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: $x = 12.5$

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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: $x = 4.2$

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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: $x = 15$

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Problem 4:

$\frac{8}{n} = \frac{8}{7}$ Since the numerators are the same, the denominators must also be equal: $n = 7$ Solution: $n = 7$

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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: $x = 6$

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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: $b = -1.5$

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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: $b \approx 5.67$

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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: $m = 22$

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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: $n = -\frac{3}{14}$

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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: $r \approx 5.67$

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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.