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Chapter 6: Problem 19

Write a proportion for each statement. 1 in, is to 7 mi as 5 in. is to 35 mi.

Short Answer

Expert verified
\[ \frac{1 \text{ inch}}{7 \text{ miles}} = \frac{5 \text{ inches}}{35 \text{ miles}} \]

Step by step solution

01

- Understand the terms

Identify the corresponding parts of the given statement. Here, 1 inch corresponds to 7 miles, and 5 inches correspond to 35 miles.
02

- Set up the proportion

Write the given relationships as fractions and then set them equal to each other to form a proportion. This results in \( \frac{1 \text{ inch}}{7 \text{ miles}} = \frac{5 \text{ inches}}{35 \text{ miles}} \).
03

- Simplify the terms

Check if the fractions are in their simplest form. Both \( \frac{1 \text{ inch}}{7 \text{ miles}} \) and \( \frac{5 \text{ inches}}{35 \text{ miles}} \) are already simplified, and their ratios indeed equal each other.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportional Relationships
Proportional relationships occur when two ratios remain constant. In this exercise, we are comparing inches to miles.
The proportional relationship given is: 1 inch is to 7 miles as 5 inches is to 35 miles.
To understand the relationship, identify the corresponding parts. Here, 1 inch and 5 inches are compared to their respective miles (7 miles and 35 miles).
If both ratios give the same value when simplified, then they are proportional.
This concept can be applied to various subjects, like converting units or creating scale models.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form.
A fraction is in its simplest form when the numerator and the denominator share no common factors besides 1.
In this exercise: \( \frac{1 \text{ inch}}{7 \text{ miles}} \) and \( \frac{5 \text{ inches}}{35 \text{ miles}} \) are fractions.
Checking these fractions, we find both are already simplified.
\( \frac{1}{7} \) contains no common factors, and \( \frac{5}{35} \) can be simplified by dividing both the numerator and the denominator by 5, which results in \( \frac{1}{7} \).
Always simplify fractions to ensure accuracy in proportional relationships.
Mathematical Ratios
A ratio compares the sizes of two or more quantities. It's a way of showing a relationship between numbers.
Ratios can be written in three ways: \( a:b \), \[ \frac{a}{b} \], or using the word 'to'.
In this problem, we have the ratios 1 inch is to 7 miles and 5 inches to 35 miles.
Writing these as fractions helps compare them easily: \( \frac{1 \text{ inch}}{7 \text{ miles}} \) and \( \frac{5 \text{ inches}}{35 \text{ miles}} \).
Simplifying these fractions provides insight into whether the proportional relationships hold true, resulting in 1:7.
Educational Step-by-Step Solutions
Step-by-step solutions simplify and break down complex problems into easy-to-understand steps.
Here's how we approached this exercise:
  • Identify corresponding parts of the statement (1 inch to 7 miles and 5 inches to 35 miles).
  • Set up the proportion by writing these relationships as fractions: \( \frac{1 \text{ inch}}{7 \text{ miles}} = \frac{5 \text{ inches}}{35 \text{ miles}} \).
  • Simplify the fractions to confirm they match.
  • Verify the ratios are equal, confirming the proportional relationship.
This method helps ensure you understand each part of the problem.

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Most popular questions from this chapter

Determine whether the given value is a solution to the proportion. $$\frac{12.4}{31}=\frac{8.2}{y} ; \quad y=20$$

Solve the proportion. Be sure to check your answers. $$\frac{15}{10}=\frac{3}{x}$$

Refer to the table that shows the average spending per person for reading (books, newspapers, magazines, etc.) by age group. Write each ratio in lowest terms. (PICTURE NOT COPY) \begin{array}{|l|c|} \hline \text { Age Group } & \text { Annual Average (\$) } \\ \hline \text { Under 25 years } & 60 \\ \hline 25 \text { to } 34 \text { years } & 111 \\ \hline 35 \text { to } 44 \text { years } & 136 \\ \hline 45 \text { to } 54 \text { years } & 172 \\ \hline 55 \text { to } 64 \text { years } & 183 \\ \hline 65 \text { to } 74 \text { years } & 159 \\ \hline 75 \text { years and over } & 128 \\ \hline \end{array} Find the ratio of spending for the group 25 to 34 years old to the spending for the group of 65 to 74 years old.

Solve the proportion. $$\frac{4}{3}=\frac{n}{5}$$

Solve the proportions. $$\frac{x-2}{4 x}=\frac{3}{8}$$
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