Algebraic fractions worksheet for simplifying expressions through multiplication and division.

Let's solve each of the algebraic fraction problems step by step. We'll simplify each expression and express the answer as a single fraction in simplest form.

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1. $\frac{z}{3} \times \frac{z}{2}$

Multiply numerators and denominators:
$$
\frac{z \cdot z}{3 \cdot 2} = \frac{z^2}{6}
$$
✔ Answer: $\boxed{\frac{z^2}{6}}$

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2. $\frac{2z}{3} \times \frac{6}{z^2}$

Multiply:
$$
\frac{2z \cdot 6}{3 \cdot z^2} = \frac{12z}{3z^2}
$$
Simplify:
- $12/3 = 4$
- $z/z^2 = 1/z$
$$
= \frac{4}{z}
$$
✔ Answer: $\boxed{\frac{4}{z}}$

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3. $\frac{a}{4} \times \frac{6b}{5}$

Multiply:
$$
\frac{a \cdot 6b}{4 \cdot 5} = \frac{6ab}{20}
$$
Simplify: divide numerator and denominator by 2:
$$
= \frac{3ab}{10}
$$
✔ Answer: $\boxed{\frac{3ab}{10}}$

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4. $\frac{2c}{5} \times \frac{10}{3c}$

Multiply:
$$
\frac{2c \cdot 10}{5 \cdot 3c} = \frac{20c}{15c}
$$
Cancel $c$ (assuming $c \neq 0$):
$$
= \frac{20}{15} = \frac{4}{3}
$$
✔ Answer: $\boxed{\frac{4}{3}}$

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5. $\frac{4y}{3} \times \frac{15z}{8y}$

Multiply:
$$
\frac{4y \cdot 15z}{3 \cdot 8y} = \frac{60yz}{24y}
$$
Cancel $y$:
$$
= \frac{60z}{24} = \frac{5z}{2}
$$
(Simplify $60/24 = 5/2$)
✔ Answer: $\boxed{\frac{5z}{2}}$

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6. $\frac{2y}{3} \div \frac{y}{6}$

Dividing fractions: multiply by reciprocal:
$$
\frac{2y}{3} \times \frac{6}{y} = \frac{2y \cdot 6}{3 \cdot y} = \frac{12y}{3y}
$$
Cancel $y$: $12/3 = 4$
$$
= 4
$$
✔ Answer: $\boxed{4}$

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7. $\frac{4c}{3} \div \frac{8y}{9}$

Multiply by reciprocal:
$$
\frac{4c}{3} \times \frac{9}{8y} = \frac{4c \cdot 9}{3 \cdot 8y} = \frac{36c}{24y}
$$
Simplify: divide numerator and denominator by 12:
$$
= \frac{3c}{2y}
$$
✔ Answer: $\boxed{\frac{3c}{2y}}$

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8. $\frac{6}{5y} \div \frac{3}{y}$

Multiply by reciprocal:
$$
\frac{6}{5y} \times \frac{y}{3} = \frac{6 \cdot y}{5y \cdot 3} = \frac{6y}{15y}
$$
Cancel $y$: $6/15 = 2/5$
$$
= \frac{2}{5}
$$
✔ Answer: $\boxed{\frac{2}{5}}$

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9. $\frac{4y}{3z} \div \frac{16y^2}{9}$

Multiply by reciprocal:
$$
\frac{4y}{3z} \times \frac{9}{16y^2} = \frac{4y \cdot 9}{3z \cdot 16y^2} = \frac{36y}{48zy^2}
$$
Simplify:
- $36/48 = 3/4$
- $y/y^2 = 1/y$
So:
$$
= \frac{3}{4zy}
$$
✔ Answer: $\boxed{\frac{3}{4zy}}$

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10. $\frac{6z}{25} \div \frac{4z^2}{5}$

Multiply by reciprocal:
$$
\frac{6z}{25} \times \frac{5}{4z^2} = \frac{6z \cdot 5}{25 \cdot 4z^2} = \frac{30z}{100z^2}
$$
Simplify:
- $30/100 = 3/10$
- $z/z^2 = 1/z$
$$
= \frac{3}{10z}
$$
✔ Answer: $\boxed{\frac{3}{10z}}$

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✔ Final Answers:

1. $\frac{z^2}{6}$
2. $\frac{4}{z}$
3. $\frac{3ab}{10}$
4. $\frac{4}{3}$
5. $\frac{5z}{2}$
6. $4$
7. $\frac{3c}{2y}$
8. $\frac{2}{5}$
9. $\frac{3}{4zy}$
10. $\frac{3}{10z}$

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