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To solve the problems involving adding and subtracting fractions with unlike denominators, we need to follow these steps:

1. Find a Common Denominator: The least common denominator (LCD) is the smallest number that is a multiple of all the denominators involved.
2. Adjust the Fractions: Rewrite each fraction with the common denominator by multiplying both the numerator and the denominator by the necessary factor.
3. Add or Subtract the Numerators: Once the denominators are the same, add or subtract the numerators as needed.
4. Simplify the Result: Reduce the resulting fraction to its simplest form if possible.

Let's solve each problem step by step.

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Problem 1: $\frac{4}{3} + \frac{2}{5}$

- Step 1: Find the LCD
The denominators are 3 and 5. The LCD is $3 \times 5 = 15$.
- Step 2: Adjust the fractions
$\frac{4}{3} = \frac{4 \times 5}{3 \times 5} = \frac{20}{15}$
$\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15}$
- Step 3: Add the numerators
$\frac{20}{15} + \frac{6}{15} = \frac{20 + 6}{15} = \frac{26}{15}$
- Step 4: Simplify
$\frac{26}{15}$ is already in simplest form.

Answer: $\boxed{\frac{26}{15}}$

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Problem 2: $\frac{7}{10} - \frac{2}{5}$

- Step 1: Find the LCD
The denominators are 10 and 5. The LCD is 10.
- Step 2: Adjust the fractions
$\frac{7}{10}$ remains $\frac{7}{10}$.
$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$
- Step 3: Subtract the numerators
$\frac{7}{10} - \frac{4}{10} = \frac{7 - 4}{10} = \frac{3}{10}$
- Step 4: Simplify
$\frac{3}{10}$ is already in simplest form.

Answer: $\boxed{\frac{3}{10}}$

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Problem 3: $\frac{5}{9} + \frac{2}{7}$

- Step 1: Find the LCD
The denominators are 9 and 7. The LCD is $9 \times 7 = 63$.
- Step 2: Adjust the fractions
$\frac{5}{9} = \frac{5 \times 7}{9 \times 7} = \frac{35}{63}$
$\frac{2}{7} = \frac{2 \times 9}{7 \times 9} = \frac{18}{63}$
- Step 3: Add the numerators
$\frac{35}{63} + \frac{18}{63} = \frac{35 + 18}{63} = \frac{53}{63}$
- Step 4: Simplify
$\frac{53}{63}$ is already in simplest form.

Answer: $\boxed{\frac{53}{63}}$

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Problem 4: $\frac{4}{8} - \frac{1}{4}$

- Step 1: Find the LCD
The denominators are 8 and 4. The LCD is 8.
- Step 2: Adjust the fractions
$\frac{4}{8}$ remains $\frac{4}{8}$.
$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
- Step 3: Subtract the numerators
$\frac{4}{8} - \frac{2}{8} = \frac{4 - 2}{8} = \frac{2}{8}$
- Step 4: Simplify
$\frac{2}{8} = \frac{1}{4}$

Answer: $\boxed{\frac{1}{4}}$

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Problem 5: $\frac{3}{9} + \frac{1}{3}$

- Step 1: Find the LCD
The denominators are 9 and 3. The LCD is 9.
- Step 2: Adjust the fractions
$\frac{3}{9}$ remains $\frac{3}{9}$.
$\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}$
- Step 3: Add the numerators
$\frac{3}{9} + \frac{3}{9} = \frac{3 + 3}{9} = \frac{6}{9}$
- Step 4: Simplify
$\frac{6}{9} = \frac{2}{3}$

Answer: $\boxed{\frac{2}{3}}$

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Problem 6: $\frac{2}{5} + \frac{1}{10}$

- Step 1: Find the LCD
The denominators are 5 and 10. The LCD is 10.
- Step 2: Adjust the fractions
$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$
$\frac{1}{10}$ remains $\frac{1}{10}$.
- Step 3: Add the numerators
$\frac{4}{10} + \frac{1}{10} = \frac{4 + 1}{10} = \frac{5}{10}$
- Step 4: Simplify
$\frac{5}{10} = \frac{1}{2}$

Answer: $\boxed{\frac{1}{2}}$

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Problem 7: $\frac{4}{5} + \frac{9}{10}$

- Step 1: Find the LCD
The denominators are 5 and 10. The LCD is 10.
- Step 2: Adjust the fractions
$\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}$
$\frac{9}{10}$ remains $\frac{9}{10}$.
- Step 3: Add the numerators
$\frac{8}{10} + \frac{9}{10} = \frac{8 + 9}{10} = \frac{17}{10}$
- Step 4: Simplify
$\frac{17}{10}$ is already in simplest form.

Answer: $\boxed{\frac{17}{10}}$

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Problem 8: $\frac{4}{6} - \frac{1}{3}$

- Step 1: Find the LCD
The denominators are 6 and 3. The LCD is 6.
- Step 2: Adjust the fractions
$\frac{4}{6}$ remains $\frac{4}{6}$.
$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$
- Step 3: Subtract the numerators
$\frac{4}{6} - \frac{2}{6} = \frac{4 - 2}{6} = \frac{2}{6}$
- Step 4: Simplify
$\frac{2}{6} = \frac{1}{3}$

Answer: $\boxed{\frac{1}{3}}$

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Problem 9: $\frac{3}{12} + \frac{2}{4}$

- Step 1: Find the LCD
The denominators are 12 and 4. The LCD is 12.
- Step 2: Adjust the fractions
$\frac{3}{12}$ remains $\frac{3}{12}$.
$\frac{2}{4} = \frac{2 \times 3}{4 \times 3} = \frac{6}{12}$
- Step 3: Add the numerators
$\frac{3}{12} + \frac{6}{12} = \frac{3 + 6}{12} = \frac{9}{12}$
- Step 4: Simplify
$\frac{9}{12} = \frac{3}{4}$

