fraction is 18. They share a common factor of 9.

- Step 2: Perform cross-cancellation.
- Divide 6 by 6: \( 6 \div 6 = 1 \)
- Divide 42 by 6: \( 42 \div 6 = 7 \)
- Divide 9 by 9: \( 9 \div 9 = 1 \)
- Divide 18 by 9: \( 18 \div 9 = 2 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 1 \times 1 = 1 \)
- Denominator: \( 2 \times 7 = 14 \)

- Final Answer: \( \boxed{\frac{1}{14}} \)

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#### 5) \( \frac{22}{15} \times \frac{45}{4} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 22, and the denominator of the second fraction is 4. They do not share any common factors.
- The numerator of the second fraction is 45, and the denominator of the first fraction is 15. They share a common factor of 15.

- Step 2: Perform cross-cancellation.
- Divide 45 by 15: \( 45 \div 15 = 3 \)
- Divide 15 by 15: \( 15 \div 15 = 1 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 22 \times 3 = 66 \)
- Denominator: \( 1 \times 4 = 4 \)

- Step 4: Simplify the fraction.
- \( \frac{66}{4} \) can be simplified by dividing both numerator and denominator by their GCD, which is 2.
- \( \frac{66 \div 2}{4 \div 2} = \frac{33}{2} \)
- Convert to a mixed number: \( \frac{33}{2} = 16 \frac{1}{2} \)

- Final Answer: \( \boxed{\frac{33}{2} \text{ or } 16 \frac{1}{2}} \)

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#### 6) \( \frac{3}{28} \times \frac{35}{6} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 3, and the denominator of the second fraction is 6. They share a common factor of 3.
- The numerator of the second fraction is 35, and the denominator of the first fraction is 28. They share a common factor of 7.

- Step 2: Perform cross-cancellation.
- Divide 3 by 3: \( 3 \div 3 = 1 \)
- Divide 6 by 3: \( 6 \div 3 = 2 \)
- Divide 35 by 7: \( 35 \div 7 = 5 \)
- Divide 28 by 7: \( 28 \div 7 = 4 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 1 \times 5 = 5 \)
- Denominator: \( 4 \times 2 = 8 \)

- Final Answer: \( \boxed{\frac{5}{8}} \)

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#### 7) \( \frac{2}{7} \times \frac{35}{12} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 2, and the denominator of the second fraction is 12. They share a common factor of 2.
- The numerator of the second fraction is 35, and the denominator of the first fraction is 7. They share a common factor of 7.

- Step 2: Perform cross-cancellation.
- Divide 2 by 2: \( 2 \div 2 = 1 \)
- Divide 12 by 2: \( 12 \div 2 = 6 \)
- Divide 35 by 7: \( 35 \div 7 = 5 \)
- Divide 7 by 7: \( 7 \div 7 = 1 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 1 \times 5 = 5 \)
- Denominator: \( 1 \times 6 = 6 \)

- Final Answer: \( \boxed{\frac{5}{6}} \)

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#### 8) \( \frac{16}{15} \times \frac{21}{24} \)

- Step 1: Identify common factors in the numerator and denominator.
- The numerator of the first fraction is 16, and the denominator of the second fraction is 24. They share a common factor of 8.
- The numerator of the second fraction is 21, and the denominator of the first fraction is 15. They share a common factor of 3.

- Step 2: Perform cross-cancellation.
- Divide 16 by 8: \( 16 \div 8 = 2 \)
- Divide 24 by 8: \( 24 \div 8 = 3 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- Divide 15 by 3: \( 15 \div 3 = 5 \)

- Step 3: Multiply the remaining numbers.
- Numerator: \( 2 \times 7 = 14 \)
- Denominator: \( 5 \times 3 = 15 \)

- Final Answer: \( \boxed{\frac{14}{15}} \)

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Final Answers:

1. \( \boxed{\frac{3}{5}} \)
2. \( \boxed{\frac{27}{16} \text{ or } 1 \frac{11}{16}} \)
3. \( \boxed{\frac{22}{7} \text{ or } 3 \frac{1}{7}} \)
4. \( \boxed{\frac{1}{14}} \)
5. \( \boxed{\frac{33}{2} \text{ or } 16 \frac{1}{2}} \)
6. \( \boxed{\frac{5}{8}} \)
7. \( \boxed{\frac{5}{6}} \)
8. \( \boxed{\frac{14}{15}} \)

Boxed Final Answer:
\[
\boxed{\frac{3}{5}, \frac{27}{16} \text{ or } 1 \frac{11}{16}, \frac{22}{7} \text{ or } 3 \frac{1}{7}, \frac{1}{14}, \frac{33}{2} \text{ or } 16 \frac{1}{2}, \frac{5}{8}, \frac{5}{6}, \frac{14}{15}}
\]