Question

directions: solve each proportion.

1. \\(\frac{9}{16} = \frac{x}{12}\\)
2. \\(\frac{x - 3}{10} = \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:

Problem 13

Step1: Cross-multiply the proportion

$9 \times 12 = 16x$

Step2: Calculate left side, solve for x

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

Problem 14

Step1: Cross-multiply the proportion

$9(x-3) = 12 \times 10$

Step2: Simplify and solve for x

$9x - 27 = 120 \implies 9x = 147 \implies x = \frac{147}{9} = \frac{49}{3}$

Problem 15

Step1: Cross-multiply the proportion

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

Step2: Simplify and solve for x

$7x + 7 = 198 \implies 7x = 191 \implies x = \frac{191}{7}$

Problem 16

Step1: Cross-multiply the proportion

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

Step2: Simplify and solve for x

$30x - 40 = 126 \implies 30x = 166 \implies x = \frac{166}{30} = \frac{83}{15}$

Problem 17

Step1: Cross-multiply the proportion

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

Step2: Simplify and solve for x

$34x - 34 = 150 \implies 34x = 184 \implies x = \frac{184}{34} = \frac{92}{17}$

Problem 18

Step1: Cross-multiply the proportion

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

Step2: Simplify and solve for x

$5x - 80 = 3x + 18 \implies 2x = 98 \implies x = 49$

Problem 19

Step1: Cross-multiply the proportion

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

Step2: Simplify and solve for x

$12x - 12 = 19x - 228 \implies -7x = -216 \implies x = \frac{216}{7}$

Problem 20

Step1: Cross-multiply the proportion

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

Step2: Simplify and solve for x

$10x - 90 = 30x - 135 \implies -20x = -45 \implies x = \frac{45}{20} = \frac{9}{4}$

Problem 21

Step1: Cross-multiply the proportion

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

Step2: Expand and simplify to quadratic

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

Step3: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1, b=-5, c=-204$
$\sqrt{(-5)^2-4(1)(-204)} = \sqrt{25+816} = \sqrt{841}=29$
$x=\frac{5\pm29}{2}$
$x=\frac{34}{2}=17$ or $x=\frac{-24}{2}=-12$

Problem 22

Step1: Cross-multiply the proportion

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

Step2: Expand and simplify to quadratic

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

Step3: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=2, b=1, c=-253$
$\sqrt{1^2-4(2)(-253)} = \sqrt{1+2024} = \sqrt{2025}=45$
$x=\frac{-1\pm45}{4}$
$x=\frac{44}{4}=11$ or $x=\frac{-46}{4}=-\frac{23}{2}$

Answer:

1. $\boldsymbol{x=\frac{27}{4}}$
2. $\boldsymbol{x=\frac{49}{3}}$
3. $\boldsymbol{x=\frac{191}{7}}$
4. $\boldsymbol{x=\frac{83}{15}}$
5. $\boldsymbol{x=\frac{92}{17}}$
6. $\boldsymbol{x=49}$
7. $\boldsymbol{x=\frac{216}{7}}$
8. $\boldsymbol{x=\frac{9}{4}}$
9. $\boldsymbol{x=17}$ or $\boldsymbol{x=-12}$
10. $\boldsymbol{x=11}$ or $\boldsymbol{x=-\frac{23}{2}}$