II. Solve for $x$ 1. $3 x: 2=2: 5$ 6. $(x-4): 6=5: 1$ 2. $4: 7=2 x: 3$ 7. $x: 3=(2 x-1): 5$ 3. $(4 x+2): 18=(6 x+3): 17$, 8. $(2 x+1): 18=(4 x-3): 21$

Let's solve each equation one by one. 

### 1. $ \frac{3x}{2} = \frac{2}{5} $

We can solve this proportion by cross-multiplying:
\[
3x \times 5 = 2 \times 2
\]
\[
15x = 4
\]
\[
x = \frac{4}{15}
\]

### 2. $ \frac{4}{7} = \frac{2x}{3} $

Again, solve by cross-multiplying:
\[
4 \times 3 = 7 \times 2x
\]
\[
12 = 14x
\]
\[
x = \frac{12}{14} = \frac{6}{7}
\]

### 3. $ \frac{4x + 2}{18} = \frac{6x + 3}{17} $

Cross-multiplying:
\[
(4x + 2) \times 17 = (6x + 3) \times 18
\]
Expanding both sides:
\[
68x + 34 = 108x + 54
\]
Simplifying:
\[
68x - 108x = 54 - 34
\]
\[
-40x = 20
\]
\[
x = \frac{20}{-40} = -\frac{1}{2}
\]

### 6. $ \frac{x - 4}{6} = \frac{5}{1} $

Cross-multiplying:
\[
(x - 4) \times 1 = 5 \times 6
\]
\[
x - 4 = 30
\]
\[
x = 30 + 4 = 34
\]

### 7. $ \frac{x}{3} = \frac{2x - 1}{5} $

Cross-multiplying:
\[
x \times 5 = (2x - 1) \times 3
\]
Expanding both sides:
\[
5x = 6x - 3
\]
Simplifying:
\[
5x - 6x = -3
\]
\[
-x = -3
\]
\[
x = 3
\]

### 8. $ \frac{2x + 1}{18} = \frac{4x - 3}{21} $

Cross-multiplying:
\[
(2x + 1) \times 21 = (4x - 3) \times 18
\]
Expanding both sides:
\[
42x + 21 = 72x - 54
\]
Simplifying:
\[
42x - 72x = -54 - 21
\]
\[
-30x = -75
\]
\[
x = \frac{75}{30} = \frac{5}{2}
\]

---

### Final solutions:

1. $ x = \frac{4}{15} $  
2. $ x = \frac{6}{7} $  
3. $ x = -\frac{1}{2} $  
6. $ x = 34 $  
7. $ x = 3 $  
8. $ x = \frac{5}{2} $

Would you like to go into more detail on any of these solutions?

---

### Related questions:
1. How do you solve equations with ratios and proportions?
2. What is cross-multiplication and when should you use it?
3. Can you explain more about simplifying equations?
4. How do negative and positive solutions change the interpretation of a problem?
5. What are common mistakes to avoid in solving proportional equations?

**Tip:** Always check your solutions by substituting them back into the original equation!

Learn how to solve proportional equations such as $3x/2 = 2/5$ and $(x - 4)/6 = 5/1$ using cross-multiplication and linear equation techniques. This problem set covers essential concepts in solving proportions, perfect for students in Grades 8-10.

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