Add and Subtract Rational expressions like denominators activity - Free Printable

To solve the given problems involving adding rational expressions, we need to follow these steps:

1. Identify the denominators and find the least common denominator (LCD).
2. Rewrite each fraction with the LCD as the denominator.
3. Combine the numerators over the common denominator.
4. Simplify the resulting expression if possible.

Let's solve each problem step by step.

---

Problem 1:

$$
\frac{n + 6n}{8n^3} + \frac{m - n}{8n^3}
$$

#### Step 1: Identify the denominators
Both fractions have the same denominator: $8n^3$.

#### Step 2: Combine the numerators
Since the denominators are the same, we can directly add the numerators:
$$
\frac{n + 6n}{8n^3} + \frac{m - n}{8n^3} = \frac{(n + 6n) + (m - n)}{8n^3}
$$

#### Step 3: Simplify the numerator
Simplify $n + 6n$ and $m - n$:
$$
n + 6n = 7n
$$
$$
(m - n) = m - n
$$
So the numerator becomes:
$$
7n + m - n = 6n + m
$$

#### Step 4: Write the final expression
$$
\frac{6n + m}{8n^3}
$$

#### Final Answer:
$$
\boxed{\frac{6n + m}{8n^3}}
$$

---

Problem 2:

$$
\frac{3 + 3v}{v^2 + v - 6}
$$

This is already a single fraction, so no further addition is needed. However, we can simplify it if possible.

#### Step 1: Factor the numerator and denominator
- Numerator: $3 + 3v = 3(1 + v)$
- Denominator: $v^2 + v - 6$. Factorize:
$$
v^2 + v - 6 = (v + 3)(v - 2)
$$

#### Step 2: Write the simplified form
$$
\frac{3 + 3v}{v^2 + v - 6} = \frac{3(1 + v)}{(v + 3)(v - 2)}
$$

#### Final Answer:
$$
\boxed{\frac{3(1 + v)}{(v + 3)(v - 2)}}
$$

---

Problem 3:

$$
\frac{a + 5b}{12a} + \frac{5a + 5b}{12a}
$$

#### Step 1: Identify the denominators
Both fractions have the same denominator: $12a$.

#### Step 2: Combine the numerators
Since the denominators are the same, we can directly add the numerators:
$$
\frac{a + 5b}{12a} + \frac{5a + 5b}{12a} = \frac{(a + 5b) + (5a + 5b)}{12a}
$$

#### Step 3: Simplify the numerator
Simplify $a + 5b + 5a + 5b$:
$$
a + 5a = 6a
$$
$$
5b + 5b = 10b
$$
So the numerator becomes:
$$
6a + 10b
$$

#### Step 4: Write the final expression
$$
\frac{6a + 10b}{12a}
$$

#### Final Answer:
$$
\boxed{\frac{6a + 10b}{12a}}
$$

---

Problem 4:

$$
\frac{7r - 2}{6r + 30} + \frac{7r + 6}{12n + 24}
$$

#### Step 1: Factor the denominators
- First denominator: $6r + 30 = 6(r + 5)$
- Second denominator: $12n + 24 = 12(n + 2)$

The denominators are different, so we need to find the LCD. The LCD is:
$$
\text{LCD} = 6 \cdot 12 \cdot (r + 5) \cdot (n + 2) = 72(r + 5)(n + 2)
$$

#### Step 2: Rewrite each fraction with the LCD
1. For $\frac{7r - 2}{6(r + 5)}$:
Multiply numerator and denominator by $12(n + 2)$:
$$
\frac{7r - 2}{6(r + 5)} = \frac{(7r - 2) \cdot 12(n + 2)}{6(r + 5) \cdot 12(n + 2)} = \frac{12(7r - 2)(n + 2)}{72(r + 5)(n + 2)}
$$

2. For $\frac{7r + 6}{12(n + 2)}$:
Multiply numerator and denominator by $6(r + 5)$:
$$
\frac{7r + 6}{12(n + 2)} = \frac{(7r + 6) \cdot 6(r + 5)}{12(n + 2) \cdot 6(r + 5)} = \frac{6(7r + 6)(r + 5)}{72(r + 5)(n + 2)}
$$

#### Step 3: Combine the fractions
Now that both fractions have the same denominator, we can add the numerators:
$$
\frac{12(7r - 2)(n + 2)}{72(r + 5)(n + 2)} + \frac{6(7r + 6)(r + 5)}{72(r + 5)(n + 2)} = \frac{12(7r - 2)(n + 2) + 6(7r + 6)(r + 5)}{72(r + 5)(n + 2)}
$$

#### Step 4: Simplify the numerator
Expand and combine like terms in the numerator:
1. Expand $12(7r - 2)(n + 2)$:
$$
12(7r - 2)(n + 2) = 12[7rn + 14r - 2n - 4] = 84rn + 168r - 24n - 48
$$

2. Expand $6(7r + 6)(r + 5)$:
$$
6(7r + 6)(r + 5) = 6[7r^2 + 35r + 6r + 30] = 6[7r^2 + 41r + 30] = 42r^2 + 246r + 180
$$

Combine the expanded terms:
$$
84rn + 168r - 24n - 48 + 42r^2 + 246r + 180 = 42r^2 + 84rn + 414r - 24n + 132
$$

