Question

directions: solve each proportion.

1. $\frac{9}{16}=\frac{x}{12}$
2. $\frac{x-3}{18}=\frac{12}{9}$
3. $\frac{7}{11}=\frac{18}{x+1}$
4. $\frac{3x-4}{14}=\frac{9}{10}$
5. $\frac{17}{15}=\frac{10}{2x-2}$
6. $\frac{x-16}{x+6}=\frac{3}{5}$
7. $\frac{6}{19}=\frac{x-12}{2x-2}$
8. $\frac{x-9}{15}=\frac{2x-9}{10}$
9. $\frac{x-9}{3}=\frac{56}{x+4}$
10. $\frac{7}{x+1}=\frac{2x-1}{36}$

Explanation:

13. Step1: Cross-multiply the proportion

$9 \times 12 = 16x$

13. Step2: Calculate left-hand side

$108 = 16x$

13. Step3: Solve for $x$

$x = \frac{108}{16} = \frac{27}{4} = 6.75$

14. Step1: Cross-multiply the proportion

$9(x-3) = 12 \times 18$

14. Step2: Calculate right-hand side

$9(x-3) = 216$

14. Step3: Divide both sides by 9

$x-3 = 24$

14. Step4: Solve for $x$

$x = 24 + 3 = 27$

15. Step1: Cross-multiply the proportion

$7(x+1) = 18 \times 11$

15. Step2: Calculate right-hand side

$7(x+1) = 198$

15. Step3: Divide both sides by 7

$x+1 = \frac{198}{7}$

15. Step4: Solve for $x$

$x = \frac{198}{7} - 1 = \frac{191}{7} \approx 27.29$

16. Step1: Cross-multiply the proportion

$10(3x-4) = 9 \times 14$

16. Step2: Calculate right-hand side

$10(3x-4) = 126$

16. Step3: Distribute left-hand side

$30x - 40 = 126$

16. Step4: Isolate $30x$

$30x = 126 + 40 = 166$

16. Step5: Solve for $x$

$x = \frac{166}{30} = \frac{83}{15} \approx 5.53$

17. Step1: Cross-multiply the proportion

$17(2x-2) = 15 \times 10$

17. Step2: Calculate right-hand side

$17(2x-2) = 150$

17. Step3: Distribute left-hand side

$34x - 34 = 150$

17. Step4: Isolate $34x$

$34x = 150 + 34 = 184$

17. Step5: Solve for $x$

$x = \frac{184}{34} = \frac{92}{17} \approx 5.41$

18. Step1: Cross-multiply the proportion

$5(x-16) = 3(x+6)$

18. Step2: Distribute both sides

$5x - 80 = 3x + 18$

18. Step3: Isolate $x$ terms

$5x - 3x = 18 + 80$

18. Step4: Solve for $x$

$2x = 98 \implies x = 49$

19. Step1: Cross-multiply the proportion

$6(2x-2) = 19(x-12)$

19. Step2: Distribute both sides

$12x - 12 = 19x - 228$

19. Step3: Isolate $x$ terms

$12x - 19x = -228 + 12$

19. Step4: Solve for $x$

$-7x = -216 \implies x = \frac{216}{7} \approx 30.86$

20. Step1: Cross-multiply the proportion

$10(x-9) = 15(2x-9)$

20. Step2: Distribute both sides

$10x - 90 = 30x - 135$

20. Step3: Isolate $x$ terms

$10x - 30x = -135 + 90$

20. Step4: Solve for $x$

$-20x = -45 \implies x = \frac{45}{20} = \frac{9}{4} = 2.25$

21. Step1: Cross-multiply the proportion

$(x-9)(x+4) = 56 \times 3$

21. Step2: Calculate right-hand side

$(x-9)(x+4) = 168$

21. Step3: Expand left-hand side

$x^2 + 4x - 9x - 36 = 168$

21. Step4: Simplify equation

$x^2 - 5x - 36 - 168 = 0 \implies x^2 - 5x - 204 = 0$

21. Step5: Solve quadratic equation

Use quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a=1, b=-5, c=-204$
$\sqrt{(-5)^2 - 4(1)(-204)} = \sqrt{25 + 816} = \sqrt{841} = 29$
$x = \frac{5 \pm 29}{2}$
$x = \frac{5+29}{2}=17$ or $x = \frac{5-29}{2}=-12$

22. Step1: Cross-multiply the proportion

$7 \times 36 = (x+1)(2x-1)$

22. Step2: Calculate left-hand side

$252 = (x+1)(2x-1)$

22. Step3: Expand right-hand side

$252 = 2x^2 - x + 2x - 1$

22. Step4: Simplify equation

$2x^2 + x - 1 - 252 = 0 \implies 2x^2 + x - 253 = 0$

22. Step5: Solve quadratic equation

Use quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a=2, b=1, c=-253$
$\sqrt{1^2 - 4(2)(-253)} = \sqrt{1 + 2024} = \sqrt{2025} = 45$
$x = \frac{-1 \pm 45}{4}$
$x = \frac{-1+45}{4}=11$ or $x = \frac{-1-45}{4}=-\frac{23}{2}=-11.5$

Answer:

1. $x = \frac{27}{4} = 6.75$
2. $x = 27$
3. $x = \frac{191}{7} \approx 27.29$
4. $x = \frac{83}{15} \approx 5.53$
5. $x = \frac{92}{17} \approx 5.41$
6. $x = 49$
7. $x = \frac{216}{7} \approx 30.86$
8. $x = \frac{9}{4} = 2.25$
9. $x = 17$ or $x = -12$
10. $x = 11$ or $x = -\frac{23}{2} = -11.5$