Answer: $\boxed{\frac{3}{4}}$

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Problem 10: $\frac{5}{12} - \frac{1}{6}$

- Step 1: Find the LCD
The denominators are 12 and 6. The LCD is 12.
- Step 2: Adjust the fractions
$\frac{5}{12}$ remains $\frac{5}{12}$.
$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$
- Step 3: Subtract the numerators
$\frac{5}{12} - \frac{2}{12} = \frac{5 - 2}{12} = \frac{3}{12}$
- Step 4: Simplify
$\frac{3}{12} = \frac{1}{4}$

Answer: $\boxed{\frac{1}{4}}$

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Problem 11: $\frac{2}{8} + \frac{3}{9}$

- Step 1: Find the LCD
The denominators are 8 and 9. The LCD is $8 \times 9 = 72$.
- Step 2: Adjust the fractions
$\frac{2}{8} = \frac{2 \times 9}{8 \times 9} = \frac{18}{72}$
$\frac{3}{9} = \frac{3 \times 8}{9 \times 8} = \frac{24}{72}$
- Step 3: Add the numerators
$\frac{18}{72} + \frac{24}{72} = \frac{18 + 24}{72} = \frac{42}{72}$
- Step 4: Simplify
$\frac{42}{72} = \frac{7}{12}$

Answer: $\boxed{\frac{7}{12}}$

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Problem 12: $\frac{1}{4} - \frac{1}{8}$

- Step 1: Find the LCD
The denominators are 4 and 8. The LCD is 8.
- Step 2: Adjust the fractions
$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$
$\frac{1}{8}$ remains $\frac{1}{8}$.
- Step 3: Subtract the numerators
$\frac{2}{8} - \frac{1}{8} = \frac{2 - 1}{8} = \frac{1}{8}$
- Step 4: Simplify
$\frac{1}{8}$ is already in simplest form.

Answer: $\boxed{\frac{1}{8}}$

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Problem 13: $\frac{6}{7} + \frac{2}{6}$

- Step 1: Find the LCD
The denominators are 7 and 6. The LCD is $7 \times 6 = 42$.
- Step 2: Adjust the fractions
$\frac{6}{7} = \frac{6 \times 6}{7 \times 6} = \frac{36}{42}$
$\frac{2}{6} = \frac{2 \times 7}{6 \times 7} = \frac{14}{42}$
- Step 3: Add the numerators
$\frac{36}{42} + \frac{14}{42} = \frac{36 + 14}{42} = \frac{50}{42}$
- Step 4: Simplify
$\frac{50}{42} = \frac{25}{21}$

Answer: $\boxed{\frac{25}{21}}$

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Problem 14: $\frac{1}{3} - \frac{1}{7}$

- Step 1: Find the LCD
The denominators are 3 and 7. The LCD is $3 \times 7 = 21$.
- Step 2: Adjust the fractions
$\frac{1}{3} = \frac{1 \times 7}{3 \times 7} = \frac{7}{21}$
$\frac{1}{7} = \frac{1 \times 3}{7 \times 3} = \frac{3}{21}$
- Step 3: Subtract the numerators
$\frac{7}{21} - \frac{3}{21} = \frac{7 - 3}{21} = \frac{4}{21}$
- Step 4: Simplify
$\frac{4}{21}$ is already in simplest form.

Answer: $\boxed{\frac{4}{21}}$

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Problem 15: $\frac{7}{8} + \frac{4}{5}$

- Step 1: Find the LCD
The denominators are 8 and 5. The LCD is $8 \times 5 = 40$.
- Step 2: Adjust the fractions
$\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}$
$\frac{4}{5} = \frac{4 \times 8}{5 \times 8} = \frac{32}{40}$
- Step 3: Add the numerators
$\frac{35}{40} + \frac{32}{40} = \frac{35 + 32}{40} = \frac{67}{40}$
- Step 4: Simplify
$\frac{67}{40}$ is already in simplest form.

Answer: $\boxed{\frac{67}{40}}$

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Final Answers:

1. $\boxed{\frac{26}{15}}$
2. $\boxed{\frac{3}{10}}$
3. $\boxed{\frac{53}{63}}$
4. $\boxed{\frac{1}{4}}$
5. $\boxed{\frac{2}{3}}$
6. $\boxed{\frac{1}{2}}$
7. $\boxed{\frac{17}{10}}$
8. $\boxed{\frac{1}{3}}$
9. $\boxed{\frac{3}{4}}$
10. $\boxed{\frac{1}{4}}$
11. $\boxed{\frac{7}{12}}$
12. $\boxed{\frac{1}{8}}$
13. $\boxed{\frac{25}{21}}$
14. $\boxed{\frac{4}{21}}$
15. $\boxed{\frac{67}{40}}$

Boxed Final Answer:
$$
\boxed{
\begin{aligned}
1. & \frac{26}{15} \\
2. & \frac{3}{10} \\
3. & \frac{53}{63} \\
4. & \frac{1}{4} \\
5. & \frac{2}{3} \\
6. & \frac{1}{2} \\
7. & \frac{17}{10} \\
8. & \frac{1}{3} \\
9. & \frac{3}{4} \\
10. & \frac{1}{4} \\
11. & \frac{7}{12} \\
12. & \frac{1}{8} \\
13. & \frac{25}{21} \\
14. & \frac{4}{21} \\
15. & \frac{67}{40}
\end{aligned}
}
$$