https://file.aiquickdraw.com/mathbot/file/1725927468666th12em7x.jpg

The image contains several proportion problems. Let's solve each one step by step.

### Problem 1:
\[
\frac{10}{8} = \frac{x}{10}
\]
Cross-multiply to solve for \( x \):
\[
10 \times 10 = 8 \times x \implies 100 = 8x \implies x = \frac{100}{8} = 12.5
\]
**Solution**: \( x = 12.5 \)

---

### Problem 2:
\[
\frac{7}{5} = \frac{x}{3}
\]
Cross-multiply to solve for \( x \):
\[
7 \times 3 = 5 \times x \implies 21 = 5x \implies x = \frac{21}{5} = 4.2
\]
**Solution**: \( x = 4.2 \)

---

### Problem 3:
\[
\frac{9}{6} = \frac{x}{10}
\]
Cross-multiply to solve for \( x \):
\[
9 \times 10 = 6 \times x \implies 90 = 6x \implies x = \frac{90}{6} = 15
\]
**Solution**: \( x = 15 \)

---

### Problem 4:
\[
\frac{8}{n} = \frac{8}{7}
\]
Since the numerators are the same, the denominators must also be equal:
\[
n = 7
\]
**Solution**: \( n = 7 \)

---

### Problem 5:
\[
\frac{4}{3} = \frac{8}{x}
\]
Cross-multiply to solve for \( x \):
\[
4 \times x = 3 \times 8 \implies 4x = 24 \implies x = \frac{24}{4} = 6
\]
**Solution**: \( x = 6 \)

---

### Problem 6:
\[
\frac{7}{b+5} = \frac{10}{5}
\]
Simplify the right-hand side:
\[
\frac{10}{5} = 2
\]
Now solve the equation:
\[
\frac{7}{b+5} = 2 \implies 7 = 2(b+5)
\]
Expand and solve for \( b \):
\[
7 = 2b + 10 \implies 2b = 7 - 10 \implies 2b = -3 \implies b = \frac{-3}{2} = -1.5
\]
**Solution**: \( b = -1.5 \)

---

### Problem 7:
\[
\frac{6}{b-1} = \frac{9}{7}
\]
Cross-multiply to solve for \( b \):
\[
6 \times 7 = 9 \times (b-1) \implies 42 = 9(b-1)
\]
Expand and solve for \( b \):
\[
42 = 9b - 9 \implies 42 + 9 = 9b \implies 51 = 9b \implies b = \frac{51}{9} = 5.67
\]
**Solution**: \( b \approx 5.67 \)

---

### Problem 8:
\[
\frac{4}{m-8} = \frac{2}{7}
\]
Cross-multiply to solve for \( m \):
\[
4 \times 7 = 2 \times (m-8) \implies 28 = 2(m-8)
\]
Expand and solve for \( m \):
\[
28 = 2m - 16 \implies 28 + 16 = 2m \implies 44 = 2m \implies m = \frac{44}{2} = 22
\]
**Solution**: \( m = 22 \)

---

### Problem 9:
\[
\frac{5}{6} = \frac{7n+9}{9}
\]
Cross-multiply to solve for \( n \):
\[
5 \times 9 = 6 \times (7n + 9) \implies 45 = 6(7n + 9)
\]
Expand and solve for \( n \):
\[
45 = 42n + 54 \implies 42n = 45 - 54 \implies 42n = -9 \implies n = \frac{-9}{42} = -\frac{3}{14}
\]
**Solution**: \( n = -\frac{3}{14} \)

---

### Problem 10:
\[
\frac{4}{9} = \frac{r-3}{6}
\]
Cross-multiply to solve for \( r \):
\[
4 \times 6 = 9 \times (r-3) \implies 24 = 9(r-3)
\]
Expand and solve for \( r \):
\[
24 = 9r - 27 \implies 24 + 27 = 9r \implies 51 = 9r \implies r = \frac{51}{9} = 5.67
\]
**Solution**: \( r \approx 5.67 \)

---

Let me know if you need further clarification or details on any of these steps!

### Relative Questions:
1. How can you verify the solutions of proportion problems?
2. What is the general method for solving proportions?
3. Can you always use cross-multiplication to solve proportions?
4. How can you apply proportions in real-life situations?
5. What are some common mistakes to avoid when solving proportions?

### Tip:
Always simplify fractions if possible before solving, as this can make calculations easier and reduce errors.

Step-by-step guide on solving proportion problems using cross-multiplication. Learn how to solve equations involving ratios and proportions, suitable for students in Grades 7-9.

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