Math worksheet for solving ratios and proportions, designed to practice solving for x in various algebraic equations.

Here are the step-by-step solutions for each problem on the worksheet.

1. $\frac{4}{x - 8} = \frac{8}{2}$
First, simplify the right side: $\frac{8}{2} = 4$.
So, $\frac{4}{x - 8} = 4$.
Multiply both sides by $(x - 8)$: $4 = 4(x - 8)$.
Divide by 4: $1 = x - 8$.
Add 8 to both sides: $x = 9$.

2. $\frac{x - 5}{x + 8} = \frac{2}{7}$
Cross-multiply: $7(x - 5) = 2(x + 8)$.
Expand: $7x - 35 = 2x + 16$.
Subtract $2x$ from both sides: $5x - 35 = 16$.
Add 35 to both sides: $5x = 51$.
Divide by 5: $x = 10.2$ (or $\frac{51}{5}$).

3. $\frac{x + 10}{x - 7} = \frac{8}{9}$
Cross-multiply: $9(x + 10) = 8(x - 7)$.
Expand: $9x + 90 = 8x - 56$.
Subtract $8x$ from both sides: $x + 90 = -56$.
Subtract 90 from both sides: $x = -146$.

4. $\frac{6}{x - 1} = \frac{9}{7}$
Cross-multiply: $6(7) = 9(x - 1)$.
Simplify: $42 = 9x - 9$.
Add 9 to both sides: $51 = 9x$.
Divide by 9: $x = \frac{51}{9}$.
Simplify the fraction (divide top and bottom by 3): $x = \frac{17}{3}$ (or $5\frac{2}{3}$).

5. $\frac{x}{12} = \frac{10}{2}$
Simplify the right side: $\frac{10}{2} = 5$.
So, $\frac{x}{12} = 5$.
Multiply both sides by 12: $x = 60$.

6. $\frac{40}{25} = \frac{x}{20}$
Simplify the left side: $\frac{40}{25}$ reduces to $\frac{8}{5}$ (divide by 5).
So, $\frac{8}{5} = \frac{x}{20}$.
Multiply both sides by 20: $x = \frac{8 \times 20}{5}$.
$x = 8 \times 4$.
$x = 32$.

7. $\frac{40}{24} = \frac{20}{x}$
Cross-multiply: $40x = 24 \times 20$.
$40x = 480$.
Divide by 40: $x = 12$.
*(Alternatively, notice that 40 is double 20, so the denominator 24 must be double x. Half of 24 is 12).*

8. $\frac{11}{10} = \frac{x}{11}$
Cross-multiply: $11 \times 11 = 10x$.
$121 = 10x$.
Divide by 10: $x = 12.1$.

9. $\frac{x - 3}{x} = \frac{9}{10}$
Cross-multiply: $10(x - 3) = 9x$.
Expand: $10x - 30 = 9x$.
Subtract $9x$ from both sides: $x - 30 = 0$.
Add 30 to both sides: $x = 30$.

10. $\frac{x}{x - 3} = \frac{2}{3}$
Cross-multiply: $3x = 2(x - 3)$.
Expand: $3x = 2x - 6$.
Subtract $2x$ from both sides: $x = -6$.

Final Answer:
1. x = 9
2. x = 10.2
3. x = -146
4. x = 17/3
5. x = 60
6. x = 32
7. x = 12
8. x = 12.1
9. x = 30
10. x = -6