Math worksheet for simplifying rational expressions with common denominators.

Let's solve each of these problems step by step. All of them involve adding or subtracting rational expressions with common denominators. When the denominators are the same, we can combine the numerators directly and then simplify if possible.

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Problem 1:

$$
\frac{x + 4y}{18y^2x^3} + \frac{x + 3y}{18y^2x^3}
$$

Step 1: Same denominator → Add the numerators:
$$
\frac{(x + 4y) + (x + 3y)}{18y^2x^3} = \frac{2x + 7y}{18y^2x^3}
$$

✔ Answer: $\boxed{\frac{2x + 7y}{18y^2x^3}}$

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Problem 2:

$$
\frac{x + 1}{12x^2 + 30x} + \frac{5x + 4}{12x^2 + 30x}
$$

Step 1: Combine numerators:
$$
\frac{(x + 1) + (5x + 4)}{12x^2 + 30x} = \frac{6x + 5}{12x^2 + 30x}
$$

Step 2: Factor denominator:
$$
12x^2 + 30x = 6x(2x + 5)
$$

Numerator: $6x + 5$ — doesn't factor nicely with denominator.

So no further simplification.

✔ Answer: $\boxed{\frac{6x + 5}{6x(2x + 5)}}$
*(or leave as $\frac{6x + 5}{12x^2 + 30x}$, but factored form is better)*

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Problem 3:

$$
\frac{x + 2}{2x^2 + 13x + 20} - \frac{x + 3}{2x^2 + 13x + 20}
$$

Same denominator → Subtract numerators:
$$
\frac{(x + 2) - (x + 3)}{2x^2 + 13x + 20} = \frac{x + 2 - x - 3}{2x^2 + 13x + 20} = \frac{-1}{2x^2 + 13x + 20}
$$

Factor denominator:
$$
2x^2 + 13x + 20 = (2x + 5)(x + 4)
$$

✔ Answer: $\boxed{\frac{-1}{(2x + 5)(x + 4)}}$

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Problem 4:

$$
\frac{5}{p^2 - 6p + 9} + \frac{4}{p^2 - 6p + 9}
$$

Same denominator → Add numerators:
$$
\frac{5 + 4}{p^2 - 6p + 9} = \frac{9}{p^2 - 6p + 9}
$$

Factor denominator:
$$
p^2 - 6p + 9 = (p - 3)^2
$$

✔ Answer: $\boxed{\frac{9}{(p - 3)^2}}$

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Problem 5:

$$
\frac{q + 5}{4q^2 + 20q} - \frac{q - 5}{4q^2 + 20q}
$$

Same denominator → Subtract numerators:
$$
\frac{(q + 5) - (q - 5)}{4q^2 + 20q} = \frac{q + 5 - q + 5}{4q^2 + 20q} = \frac{10}{4q^2 + 20q}
$$

Factor denominator:
$$
4q^2 + 20q = 4q(q + 5)
$$

Simplify:
$$
\frac{10}{4q(q + 5)} = \frac{5}{2q(q + 5)}
$$

✔ Answer: $\boxed{\frac{5}{2q(q + 5)}}$

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Problem 6:

$$
\frac{x - 1}{12x^2 + 8x} + \frac{x + 5}{12x^2 + 8x}
$$

Add numerators:
$$
\frac{(x - 1) + (x + 5)}{12x^2 + 8x} = \frac{2x + 4}{12x^2 + 8x}
$$

Factor numerator and denominator:

Numerator: $2x + 4 = 2(x + 2)$

Denominator: $12x^2 + 8x = 4x(3x + 2)$

So:
$$
\frac{2(x + 2)}{4x(3x + 2)} = \frac{x + 2}{2x(3x + 2)}
$$

✔ Answer: $\boxed{\frac{x + 2}{2x(3x + 2)}}$

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Problem 7:

$$
\frac{6n}{12n + 24} + \frac{n + 6}{12n + 24}
$$

Add numerators:
$$
\frac{6n + (n + 6)}{12n + 24} = \frac{7n + 6}{12n + 24}
$$

Factor denominator:
$$
12n + 24 = 12(n + 2)
$$

Check if numerator factors: $7n + 6$ — doesn’t factor nicely with denominator.

No common factors → cannot simplify further.

✔ Answer: $\boxed{\frac{7n + 6}{12(n + 2)}}$

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Problem 8:

$$
\frac{4x + 5y}{15x} - \frac{x + 5y}{15x}
$$

Subtract numerators:
$$
\frac{(4x + 5y) - (x + 5y)}{15x} = \frac{4x + 5y - x - 5y}{15x} = \frac{3x}{15x}
$$

Simplify:
$$
\frac{3x}{15x} = \frac{3}{15} = \frac{1}{5}, \quad \text{for } x \ne 0
$$

✔ Answer: $\boxed{\frac{1}{5}}$

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✔ Final Answers Summary:

1. $\boxed{\frac{2x + 7y}{18y^2x^3}}$
2. $\boxed{\frac{6x + 5}{6x(2x + 5)}}$
3. $\boxed{\frac{-1}{(2x + 5)(x + 4)}}$
4. $\boxed{\frac{9}{(p - 3)^2}}$
5. $\boxed{\frac{5}{2q(q + 5)}}$
6. $\boxed{\frac{x + 2}{2x(3x + 2)}}$
7. $\boxed{\frac{7n + 6}{12(n + 2)}}$
8. $\boxed{\frac{1}{5}}$

Let me know if you'd like explanations in a different format or need help with any specific step!