Worksheet for solving proportions with 16 algebraic equations.

Here are the step-by-step solutions for each proportion problem on the assignment. To solve these, we use cross-multiplication: multiply the top of one fraction by the bottom of the other, set them equal to each other, and then solve for $x$.

1. $\frac{1}{x} = \frac{4}{8}$
* Cross-multiply: $1 \cdot 8 = 4 \cdot x$
* Equation: $8 = 4x$
* Divide by 4: $x = 2$

2. $\frac{x}{6} = \frac{21}{x}$
* Cross-multiply: $x \cdot x = 6 \cdot 21$
* Equation: $x^2 = 126$
* Take the square root: $x = \sqrt{126}$
* Simplify: $\sqrt{126} = \sqrt{9 \cdot 14} = 3\sqrt{14}$ (Approx. $11.22$)

3. $\frac{5}{10} = \frac{10}{x}$
* Cross-multiply: $5 \cdot x = 10 \cdot 10$
* Equation: $5x = 100$
* Divide by 5: $x = 20$

4. $\frac{14}{3} = \frac{5}{6}$
* *Note: This appears to be a typo in the worksheet because there is no variable $x$ to solve for, and the statement $14/3 = 5/6$ is mathematically false.*

5. $\frac{x - 3}{8} = \frac{5}{10}$
* Simplify right side first: $\frac{5}{10} = \frac{1}{2}$
* New equation: $\frac{x - 3}{8} = \frac{1}{2}$
* Cross-multiply: $2(x - 3) = 8 \cdot 1$
* Expand: $2x - 6 = 8$
* Add 6: $2x = 14$
* Divide by 2: $x = 7$

6. $\frac{12}{x - 4} = \frac{7}{14}$
* Simplify right side first: $\frac{7}{14} = \frac{1}{2}$
* New equation: $\frac{12}{x - 4} = \frac{1}{2}$
* Cross-multiply: $12 \cdot 2 = 1(x - 4)$
* Equation: $24 = x - 4$
* Add 4: $x = 28$

7. $\frac{12}{6} = \frac{x - 5}{5}$
* Simplify left side first: $\frac{12}{6} = 2$
* New equation: $2 = \frac{x - 5}{5}$
* Multiply by 5: $10 = x - 5$
* Add 5: $x = 15$

8. $\frac{9}{3} = \frac{x + 2}{9}$
* Simplify left side first: $\frac{9}{3} = 3$
* New equation: $3 = \frac{x + 2}{9}$
* Multiply by 9: $27 = x + 2$
* Subtract 2: $x = 25$

9. $\frac{x}{10} = \frac{x + 6}{11}$
* Cross-multiply: $11 \cdot x = 10(x + 6)$
* Expand: $11x = 10x + 60$
* Subtract $10x$: $x = 60$

10. $\frac{x}{x + 8} = \frac{12}{9}$
* Simplify right side first: $\frac{12}{9} = \frac{4}{3}$
* New equation: $\frac{x}{x + 8} = \frac{4}{3}$
* Cross-multiply: $3 \cdot x = 4(x + 8)$
* Expand: $3x = 4x + 32$
* Subtract $4x$: $-x = 32$
* Multiply by -1: $x = -32$

11. $\frac{10}{x} = \frac{9}{x + 7}$
* Cross-multiply: $10(x + 7) = 9 \cdot x$
* Expand: $10x + 70 = 9x$
* Subtract $10x$: $70 = -x$
* Multiply by -1: $x = -70$

12. $\frac{3}{6} = \frac{x}{x - 8}$
* Simplify left side first: $\frac{3}{6} = \frac{1}{2}$
* New equation: $\frac{1}{2} = \frac{x}{x - 8}$
* Cross-multiply: $1(x - 8) = 2 \cdot x$
* Equation: $x - 8 = 2x$
* Subtract $x$: $-8 = x$

13. $\frac{2}{3} = \frac{x - 10}{x + 1}$
* Cross-multiply: $2(x + 1) = 3(x - 10)$
* Expand: $2x + 2 = 3x - 30$
* Subtract $2x$: $2 = x - 30$
* Add 30: $x = 32$

14. $\frac{8}{x + 1} = \frac{7}{x + 9}$
* Cross-multiply: $8(x + 9) = 7(x + 1)$
* Expand: $8x + 72 = 7x + 7$
* Subtract $7x$: $x + 72 = 7$
* Subtract 72: $x = -65$

15. $\frac{x - 1}{3} = \frac{x + 4}{2}$
* Cross-multiply: $2(x - 1) = 3(x + 4)$
* Expand: $2x - 2 = 3x + 12$
* Subtract $2x$: $-2 = x + 12$
* Subtract 12: $x = -14$

16. $\frac{x + 3}{3} = \frac{x - 2}{2}$
* Cross-multiply: $2(x + 3) = 3(x - 2)$
* Expand: $2x + 6 = 3x - 6$
* Subtract $2x$: $6 = x - 6$
* Add 6: $x = 12$

Final Answer:
1. $x = 2$
2. $x = 3\sqrt{14}$ (or approx $11.22$)
3. $x = 20$
4. No solution (Typo in problem)
5. $x = 7$
6. $x = 28$
7. $x = 15$
8. $x = 25$
9. $x = 60$
10. $x = -32$
11. $x = -70$
12. $x = -8$
13. $x = 32$
14. $x = -65$
15. $x = -14$
16. $x = 12$