Simplify the given complex fractions in this math worksheet.

Explanation:
Let's simplify each complex fraction step by step.

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① $\frac{5 - 2x}{x^2 + 2x}$
Factor the denominator: $x^2 + 2x = x(x + 2)$
Numerator: $5 - 2x$ — no common factors with denominator
So the expression is: $\frac{5 - 2x}{x(x + 2)}$
No simplification possible.
Final: $\frac{5 - 2x}{x(x + 2)}$

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② $\frac{xy(x + y)}{3y^2 + 5x^2}$
Numerator: $xy(x + y)$
Denominator: $3y^2 + 5x^2$ — cannot factor easily, and no common terms with numerator
No common factors
Final: $\frac{xy(x + y)}{3y^2 + 5x^2}$

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③ $\frac{\frac{a}{b} + \frac{x}{y}}{\frac{a}{z} + \frac{x}{c}}$
This is a complex fraction. To simplify, multiply numerator and denominator by the least common denominator (LCD) of all the small fractions.

LCD of $b, y, z, c$ is $b y z c$
Multiply numerator and denominator by $b y z c$:

Numerator:
$\left(\frac{a}{b} + \frac{x}{y}\right) \cdot b y z c = a y z c + x b z c$

Denominator:
$\left(\frac{a}{z} + \frac{x}{c}\right) \cdot b y z c = a b y c + x b y z$

So the expression becomes:
$\frac{a y z c + x b z c}{a b y c + x b y z}$

Factor numerator and denominator:

Numerator: $z c (a y + x b)$
Denominator: $b y (a c + x z)$

So: $\frac{z c (a y + b x)}{b y (a c + x z)}$

No further simplification.
Final: $\frac{z c (a y + b x)}{b y (a c + x z)}$

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④ $\frac{\frac{1}{x} + \frac{2}{x}}{x + \frac{2}{x^2}}$
Simplify numerator: $\frac{1}{x} + \frac{2}{x} = \frac{3}{x}$
Denominator: $x + \frac{2}{x^2} = \frac{x^3 + 2}{x^2}$

Now: $\frac{3/x}{(x^3 + 2)/x^2} = \frac{3}{x} \cdot \frac{x^2}{x^3 + 2} = \frac{3x}{x^3 + 2}$

Final: $\frac{3x}{x^3 + 2}$

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⑤ $\frac{x + \frac{2d}{3ac}}{x + \frac{3d}{2ac}}$
Multiply numerator and denominator by $6ac$ (LCM of denominators 3ac and 2ac)

Numerator: $(x + \frac{2d}{3ac}) \cdot 6ac = 6ac \cdot x + 6ac \cdot \frac{2d}{3ac} = 6acx + 4d$

Denominator: $(x + \frac{3d}{2ac}) \cdot 6ac = 6acx + 6ac \cdot \frac{3d}{2ac} = 6acx + 9d$

So: $\frac{6acx + 4d}{6acx + 9d}$

No common factors, so this is simplified.

Final: $\frac{6acx + 4d}{6acx + 9d}$

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⑥ $\frac{5a}{2(a + 2)}$
Numerator: $5a$
Denominator: $2(a + 2)$
No common factors
Final: $\frac{5a}{2(a + 2)}$

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⑦ $\frac{x}{2 - 3x}$
This is already simplified.
No common factors.
Final: $\frac{x}{2 - 3x}$

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⑧ $\frac{\frac{x + 1}{3}}{\frac{2x - 1}{3}}$
Divide fractions: $\frac{x + 1}{3} \div \frac{2x - 1}{3} = \frac{x + 1}{3} \cdot \frac{3}{2x - 1} = \frac{x + 1}{2x - 1}$

Final: $\frac{x + 1}{2x - 1}$

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⑨ $\frac{1 - \frac{x}{y}}{\frac{x^2}{y^2} - 1}$
Simplify numerator: $1 - \frac{x}{y} = \frac{y - x}{y}$
Denominator: $\frac{x^2}{y^2} - 1 = \frac{x^2 - y^2}{y^2} = \frac{(x - y)(x + y)}{y^2}$

Now: $\frac{(y - x)/y}{(x^2 - y^2)/y^2} = \frac{y - x}{y} \cdot \frac{y^2}{x^2 - y^2}$

Note: $y - x = -(x - y)$, and $x^2 - y^2 = (x - y)(x + y)$

So: $\frac{-(x - y)}{y} \cdot \frac{y^2}{(x - y)(x + y)} = -\frac{y}{x + y}$

Final: $-\frac{y}{x + y}$

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⑩ $\frac{\frac{x - 3}{x^2 - 25}}{\frac{x^2 - 9}{x - 5}}$
First, simplify the complex fraction:
$\frac{x - 3}{x^2 - 25} \div \frac{x^2 - 9}{x - 5} = \frac{x - 3}{x^2 - 25} \cdot \frac{x - 5}{x^2 - 9}$

Now factor:

$x^2 - 25 = (x - 5)(x + 5)$
$x^2 - 9 = (x - 3)(x + 3)$

So:
$\frac{x - 3}{(x - 5)(x + 5)} \cdot \frac{x - 5}{(x - 3)(x + 3)}$

Cancel common terms:
$(x - 3)$ and $(x - 5)$ cancel

Left with: $\frac{1}{(x + 5)(x + 3)}$

Final: $\frac{1}{(x + 5)(x + 3)}$

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Final Answer:
① $\frac{5 - 2x}{x(x + 2)}$
② $\frac{xy(x + y)}{3y^2 + 5x^2}$
③ $\frac{z c (a y + b x)}{b y (a c + x z)}$
④ $\frac{3x}{x^3 + 2}$
⑤ $\frac{6acx + 4d}{6acx + 9d}$
⑥ $\frac{5a}{2(a + 2)}$
⑦ $\frac{x}{2 - 3x}$
⑧ $\frac{x + 1}{2x - 1}$
⑨ $-\frac{y}{x + y}$
⑩ $\frac{1}{(x + 5)(x + 3)}$