Proportions: Practice

Cross-products run along the diagonals: top-left times bottom-right, and bottom-left times top-right.

$2 \times 10 \quad \text{and} \quad 5 \times 4$

Multiplying straight across ($2 \times 4$) is not a cross-product.

Compare the cross-products, the diagonal products.

$2 \times 5 = 10 \qquad 3 \times 4 = 12$

Since $10 \neq 12$, the ratios are not equal, so the proportion is false. Adding $2$ to each part does not keep a ratio equal.

A proportion $\frac{a}{b} = \frac{c}{d}$ is read "$a$ is to $b$ as $c$ is to $d$," taking each ratio in order.

$\frac{4}{9} = \frac{8}{18} \;\Rightarrow\; 4 \text{ is to } 9 \text{ as } 8 \text{ is to } 18$

The number of pens doubled from $2$ to $4$, so the cost doubles too. Set up pens over dollars: $\frac{2}{6} = \frac{4}{c}$, scale factor $2$.

$c = 6 \times 2 = 12$

So $4$ pens cost $12$ dollars.

Simplify $\frac{6}{9}$ by dividing both parts by their greatest common factor, $3$.

$\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}$

Both ratios equal $\frac{2}{3}$, so the proportion is true.

The top went from $2$ to $8$, so the scale factor is $8 \div 2 = 4$. Multiply the bottom by $4$.

$7 \times 4 = 28$

So the blank is $28$, giving $\frac{2}{7} = \frac{8}{28}$.

Set up eggs over cups on both sides: $\frac{1}{2} = \frac{e}{8}$. The bottom went from $2$ to $8$, a scale factor of $8 \div 2 = 4$.

$e = 1 \times 4 = 4$

So $4$ eggs are needed.

The top went from $4$ to $12$, so the scale factor is $12 \div 4 = 3$. Multiply the bottom by $3$.

$n = 5 \times 3 = 15$

So $\frac{4}{5} = \frac{12}{15}$.

Check $\frac{3}{5} = \frac{6}{9}$ with cross-products.

$3 \times 9 = 27 \qquad 5 \times 6 = 30$

Since $27 \neq 30$, this one is false. The other three have equal cross-products.

The top went from $5$ to $10$, a scale factor of $10 \div 5 = 2$. Multiply the bottom $3$ by the same factor.

$3 \times 2 = 6$

So the blank is $6$, giving $\frac{10}{6} = \frac{5}{3}$.

The bottom went from $4$ to $8$, a scale factor of $8 \div 4 = 2$. Multiply the top by $2$.

$n = 3 \times 2 = 6$

So $\frac{6}{8} = \frac{3}{4}$.

A ratio is in proportion with $\frac{2}{3}$ when it reduces to $\frac{2}{3}$. Check $\frac{6}{9}$ by dividing both parts by $3$.

$\frac{6}{9} = \frac{2}{3}$

So $\frac{6}{9}$ forms a true proportion with $\frac{2}{3}$.