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Chapter 6: Problem 34

Determine whether the pairs of numbers are proportional. Are the numbers \(-7.1\) and 2.4 proportional to the numbers \(-35.5\) and \(10 ?\)

Short Answer

Expert verified
No, the pairs are not proportional.

Step by step solution

01

- Understand the Problem

To determine if two pairs of numbers are proportional, compare their ratios. If the ratios are equal, the pairs are proportional.
02

- Calculate the First Ratio

Find the ratio of the first pair of numbers \(-7.1\) and \(2.4\). The ratio is given by \(\frac{-7.1}{2.4}\).
03

- Simplify the First Ratio

Use a calculator to simplify \(\frac{-7.1}{2.4} \approx -2.9583\).
04

- Calculate the Second Ratio

Find the ratio of the second pair of numbers \(-35.5\) and \(10\). The ratio is given by \(\frac{-35.5}{10}\).
05

- Simplify the Second Ratio

Use a calculator to simplify \(\frac{-35.5}{10}\ \approx -3.55\).
06

- Compare the Ratios

Compare the two ratios: \(-2.9583\) and \(-3.55\). Since they are not equal, the pairs are not proportional.

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Key Concepts

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

Ratios
A ratio compares two quantities, showing the relative size of one quantity to another. Imagine you have a ratio of apples to oranges. If you have 3 apples for every 2 oranges, the ratio is written as \(\frac{3}{2}\) or 3:2.

Here are some key points:
  • A ratio is essentially a fraction.
  • It tells us how much one thing exists compared to another.
  • Ratios can be simplified just like fractions to their simplest form.


For example, in the exercise, the first ratio is between \-7.1\ and \2.4\. This is written as \(\frac{-7.1}{2.4}\). In a real-world context, ratios are used in recipes, maps, and even currency exchange.
Proportional Relationships
Two pairs of numbers are said to be in a proportional relationship if their ratios are equal. For instance, if you have a ratio of \ \frac{a}{b} \ and another ratio \ \frac{c}{d} \, they are proportional if \(\frac{a}{b} = \frac{c}{d}\).

Consider the exercise where we need to check if \ [-7.1, 2.4] \ is proportional to \ [-35.5, 10] \. First, we find each pair's ratios: \ \frac{-7.1}{2.4} \ and \ \frac{-35.5}{10} \ and then simplify them.
  • The first ratio simplifies to approximately \ -2.9583 \.
  • The second ratio simplifies to \ -3.55 \.


Since these ratios are not equal, the pairs \ [-7.1, 2.4] \ and \ [-35.5, 10] \ are not proportional.
Comparing Fractions
Comparing fractions or ratios helps determine if pairs of numbers share a proportional relationship. When comparing fractions, the key is getting a common form or simplifying each fraction as much as possible.

Here鈥檚 how you can do it:
  • Find each fraction鈥檚 simplest form by dividing the numerator by the denominator.
  • Use tools like calculators for complex decimals or fractions.
  • Compare the simplified forms directly.


In this exercise, we compare \( \frac{-7.1}{2.4} \) and \( \frac{-35.5}{10} \) by simplifying them to \ -2.9583 \ and \ -3.55 \, respectively. Since they are not equal, the original pairs do not have a proportional relationship.

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Most popular questions from this chapter

Solve the proportion. Be sure to check your answers. $$\frac{16}{12}=\frac{21}{a}$$

Solve the proportion. $$\frac{3}{2.1}=\frac{7}{y}$$

Write a proportion for each statement. Then solve for the variable. 65 is to 15 as 26 is to \(y\)

Pierre bought 480 Euros with \(\$ 750\) American. At this rate, how many Euros can he buy with \(\$ 900\) American?

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