Determining Proportions - Pre Algebra

1. Using cross products, explain why $\frac{5}{8}$ and $\frac{15}{25}$ do not form a proportion.

   $5 \cdot 25 = 125$ but $8 \cdot 15 = 120$. Unequal cross products mean the ratios are not equal.

2. What is a proportion?

   An equation stating that two ratios are equal, such as $\frac{2}{3} = \frac{4}{6}$.

3. What unit price shows that 6 pens for 3 dollars and 10 pens for 5 dollars are proportional?

   Half a dollar per pen in both cases. Equal unit rates mean the ratios form a proportion.

4. To what simplest form do $\frac{14}{21}$ and $\frac{10}{15}$ both reduce, showing they form a proportion?

   $\frac{2}{3}$. Divide $\frac{14}{21}$ by $7$ and $\frac{10}{15}$ by $5$.

5. Recipe A uses $2$ cups flour to $3$ cups milk, B uses $4$ to $6$, and C uses $4$ to $5$. Which recipe is not in A's ratio?

   Recipe C. B doubles both terms of $2:3$, but $4:5$ is not a multiple of $2:3$.

6. By what factor is $3:4$ scaled to give the equivalent ratio $12:16$?

   $4$. Both terms are multiplied by $4$: $3 \cdot 4 = 12$ and $4 \cdot 4 = 16$.

7. Why is $3:5$ not equivalent to $4:6$, even though $1$ was added to each term?

   Equivalent ratios come from multiplying or dividing both terms by the same number, not adding. $\frac{3}{5} = 0.6$ but $\frac{4}{6} \approx 0.67$.

8. Which ratio is not equivalent to $2:3$: $4:6$, $6:9$ or $8:10$?

   $8:10$. It simplifies to $4:5$, while the others simplify to $2:3$.

9. What does the cross-products test say about $\frac{a}{b} = \frac{c}{d}$, with $b$ and $d$ nonzero?

   The ratios form a proportion exactly when $ad = bc$.

10. What are the cross products of $\frac{4}{6}$ and $\frac{10}{15}$, and what do they show?

    $4 \cdot 15 = 60$ and $6 \cdot 10 = 60$. They are equal, so the ratios form a proportion.

11. Which ratio forms a proportion with $\frac{3}{4}$: $\frac{6}{9}$, $\frac{9}{12}$ or $\frac{12}{15}$?

    $\frac{9}{12}$. Multiplying both terms of $\frac{3}{4}$ by $3$ gives $\frac{9}{12}$.

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