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Let's solve the given set of fraction problems step by step, and I'll provide the simplified answers for each.

### a. \(\frac{7}{8} + \frac{1}{2} - \frac{3}{4}\)
Find the least common denominator (LCD) of 8, 2, and 4. The LCD is 8.

\[
\frac{7}{8} + \frac{1}{2} - \frac{3}{4} = \frac{7}{8} + \frac{4}{8} - \frac{6}{8} = \frac{7 + 4 - 6}{8} = \frac{5}{8}
\]

Answer: \(\frac{5}{8}\)

### b. \(25 \frac{1}{2} - 18 \frac{5}{8}\)
Convert mixed numbers to improper fractions:
\[
25 \frac{1}{2} = \frac{51}{2}, \quad 18 \frac{5}{8} = \frac{149}{8}
\]
Find the LCD of 2 and 8, which is 8:
\[
\frac{51}{2} = \frac{204}{8}
\]
Now subtract:
\[
\frac{204}{8} - \frac{149}{8} = \frac{204 - 149}{8} = \frac{55}{8} = 6 \frac{7}{8}
\]

Answer: \(6 \frac{7}{8}\)

### c. \(\frac{1}{3} \times \frac{2}{3}\)
Multiply the numerators and denominators:
\[
\frac{1}{3} \times \frac{2}{3} = \frac{1 \times 2}{3 \times 3} = \frac{2}{9}
\]

Answer: \(\frac{2}{9}\)

### d. \(\frac{4}{5} \times \frac{1}{2}\)
Multiply the numerators and denominators:
\[
\frac{4}{5} \times \frac{1}{2} = \frac{4 \times 1}{5 \times 2} = \frac{4}{10} = \frac{2}{5}
\]

Answer: \(\frac{2}{5}\)

### e. \(34 \frac{5}{6} \div 4 \frac{5}{8}\)
Convert mixed numbers to improper fractions:
\[
34 \frac{5}{6} = \frac{209}{6}, \quad 4 \frac{5}{8} = \frac{37}{8}
\]
Now divide by multiplying by the reciprocal:
\[
\frac{209}{6} \div \frac{37}{8} = \frac{209}{6} \times \frac{8}{37} = \frac{209 \times 8}{6 \times 37} = \frac{1672}{222} = \frac{836}{111}
\]

Answer: \(\frac{836}{111}\)

### f. \(\frac{7}{8} + \left( \frac{3}{4} \times \frac{3}{5} \right) + \frac{1}{2}\)
First, solve the multiplication:
\[
\frac{3}{4} \times \frac{3}{5} = \frac{9}{20}
\]
Now add the fractions:
The LCD of 8, 20, and 2 is 40:
\[
\frac{7}{8} = \frac{35}{40}, \quad \frac{9}{20} = \frac{18}{40}, \quad \frac{1}{2} = \frac{20}{40}
\]
Now add:
\[
\frac{35}{40} + \frac{18}{40} + \frac{20}{40} = \frac{73}{40} = 1 \frac{33}{40}
\]

Answer: \(1 \frac{33}{40}\)

### g. \(\left( \frac{3}{8} \times \frac{1}{3} \right) + \left( \frac{5}{8} \div \frac{3}{5} \right) - \frac{3}{8} + \frac{1}{4}\)
First, solve the operations inside parentheses:
\[
\frac{3}{8} \times \frac{1}{3} = \frac{1}{8}, \quad \frac{5}{8} \div \frac{3}{5} = \frac{5}{8} \times \frac{5}{3} = \frac{25}{24}
\]
Now add and subtract:
The LCD of 8, 24, and 4 is 24:
\[
\frac{1}{8} = \frac{3}{24}, \quad \frac{3}{8} = \frac{9}{24}, \quad \frac{1}{4} = \frac{6}{24}
\]
Now combine:
\[
\frac{3}{24} + \frac{25}{24} - \frac{9}{24} + \frac{6}{24} = \frac{25}{24} = 1 \frac{1}{24}
\]

Answer: \(1 \frac{1}{24}\)

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Would you like any further details on these calculations?

Here are 5 related questions:
1. How do you find the least common denominator in fraction addition?
2. What is the method to convert mixed numbers to improper fractions?
3. Can you explain how multiplying and dividing fractions differ?
4. How do you simplify complex fraction expressions?
5. How do you handle operations involving multiple fractions and mixed numbers?

**Tip:** When dealing with mixed numbers in operations, always convert them into improper fractions first for easier calculation.

Calculez et donnez les résultats en fraction simplifiée pour les expressions données (fraction arithmetic and operations).

Learn how to solve fraction problems including addition, subtraction, multiplication, and division, as well as working with mixed numbers. This step-by-step guide includes concepts like finding the least common denominator, converting mixed numbers to improper fractions, and simplifying the final answers.

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