Fraction operations with variables: multiplication and division exercises.

Since both ExplainAnswer and TeachTopic are false, we skip those sections entirely.

We will now solve each expression in the image step by step, simplify carefully, and then provide the final answers in order.

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Let’s go one by one:

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1. $\frac{3}{5} \times \frac{x}{y}$

Multiply numerators: $3 \cdot x = 3x$
Multiply denominators: $5 \cdot y = 5y$
→ Answer: $\boxed{\frac{3x}{5y}}$

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2. $\frac{7x}{xy} \times \frac{3}{6y}$

First, simplify $\frac{7x}{xy}$ → cancel $x$: $\frac{7}{y}$
Now multiply: $\frac{7}{y} \times \frac{3}{6y} = \frac{7 \cdot 3}{y \cdot 6y} = \frac{21}{6y^2}$
Simplify $\frac{21}{6}$ → divide numerator and denominator by 3: $\frac{7}{2}$
→ Final: $\boxed{\frac{7}{2y^2}}$

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3. $\frac{x}{y} \times \frac{3}{5}$

Multiply: $\frac{x \cdot 3}{y \cdot 5} = \frac{3x}{5y}$
→ Answer: $\boxed{\frac{3x}{5y}}$

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4. $\frac{7x}{xy} \div \frac{3}{6y}$

Dividing by a fraction = multiplying by its reciprocal:
$\frac{7x}{xy} \times \frac{6y}{3}$

Simplify $\frac{7x}{xy}$ → cancel $x$: $\frac{7}{y}$
Now: $\frac{7}{y} \times \frac{6y}{3}$

Cancel $y$: $\frac{7}{1} \times \frac{6}{3} = 7 \times 2 = 14$
→ Answer: $\boxed{14}$

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5. $\frac{x}{y} \times \frac{3}{6}$

Simplify $\frac{3}{6} = \frac{1}{2}$
So: $\frac{x}{y} \times \frac{1}{2} = \frac{x}{2y}$
→ Answer: $\boxed{\frac{x}{2y}}$

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6. $\frac{7x}{xy} \div \frac{6y}{3}$

Reciprocal of $\frac{6y}{3}$ is $\frac{3}{6y}$
So: $\frac{7x}{xy} \times \frac{3}{6y}$

Simplify $\frac{7x}{xy}$ → $\frac{7}{y}$
Then: $\frac{7}{y} \times \frac{3}{6y} = \frac{21}{6y^2} = \frac{7}{2y^2}$ (same as #2)
→ Answer: $\boxed{\frac{7}{2y^2}}$

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7. $\frac{x}{y} \times \frac{3}{6x}$

Simplify $\frac{3}{6x} = \frac{1}{2x}$
So: $\frac{x}{y} \times \frac{1}{2x}$

Cancel $x$: $\frac{1}{y} \times \frac{1}{2} = \frac{1}{2y}$
→ Answer: $\boxed{\frac{1}{2y}}$

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8. $\frac{7x}{xy} \times \frac{6y}{3}$

Simplify $\frac{7x}{xy}$ → $\frac{7}{y}$
Then: $\frac{7}{y} \times \frac{6y}{3}$

Cancel $y$: $\frac{7}{1} \times \frac{6}{3} = 7 \times 2 = 14$
→ Answer: $\boxed{14}$

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9. $\frac{x}{xy} \times \frac{3}{6}$

Simplify $\frac{x}{xy}$ → $\frac{1}{y}$
Then: $\frac{1}{y} \times \frac{3}{6} = \frac{1}{y} \times \frac{1}{2} = \frac{1}{2y}$
→ Answer: $\boxed{\frac{1}{2y}}$

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10. $\frac{7x}{xy} \times \frac{6y}{3} \times \frac{4y}{14}$

Step 1: Simplify $\frac{7x}{xy}$ → $\frac{7}{y}$
Step 2: $\frac{6y}{3} = 2y$
Step 3: So far: $\frac{7}{y} \times 2y \times \frac{4y}{14}$

Cancel $y$ in first two: $\frac{7}{1} \times 2 \times \frac{4y}{14} = 14 \times \frac{4y}{14}$

Cancel 14: $1 \times 4y = 4y$
→ Answer: $\boxed{4y}$

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11. $\frac{7x}{xy} \times \frac{3}{6}$

Simplify $\frac{7x}{xy}$ → $\frac{7}{y}$
Then: $\frac{7}{y} \times \frac{3}{6} = \frac{7}{y} \times \frac{1}{2} = \frac{7}{2y}$
→ Answer: $\boxed{\frac{7}{2y}}$

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12. $\frac{7x}{xy} \div \frac{6y}{3} \times \frac{4y}{14}$

First, handle division: $\frac{7x}{xy} \div \frac{6y}{3} = \frac{7x}{xy} \times \frac{3}{6y}$

As before, $\frac{7x}{xy} = \frac{7}{y}$, so: $\frac{7}{y} \times \frac{3}{6y} = \frac{21}{6y^2} = \frac{7}{2y^2}$

Now multiply by $\frac{4y}{14}$:
$\frac{7}{2y^2} \times \frac{4y}{14}$

Multiply numerators: $7 \cdot 4y = 28y$
Denominators: $2y^2 \cdot 14 = 28y^2$
So: $\frac{28y}{28y^2} = \frac{1}{y}$

→ Answer: $\boxed{\frac{1}{y}}$

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Final Answers in order (top to bottom, left to right):

1. $\frac{3x}{5y}$
2. $\frac{7}{2y^2}$
3. $\frac{3x}{5y}$
4. $14$
5. $\frac{x}{2y}$
6. $\frac{7}{2y^2}$
7. $\frac{1}{2y}$
8. $14$
9. $\frac{1}{2y}$
10. $4y$
11. $\frac{7}{2y}$
12. $\frac{1}{y}$

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Final Answer:
$\frac{3x}{5y},\ \frac{7}{2y^2},\ \frac{3x}{5y},\ 14,\ \frac{x}{2y},\ \frac{7}{2y^2},\ \frac{1}{2y},\ 14,\ \frac{1}{2y},\ 4y,\ \frac{7}{2y},\ \frac{1}{y}$