(1) If $$ 2 a-5 b=0 $$, find: $$ \frac{a}{b} $$ (2) If $$ \frac{x+2 y}{4 x-3 y}=\frac{7}{6} $$, then prove that: $$ \frac{x}{y}=\frac{3}{2} $$ 3 If $$ 4 a^{2}-9 b^{2}=0 $$, find $$ : a: b $$

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Let's solve each problem step by step.

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### **Problem (1)**
**Given:** $$ 2a - 5b = 0 $$  
**Find:** $$ \frac{a}{b} $$

**Solution:**

1. Start with the equation:
   $$
   2a - 5b = 0
   $$
2. Rearrange to solve for $$ a $$:
   $$
   2a = 5b
   $$
3. Divide both sides by 2:
   $$
   a = \frac{5b}{2}
   $$
4. Now, find $$ \frac{a}{b} $$:
   $$
   \frac{a}{b} = \frac{\frac{5b}{2}}{b} = \frac{5}{2}
   $$

**Answer:** $$ \frac{a}{b} = \frac{5}{2} $$

---

### **Problem (2)**
**Given:** 
$$
\frac{x + 2y}{4x - 3y} = \frac{7}{6}
$$  
**Prove that:** $$ \frac{x}{y} = \frac{3}{2} $$

**Solution:**

1. Start with the given proportion:
   $$
   \frac{x + 2y}{4x - 3y} = \frac{7}{6}
   $$
2. Cross-multiply to eliminate the fractions:
   $$
   6(x + 2y) = 7(4x - 3y)
   $$
3. Expand both sides:
   $$
   6x + 12y = 28x - 21y
   $$
4. Bring like terms to one side:
   $$
   6x + 12y - 28x + 21y = 0
   $$
   $$
   -22x + 33y = 0
   $$
5. Simplify the equation:
   $$
   22x = 33y
   $$
6. Divide both sides by 11:
   $$
   2x = 3y
   $$
7. Solve for $$ \frac{x}{y} $$:
   $$
   \frac{x}{y} = \frac{3}{2}
   $$

**Conclusion:** $$ \frac{x}{y} = \frac{3}{2} $$

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### **Problem (3)**
**Given:** $$ 4a^{2} - 9b^{2} = 0 $$  
**Find:** The ratio $$ a : b $$

**Solution:**

1. Start with the equation:
   $$
   4a^{2} - 9b^{2} = 0
   $$
2. Rearrange the equation:
   $$
   4a^{2} = 9b^{2}
   $$
3. Divide both sides by $$ b^{2} $$ (assuming $$ b \neq 0 $$):
   $$
   \frac{4a^{2}}{b^{2}} = 9
   $$
4. Take the square root of both sides:
   $$
   \frac{2a}{b} = 3
   $$
5. Solve for $$ \frac{a}{b} $$:
   $$
   \frac{a}{b} = \frac{3}{2}
   $$
6. Therefore, the ratio $$ a : b $$ is:
   $$
   a : b = 3 : 2
   $$

**Answer:** The ratio $$ a : b $$ is $$ 3 : 2 $$.

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**Solutions:** 1. $$ \frac{a}{b} = \frac{5}{2} $$ 2. $$ \frac{x}{y} = \frac{3}{2} $$ 3. $$ a : b = 3 : 2 $$

(1) If , find:
(2) If , then prove that:
3 If , find

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