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### Problem Transcription:

1. **For \( \triangle \) DEF, find the value of \( x \).**

   a. \( \frac{10}{12} = \frac{x}{8} \)

   b. \( \frac{9}{x} = \frac{15}{10} \)

   c. \( \frac{x}{14} = \frac{5}{10} \)

2. **Solve for \( x \) in the following equations:**

   a. \( \frac{12}{4} = \frac{15}{x} \)

   b. \( \frac{14}{x} = \frac{21}{9} \)

   c. \( \frac{x}{18} = \frac{16}{12} \)

3. **Find the value of \( x \):**

   a. \( \frac{x}{3} = \frac{4}{5} \)

   b. \( \frac{3x}{7} = \frac{6}{14} \)

   c. \( \frac{45}{x} = \frac{36}{24} \)

### Let's start solving each one:

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#### Problem 1:

**a.** \( \frac{10}{12} = \frac{x}{8} \)

- Cross multiply: \( 10 \times 8 = 12 \times x \)
- \( 80 = 12x \)
- \( x = \frac{80}{12} = \frac{20}{3} \approx 6.67 \)

**b.** \( \frac{9}{x} = \frac{15}{10} \)

- Cross multiply: \( 9 \times 10 = 15 \times x \)
- \( 90 = 15x \)
- \( x = \frac{90}{15} = 6 \)

**c.** \( \frac{x}{14} = \frac{5}{10} \)

- Cross multiply: \( x \times 10 = 5 \times 14 \)
- \( 10x = 70 \)
- \( x = \frac{70}{10} = 7 \)

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#### Problem 2:

**a.** \( \frac{12}{4} = \frac{15}{x} \)

- Simplify: \( \frac{12}{4} = 3 \)
- \( 3 = \frac{15}{x} \)
- Cross multiply: \( 3x = 15 \)
- \( x = \frac{15}{3} = 5 \)

**b.** \( \frac{14}{x} = \frac{21}{9} \)

- Cross multiply: \( 14 \times 9 = 21 \times x \)
- \( 126 = 21x \)
- \( x = \frac{126}{21} = 6 \)

**c.** \( \frac{x}{18} = \frac{16}{12} \)

- Simplify \( \frac{16}{12} = \frac{4}{3} \)
- \( \frac{x}{18} = \frac{4}{3} \)
- Cross multiply: \( 3x = 72 \)
- \( x = \frac{72}{3} = 24 \)

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#### Problem 3:

**a.** \( \frac{x}{3} = \frac{4}{5} \)

- Cross multiply: \( 5x = 12 \)
- \( x = \frac{12}{5} = 2.4 \)

**b.** \( \frac{3x}{7} = \frac{6}{14} \)

- Simplify \( \frac{6}{14} = \frac{3}{7} \)
- \( \frac{3x}{7} = \frac{3}{7} \)
- \( 3x = 3 \)
- \( x = 1 \)

**c.** \( \frac{45}{x} = \frac{36}{24} \)

- Simplify \( \frac{36}{24} = \frac{3}{2} \)
- \( \frac{45}{x} = \frac{3}{2} \)
- Cross multiply: \( 3x = 90 \)
- \( x = \frac{90}{3} = 30 \)

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Would you like to go over any of these problems in more detail or need help with related questions? 

### Relative Questions:

1. How does cross-multiplication work in solving proportions?
2. Can you simplify the proportion \( \frac{36}{24} \) before solving?
3. What are some common mistakes to avoid when solving proportions?
4. How can you verify your solution to a proportion problem?
5. How would the steps change if the proportion involved decimals?

**Tip:** Always check your final answer by substituting it back into the original equation to ensure it satisfies the proportion.

Explore how to solve proportional equations with detailed step-by-step solutions. This problem set covers key concepts such as proportions, cross multiplication, and fractional equations. Suitable for students in Grades 6-8.

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