#### Step 5: Write the final expression
$$
\frac{42r^2 + 84rn + 414r - 24n + 132}{72(r + 5)(n + 2)}
$$

#### Final Answer:
$$
\boxed{\frac{42r^2 + 84rn + 414r - 24n + 132}{72(r + 5)(n + 2)}}
$$

---

Problem 5:

$$
\frac{3}{v^2 + v - 6} + \frac{3v}{v^2 + v - 6}
$$

#### Step 1: Identify the denominators
Both fractions have the same denominator: $v^2 + v - 6$.

#### Step 2: Combine the numerators
Since the denominators are the same, we can directly add the numerators:
$$
\frac{3}{v^2 + v - 6} + \frac{3v}{v^2 + v - 6} = \frac{3 + 3v}{v^2 + v - 6}
$$

#### Step 3: Simplify the numerator
Factor out $3$ from the numerator:
$$
3 + 3v = 3(1 + v)
$$

#### Step 4: Write the final expression
$$
\frac{3(1 + v)}{v^2 + v - 6}
$$

Factor the denominator:
$$
v^2 + v - 6 = (v + 3)(v - 2)
$$

So the expression becomes:
$$
\frac{3(1 + v)}{(v + 3)(v - 2)}
$$

#### Final Answer:
$$
\boxed{\frac{3(1 + v)}{(v + 3)(v - 2)}}
$$

---

Problem 6:

$$
\frac{7n + 6}{12n + 24}
$$

#### Step 1: Factor the numerator and denominator
- Numerator: $7n + 6$ (already factored)
- Denominator: $12n + 24 = 12(n + 2)$

#### Step 2: Write the simplified form
$$
\frac{7n + 6}{12n + 24} = \frac{7n + 6}{12(n + 2)}
$$

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

---

Problem 7:

$$
\frac{6*}{6r + 30} + \frac{r - 2}{6r + 30}
$$

#### Step 1: Identify the denominators
Both fractions have the same denominator: $6r + 30$.

#### Step 2: Combine the numerators
Since the denominators are the same, we can directly add the numerators:
$$
\frac{6*}{6r + 30} + \frac{r - 2}{6r + 30} = \frac{6* + (r - 2)}{6r + 30}
$$

#### Step 3: Simplify the numerator
The numerator is $6* + r - 2$. Since $6*$ is not clear, let's assume it is $6$. Then:
$$
6 + r - 2 = r + 4
$$

#### Step 4: Write the final expression
$$
\frac{r + 4}{6r + 30}
$$

Factor the denominator:
$$
6r + 30 = 6(r + 5)
$$

So the expression becomes:
$$
\frac{r + 4}{6(r + 5)}
$$

#### Final Answer:
$$
\boxed{\frac{r + 4}{6(r + 5)}}
$$

---

Problem 8:

$$
\frac{2m + 5n}{8m^2}
$$

This is already a single fraction, so no further addition is needed.

#### Final Answer:
$$
\boxed{\frac{2m + 5n}{8m^2}}
$$

---

Problem 9:

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

#### Step 1: Identify the denominators
Both fractions have the same denominator: $12n + 24$.

#### Step 2: Combine the numerators
Since the denominators are the same, we can directly add the numerators:
$$
\frac{6n + 6}{12n + 24} + \frac{n + 6}{12n + 24} = \frac{(6n + 6) + (n + 6)}{12n + 24}
$$

#### Step 3: Simplify the numerator
Simplify $6n + 6 + n + 6$:
$$
6n + n = 7n
$$
$$
6 + 6 = 12
$$
So the numerator becomes:
$$
7n + 12
$$

#### Step 4: Write the final expression
$$
\frac{7n + 12}{12n + 24}
$$

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

So the expression becomes:
$$
\frac{7n + 12}{12(n + 2)}
$$

#### Final Answer:
$$
\boxed{\frac{7n + 12}{12(n + 2)}}
$$

---

Problem 10:

$$
\frac{b - 12}{3b^2 + 16b + 16}
$$

This is already a single fraction, so no further addition is needed.

#### Final Answer:
$$
\boxed{\frac{b - 12}{3b^2 + 16b + 16}}
$$

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Final Answers Summary:

1. $\boxed{\frac{6n + m}{8n^3}}$
2. $\boxed{\frac{3(1 + v)}{(v + 3)(v - 2)}}$
3. $\boxed{\frac{6a + 10b}{12a}}$
4. $\boxed{\frac{42r^2 + 84rn + 414r - 24n + 132}{72(r + 5)(n + 2)}}$
5. $\boxed{\frac{3(1 + v)}{(v + 3)(v - 2)}}$
6. $\boxed{\frac{7n + 6}{12(n + 2)}}$
7. $\boxed{\frac{r + 4}{6(r + 5)}}$
8. $\boxed{\frac{2m + 5n}{8m^2}}$
9. $\boxed{\frac{7n + 12}{12(n + 2)}}$
10. $\boxed{\frac{b - 12}{3b^2 + 16b + 16}}$