Question

retrieval
solve each proportion. show your work and check your solution.

1. $\frac{2}{5} = \frac{x}{20}$
2. $\frac{3}{7} = \frac{18}{b}$
3. $\frac{4}{x + 2} = \frac{1}{3}$
4. create three equivalent ratios for the similar polygons.

(image of two similar polygons with side lengths: smaller polygon has sides 2, $a$, 1, $x$, 3; larger polygon has sides 4, 5, $y$, 5, 6)
additional notes:

Explanation:

1. Solve $\frac{2}{5}=\frac{x}{20}$

Step1: Cross-multiply to isolate $x$

$5x = 2 \times 20$

Step2: Calculate right-hand side

$5x = 40$

Step3: Divide by 5 to solve $x$

$x = \frac{40}{5} = 8$

Step4: Check the solution

Substitute $x=8$ into original proportion: $\frac{2}{5} = \frac{8}{20}$, and $\frac{8}{20} = \frac{2}{5}$, so the solution is valid.

2. Solve $\frac{3}{7}=\frac{18}{b}$

Step1: Cross-multiply to isolate $b$

$3b = 7 \times 18$

Step2: Calculate right-hand side

$3b = 126$

Step3: Divide by 3 to solve $b$

$b = \frac{126}{3} = 42$

Step4: Check the solution

Substitute $b=42$ into original proportion: $\frac{3}{7} = \frac{18}{42}$, and $\frac{18}{42} = \frac{3}{7}$, so the solution is valid.

3. Solve $\frac{4}{x+2}=\frac{1}{3}$

Step1: Cross-multiply to isolate $x$

$1 \times (x+2) = 4 \times 3$

Step2: Calculate right-hand side

$x+2 = 12$

Step3: Subtract 2 to solve $x$

$x = 12 - 2 = 10$

Step4: Check the solution

Substitute $x=10$ into original proportion: $\frac{4}{10+2} = \frac{4}{12} = \frac{1}{3}$, which matches the right-hand side, so the solution is valid.

4. Create equivalent ratios for similar polygons

Step1: Match corresponding sides

Small polygon sides: $2, a, 1, 3, x$; Large polygon sides: $4, 5, y, 6, 5$

Step2: Write simplified side ratios

Corresponding pairs give equivalent ratios:
$\frac{2}{4} = \frac{1}{2}$, $\frac{3}{6} = \frac{1}{2}$, $\frac{x}{5} = \frac{1}{2}$
These are three equivalent ratios (we can also use $\frac{a}{5} = \frac{1}{2}$ or $\frac{1}{y} = \frac{1}{2}$ as valid options).

Answer:

1. $x = 8$
2. $b = 42$
3. $x = 10$
4. Three equivalent ratios: $\frac{2}{4}$, $\frac{3}{6}$, $\frac{x}{5}$ (all simplify to $\frac{1}{2}$)