Addition and subtraction of algebraic fractions practice problems.

Let's solve each of these algebraic fraction problems step by step. The goal is to add or subtract the given algebraic fractions. We'll follow the standard rules:

- If the denominators are the same, simply add or subtract the numerators.
- If the denominators are different, find a common denominator (LCD), rewrite each fraction with that denominator, then combine.

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1) $\frac{1}{a} + \frac{4}{a}$

Same denominator:
$$
\frac{1 + 4}{a} = \frac{5}{a}
$$

✔ Answer: $\boxed{\frac{5}{a}}$

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2) $\frac{7}{b} - \frac{4}{b}$

Same denominator:
$$
\frac{7 - 4}{b} = \frac{3}{b}
$$

✔ Answer: $\boxed{\frac{3}{b}}$

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3) $\frac{7}{3c} + \frac{4}{3c}$

Same denominator:
$$
\frac{7 + 4}{3c} = \frac{11}{3c}
$$

✔ Answer: $\boxed{\frac{11}{3c}}$

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4) $\frac{7}{3d} - \frac{4}{3d}$

Same denominator:
$$
\frac{7 - 4}{3d} = \frac{3}{3d} = \frac{1}{d}
$$

✔ Answer: $\boxed{\frac{1}{d}}$

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5) $\frac{7}{e} + \frac{4}{3e}$

Different denominators: $e$ and $3e$. LCD is $3e$.

Convert $\frac{7}{e} = \frac{7 \cdot 3}{3e} = \frac{21}{3e}$

Now:
$$
\frac{21}{3e} + \frac{4}{3e} = \frac{25}{3e}
$$

✔ Answer: $\boxed{\frac{25}{3e}}$

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6) $\frac{7}{1+f} + \frac{4}{1+f}$

Same denominator:
$$
\frac{7 + 4}{1+f} = \frac{11}{1+f}
$$

✔ Answer: $\boxed{\frac{11}{1+f}}$

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7) $\frac{7}{g+h} + \frac{4}{g+h}$

Same denominator:
$$
\frac{7 + 4}{g+h} = \frac{11}{g+h}
$$

✔ Answer: $\boxed{\frac{11}{g+h}}$

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8) $\frac{7}{g+h} - \frac{4}{g+h}$

Same denominator:
$$
\frac{7 - 4}{g+h} = \frac{3}{g+h}
$$

✔ Answer: $\boxed{\frac{3}{g+h}}$

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9) $\frac{7}{g-h} - \frac{4}{g-h}$

Same denominator:
$$
\frac{7 - 4}{g-h} = \frac{3}{g-h}
$$

✔ Answer: $\boxed{\frac{3}{g-h}}$

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10) $\frac{7}{i^2} - \frac{4}{i^2}$

Same denominator:
$$
\frac{7 - 4}{i^2} = \frac{3}{i^2}
$$

✔ Answer: $\boxed{\frac{3}{i^2}}$

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11) $\frac{j}{2} + \frac{j}{3}$

Different denominators: 2 and 3 → LCD = 6

$$
\frac{j}{2} = \frac{3j}{6},\quad \frac{j}{3} = \frac{2j}{6}
$$
$$
\frac{3j}{6} + \frac{2j}{6} = \frac{5j}{6}
$$

✔ Answer: $\boxed{\frac{5j}{6}}$

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12) $\frac{k}{3} - \frac{k}{5}$

LCD of 3 and 5 is 15

$$
\frac{k}{3} = \frac{5k}{15},\quad \frac{k}{5} = \frac{3k}{15}
$$
$$
\frac{5k - 3k}{15} = \frac{2k}{15}
$$

✔ Answer: $\boxed{\frac{2k}{15}}$

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13) $\frac{l}{5} + \frac{2l}{3}$

LCD of 5 and 3 is 15

$$
\frac{l}{5} = \frac{3l}{15},\quad \frac{2l}{3} = \frac{10l}{15}
$$
$$
\frac{3l + 10l}{15} = \frac{13l}{15}
$$

✔ Answer: $\boxed{\frac{13l}{15}}$

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14) $\frac{2m}{5} + \frac{2m}{3}$

