Question

lesson 5-4
determine whether the ratios are proportional.

1. \\(\frac{25}{40}, \frac{30}{48}\\) 14. \\(\frac{32}{36}, \frac{24}{28}\\) 15. \\(\frac{5}{6}, \frac{15}{18}\\) 16. \\(\frac{21}{49}, \frac{18}{42}\\)

find a ratio equivalent to each ratio. then use the ratios to write a proportion.

1. \\(\frac{72}{81}\\) 18. \\(\frac{15}{40}\\) 19. \\(\frac{24}{32}\\) 20. \\(\frac{5}{13}\\)

lesson 5-5

Explanation:

Let's solve these problems one by one. We'll start with determining if the ratios are proportional (problems 13 - 16) and then finding equivalent ratios and writing proportions (problems 17 - 20).

Problem 13: Determine if $\boldsymbol{\frac{25}{40}}$ and $\boldsymbol{\frac{30}{48}}$ are proportional.

To check if two ratios are proportional, we can simplify both or cross - multiply.

• Simplify $\frac{25}{40}$: Divide numerator and denominator by 5. $\frac{25\div5}{40\div5}=\frac{5}{8}$.
• Simplify $\frac{30}{48}$: Divide numerator and denominator by 6. $\frac{30\div6}{48\div6}=\frac{5}{8}$.

Since both simplify to $\frac{5}{8}$, the ratios are proportional.

Problem 14: Determine if $\boldsymbol{\frac{32}{36}}$ and $\boldsymbol{\frac{24}{28}}$ are proportional.

• Simplify $\frac{32}{36}$: Divide numerator and denominator by 4. $\frac{32\div4}{36\div4}=\frac{8}{9}$.
• Simplify $\frac{24}{28}$: Divide numerator and denominator by 4. $\frac{24\div4}{28\div4}=\frac{6}{7}$.

Since $\frac{8}{9}
eq\frac{6}{7}$, the ratios are not proportional.

Problem 15: Determine if $\boldsymbol{\frac{5}{6}}$ and $\boldsymbol{\frac{15}{18}}$ are proportional.

• Simplify $\frac{15}{18}$: Divide numerator and denominator by 3. $\frac{15\div3}{18\div3}=\frac{5}{6}$.

Since $\frac{5}{6}=\frac{5}{6}$, the ratios are proportional.

Problem 16: Determine if $\boldsymbol{\frac{21}{49}}$ and $\boldsymbol{\frac{18}{42}}$ are proportional.

• Simplify $\frac{21}{49}$: Divide numerator and denominator by 7. $\frac{21\div7}{49\div7}=\frac{3}{7}$.
• Simplify $\frac{18}{42}$: Divide numerator and denominator by 6. $\frac{18\div6}{42\div6}=\frac{3}{7}$.

Since both simplify to $\frac{3}{7}$, the ratios are proportional.

Problem 17: Find an equivalent ratio to $\boldsymbol{\frac{72}{81}}$ and write a proportion.

To find an equivalent ratio, we can divide numerator and denominator by a common factor. Let's divide by 9.

• $\frac{72\div9}{81\div9}=\frac{8}{9}$.
• The proportion is $\frac{72}{81}=\frac{8}{9}$.

Problem 18: Find an equivalent ratio to $\boldsymbol{\frac{15}{40}}$ and write a proportion.

We can divide numerator and denominator by 5.

• $\frac{15\div5}{40\div5}=\frac{3}{8}$.
• The proportion is $\frac{15}{40}=\frac{3}{8}$.

Problem 19: Find an equivalent ratio to $\boldsymbol{\frac{24}{32}}$ and write a proportion.

We can divide numerator and denominator by 8.

• $\frac{24\div8}{32\div8}=\frac{3}{4}$.
• The proportion is $\frac{24}{32}=\frac{3}{4}$.

Problem 20: Find an equivalent ratio to $\boldsymbol{\frac{5}{13}}$ and write a proportion.

We can multiply numerator and denominator by 2.

• $\frac{5\times2}{13\times2}=\frac{10}{26}$.
• The proportion is $\frac{5}{13}=\frac{10}{26}$.

Summary of Answers:

1. The ratios $\frac{25}{40}$ and $\frac{30}{48}$ are proportional.
2. The ratios $\frac{32}{36}$ and $\frac{24}{28}$ are not proportional.
3. The ratios $\frac{5}{6}$ and $\frac{15}{18}$ are proportional.
4. The ratios $\frac{21}{49}$ and $\frac{18}{42}$ are proportional.
5. Equivalent ratio: $\frac{8}{9}$, Proportion: $\frac{72}{81}=\frac{8}{9}$
6. Equivalent ratio: $\frac{3}{8}$, Proportion: $\frac{15}{40}=\frac{3}{8}$
7. Equivalent ratio: $\frac{3}{4}$, Proportion: $\frac{24}{32}=\frac{3}{4}$
8. Equivalent ratio: $\frac{10}{26}$, Proportion: $\frac{5}{13}=\frac{10}{26}$

Answer:

Let's solve these problems one by one. We'll start with determining if the ratios are proportional (problems 13 - 16) and then finding equivalent ratios and writing proportions (problems 17 - 20).

