Multiplying Algebraic Fractions worksheet for simplifying expressions with variables and fractions.

To solve the problem of simplifying the given algebraic fractions, we will multiply the numerators together and the denominators together, then simplify the resulting expression by canceling out common factors. Let's go through each problem step by step.

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Problem 1:

$$
\frac{5x}{2} \times \frac{6}{x}
$$

1. Multiply the numerators: \( 5x \times 6 = 30x \).
2. Multiply the denominators: \( 2 \times x = 2x \).
3. Simplify the fraction:
$$
\frac{30x}{2x} = \frac{30}{2} = 15
$$

Answer:
$$
\boxed{15}
$$

---

Problem 2:

$$
\frac{x}{3} \times \frac{21}{x+1}
$$

1. Multiply the numerators: \( x \times 21 = 21x \).
2. Multiply the denominators: \( 3 \times (x+1) = 3(x+1) \).
3. Simplify the fraction:
$$
\frac{21x}{3(x+1)} = \frac{7x}{x+1}
$$

Answer:
$$
\boxed{\frac{7x}{x+1}}
$$

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Problem 3:

$$
\frac{4x}{7} \times \frac{35}{6x}
$$

1. Multiply the numerators: \( 4x \times 35 = 140x \).
2. Multiply the denominators: \( 7 \times 6x = 42x \).
3. Simplify the fraction:
$$
\frac{140x}{42x} = \frac{140}{42} = \frac{10}{3}
$$

Answer:
$$
\boxed{\frac{10}{3}}
$$

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Problem 4:

$$
\frac{x+5}{8} \times \frac{4}{2x+10}
$$

1. Multiply the numerators: \( (x+5) \times 4 = 4(x+5) \).
2. Multiply the denominators: \( 8 \times (2x+10) = 8(2x+10) \).
3. Factorize \( 2x+10 \) as \( 2(x+5) \):
$$
\frac{4(x+5)}{8 \cdot 2(x+5)} = \frac{4(x+5)}{16(x+5)}
$$
4. Cancel out the common factor \( x+5 \):
$$
\frac{4}{16} = \frac{1}{4}
$$

Answer:
$$
\boxed{\frac{1}{4}}
$$

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Problem 5:

$$
\frac{x+3}{4} \times \frac{3}{2x+6}
$$

1. Multiply the numerators: \( (x+3) \times 3 = 3(x+3) \).
2. Multiply the denominators: \( 4 \times (2x+6) = 4(2x+6) \).
3. Factorize \( 2x+6 \) as \( 2(x+3) \):
$$
\frac{3(x+3)}{4 \cdot 2(x+3)} = \frac{3(x+3)}{8(x+3)}
$$
4. Cancel out the common factor \( x+3 \):
$$
\frac{3}{8}
$$

Answer:
$$
\boxed{\frac{3}{8}}
$$

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Problem 6:

$$
\frac{14x}{7} \times \frac{7}{2x}
$$

1. Multiply the numerators: \( 14x \times 7 = 98x \).
2. Multiply the denominators: \( 7 \times 2x = 14x \).
3. Simplify the fraction:
$$
\frac{98x}{14x} = \frac{98}{14} = 7
$$

Answer:
$$
\boxed{7}
$$

---

Problem 7:

$$
\frac{3}{2x} \times \frac{x}{15}
$$

1. Multiply the numerators: \( 3 \times x = 3x \).
2. Multiply the denominators: \( 2x \times 15 = 30x \).
3. Simplify the fraction:
$$
\frac{3x}{30x} = \frac{3}{30} = \frac{1}{10}
$$

Answer:
$$
\boxed{\frac{1}{10}}
$$

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Problem 8:

$$
\frac{2x+4}{18} \times \frac{3}{5}
$$

1. Multiply the numerators: \( (2x+4) \times 3 = 3(2x+4) \).
2. Multiply the denominators: \( 18 \times 5 = 90 \).
3. Factorize \( 2x+4 \) as \( 2(x+2) \):
$$
\frac{3 \cdot 2(x+2)}{90} = \frac{6(x+2)}{90}
$$
4. Simplify the fraction:
$$
\frac{6(x+2)}{90} = \frac{x+2}{15}
$$

Answer:
$$
\boxed{\frac{x+2}{15}}
$$

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Problem 9:

$$
\frac{x}{4} \times \frac{16}{9x}
$$

1. Multiply the numerators: \( x \times 16 = 16x \).
2. Multiply the denominators: \( 4 \times 9x = 36x \).
3. Simplify the fraction:
$$
\frac{16x}{36x} = \frac{16}{36} = \frac{4}{9}
$$

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

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Problem 10:

$$
\frac{6x}{3} \times \frac{7}{4x}
$$

1. Multiply the numerators: \( 6x \times 7 = 42x \).
2. Multiply the denominators: \( 3 \times 4x = 12x \).
3. Simplify the fraction:
$$
\frac{42x}{12x} = \frac{42}{12} = \frac{7}{2}
$$

Answer:
$$
\boxed{\frac{7}{2}}
$$

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Problem 11:

$$
\frac{2x}{x+5} \times \frac{3x+15}{4}
$$

1. Multiply the numerators: \( 2x \times (3x+15) = 2x(3x+15) \).
2. Multiply the denominators: \( (x+5) \times 4 = 4(x+5) \).
3. Factorize \( 3x+15 \) as \( 3(x+5) \):
$$
\frac{2x \cdot 3(x+5)}{4(x+5)} = \frac{6x(x+5)}{4(x+5)}
$$
4. Cancel out the common factor \( x+5 \):
$$
\frac{6x}{4} = \frac{3x}{2}
$$

Answer:
$$
\boxed{\frac{3x}{2}}
$$

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Problem 12:

$$
\frac{x}{x-4} \times \frac{2x-8}{3}
$$

1. Multiply the numerators: \( x \times (2x-8) = x(2x-8) \).
2. Multiply the denominators: \( (x-4) \times 3 = 3(x-4) \).
3. Factorize \( 2x-8 \) as \( 2(x-4) \):
$$
\frac{x \cdot 2(x-4)}{3(x-4)} = \frac{2x(x-4)}{3(x-4)}
$$
4. Cancel out the common factor \( x-4 \):
$$
\frac{2x}{3}
$$

Answer:
$$
\boxed{\frac{2x}{3}}
$$

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Final Answers:

1. \( \boxed{15} \)
2. \( \boxed{\frac{7x}{x+1}} \)
3. \( \boxed{\frac{10}{3}} \)
4. \( \boxed{\frac{1}{4}} \)
5. \( \boxed{\frac{3}{8}} \)
6. \( \boxed{7} \)
7. \( \boxed{\frac{1}{10}} \)
8. \( \boxed{\frac{x+2}{15}} \)
9. \( \boxed{\frac{4}{9}} \)
10. \( \boxed{\frac{7}{2}} \)
11. \( \boxed{\frac{3x}{2}} \)
12. \( \boxed{\frac{2x}{3}} \)