LCD = 15

$$
\frac{2m}{5} = \frac{6m}{15},\quad \frac{2m}{3} = \frac{10m}{15}
$$
$$
\frac{6m + 10m}{15} = \frac{16m}{15}
$$

✔ Answer: $\boxed{\frac{16m}{15}}$

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15) $\frac{2p}{7} - \frac{4p}{3}$

LCD of 7 and 3 is 21

$$
\frac{2p}{7} = \frac{6p}{21},\quad \frac{4p}{3} = \frac{28p}{21}
$$
$$
\frac{6p - 28p}{21} = \frac{-22p}{21}
$$

✔ Answer: $\boxed{-\frac{22p}{21}}$

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16) $\frac{q+1}{2} + \frac{2}{3}$

LCD = 6

$$
\frac{q+1}{2} = \frac{3(q+1)}{6} = \frac{3q+3}{6},\quad \frac{2}{3} = \frac{4}{6}
$$
$$
\frac{3q+3 + 4}{6} = \frac{3q + 7}{6}
$$

✔ Answer: $\boxed{\frac{3q + 7}{6}}$

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17) $\frac{5}{2} + \frac{2r+2}{3}$

LCD = 6

$$
\frac{5}{2} = \frac{15}{6},\quad \frac{2r+2}{3} = \frac{2(2r+2)}{6} = \frac{4r+4}{6}
$$
$$
\frac{15 + 4r + 4}{6} = \frac{4r + 19}{6}
$$

✔ Answer: $\boxed{\frac{4r + 19}{6}}$

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18) $\frac{5}{2} + \frac{2(s+2)}{3}$

First simplify: $2(s+2) = 2s + 4$, so:

$$
\frac{5}{2} + \frac{2s+4}{3}
$$

LCD = 6

$$
\frac{5}{2} = \frac{15}{6},\quad \frac{2s+4}{3} = \frac{2(2s+4)}{6} = \frac{4s+8}{6}
$$
$$
\frac{15 + 4s + 8}{6} = \frac{4s + 23}{6}
$$

✔ Answer: $\boxed{\frac{4s + 23}{6}}$

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19) $\frac{t+1}{2} + \frac{t+2}{3}$

LCD = 6

$$
\frac{t+1}{2} = \frac{3(t+1)}{6} = \frac{3t+3}{6},\quad \frac{t+2}{3} = \frac{2(t+2)}{6} = \frac{2t+4}{6}
$$
$$
\frac{3t+3 + 2t+4}{6} = \frac{5t + 7}{6}
$$

✔ Answer: $\boxed{\frac{5t + 7}{6}}$

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20) $\frac{v+1}{2} - \frac{v+2}{3}$

LCD = 6

$$
\frac{v+1}{2} = \frac{3(v+1)}{6} = \frac{3v+3}{6},\quad \frac{v+2}{3} = \frac{2(v+2)}{6} = \frac{2v+4}{6}
$$
$$
\frac{3v+3 - (2v+4)}{6} = \frac{3v+3 - 2v - 4}{6} = \frac{v - 1}{6}
$$

✔ Answer: $\boxed{\frac{v - 1}{6}}$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1) | $\frac{5}{a}$ |
| 2) | $\frac{3}{b}$ |
| 3) | $\frac{11}{3c}$ |
| 4) | $\frac{1}{d}$ |
| 5) | $\frac{25}{3e}$ |
| 6) | $\frac{11}{1+f}$ |
| 7) | $\frac{11}{g+h}$ |
| 8) | $\frac{3}{g+h}$ |
| 9) | $\frac{3}{g-h}$ |
| 10) | $\frac{3}{i^2}$ |
| 11) | $\frac{5j}{6}$ |
| 12) | $\frac{2k}{15}$ |
| 13) | $\frac{13l}{15}$ |
| 14) | $\frac{16m}{15}$ |
| 15) | $-\frac{22p}{21}$ |
| 16) | $\frac{3q + 7}{6}$ |
| 17) | $\frac{4r + 19}{6}$ |
| 18) | $\frac{4s + 23}{6}$ |
| 19) | $\frac{5t + 7}{6}$ |
| 20) | $\frac{v - 1}{6}$ |

Let me know if you'd like these simplified further or explained in more detail!