Problem 13: Determine if $\boldsymbol{\frac{25}{40}}$ and $\boldsymbol{\frac{30}{48}}$ are proportional.

To check if two ratios are proportional, we can simplify both or cross - multiply.

• Simplify $\frac{25}{40}$: Divide numerator and denominator by 5. $\frac{25\div5}{40\div5}=\frac{5}{8}$.
• Simplify $\frac{30}{48}$: Divide numerator and denominator by 6. $\frac{30\div6}{48\div6}=\frac{5}{8}$.

Since both simplify to $\frac{5}{8}$, the ratios are proportional.

Problem 14: Determine if $\boldsymbol{\frac{32}{36}}$ and $\boldsymbol{\frac{24}{28}}$ are proportional.

• Simplify $\frac{32}{36}$: Divide numerator and denominator by 4. $\frac{32\div4}{36\div4}=\frac{8}{9}$.
• Simplify $\frac{24}{28}$: Divide numerator and denominator by 4. $\frac{24\div4}{28\div4}=\frac{6}{7}$.

Since $\frac{8}{9}
eq\frac{6}{7}$, the ratios are not proportional.

Problem 15: Determine if $\boldsymbol{\frac{5}{6}}$ and $\boldsymbol{\frac{15}{18}}$ are proportional.

• Simplify $\frac{15}{18}$: Divide numerator and denominator by 3. $\frac{15\div3}{18\div3}=\frac{5}{6}$.

Since $\frac{5}{6}=\frac{5}{6}$, the ratios are proportional.

Problem 16: Determine if $\boldsymbol{\frac{21}{49}}$ and $\boldsymbol{\frac{18}{42}}$ are proportional.

• Simplify $\frac{21}{49}$: Divide numerator and denominator by 7. $\frac{21\div7}{49\div7}=\frac{3}{7}$.
• Simplify $\frac{18}{42}$: Divide numerator and denominator by 6. $\frac{18\div6}{42\div6}=\frac{3}{7}$.

Since both simplify to $\frac{3}{7}$, the ratios are proportional.

Problem 17: Find an equivalent ratio to $\boldsymbol{\frac{72}{81}}$ and write a proportion.

To find an equivalent ratio, we can divide numerator and denominator by a common factor. Let's divide by 9.

• $\frac{72\div9}{81\div9}=\frac{8}{9}$.
• The proportion is $\frac{72}{81}=\frac{8}{9}$.

Problem 18: Find an equivalent ratio to $\boldsymbol{\frac{15}{40}}$ and write a proportion.

We can divide numerator and denominator by 5.

• $\frac{15\div5}{40\div5}=\frac{3}{8}$.
• The proportion is $\frac{15}{40}=\frac{3}{8}$.

Problem 19: Find an equivalent ratio to $\boldsymbol{\frac{24}{32}}$ and write a proportion.

We can divide numerator and denominator by 8.

• $\frac{24\div8}{32\div8}=\frac{3}{4}$.
• The proportion is $\frac{24}{32}=\frac{3}{4}$.

Problem 20: Find an equivalent ratio to $\boldsymbol{\frac{5}{13}}$ and write a proportion.

We can multiply numerator and denominator by 2.

• $\frac{5\times2}{13\times2}=\frac{10}{26}$.
• The proportion is $\frac{5}{13}=\frac{10}{26}$.

Summary of Answers:

1. The ratios $\frac{25}{40}$ and $\frac{30}{48}$ are proportional.
2. The ratios $\frac{32}{36}$ and $\frac{24}{28}$ are not proportional.
3. The ratios $\frac{5}{6}$ and $\frac{15}{18}$ are proportional.
4. The ratios $\frac{21}{49}$ and $\frac{18}{42}$ are proportional.
5. Equivalent ratio: $\frac{8}{9}$, Proportion: $\frac{72}{81}=\frac{8}{9}$
6. Equivalent ratio: $\frac{3}{8}$, Proportion: $\frac{15}{40}=\frac{3}{8}$
7. Equivalent ratio: $\frac{3}{4}$, Proportion: $\frac{24}{32}=\frac{3}{4}$
8. Equivalent ratio: $\frac{10}{26}$, Proportion: $\frac{5}{13}=\frac{10}{26}$