Problem 7 - Prealgebra

Question

For each of the following proportions, name the means, name the extremes, and show that the product of the means is equal to the product of the extremes. $$\frac{0.5}{5}=\frac{1}{10}$$

Step-by-Step Solution

Answer

Means: 5 and 1; Extremes: 0.5 and 10; both products equal 5.

Step 1: Identify the Proportion

The given proportion is \( \frac{0.5}{5} = \frac{1}{10} \). A proportion is an equation that states that two ratios are equivalent.

Step 2: Identify Means and Extremes

In the given proportion \( \frac{0.5}{5} = \frac{1}{10} \), the 'means' are the inner terms 5 and 1, and the 'extremes' are the outer terms 0.5 and 10.

Step 3: Calculate the Product of the Means

Multiply the means: \( 5 \times 1 = 5 \). This is the product of the means.

Step 4: Calculate the Product of the Extremes

Multiply the extremes: \( 0.5 \times 10 = 5 \). This is the product of the extremes.

Step 5: Compare Products

Compare the product of the means with the product of the extremes: both are equal to 5, confirming the given proportion.

Key Concepts

Means and Extremes

When discussing proportions like \( \frac{0.5}{5} = \frac{1}{10} \), it's important to understand the terms "means" and "extremes." In a proportion written as \( \frac{a}{b} = \frac{c}{d} \), the means are the middle terms, \(b\) and \(c\), while the extremes are the outer terms, \(a\) and \(d\). Knowing which terms are means and extremes helps you to use the cross-multiplication method. In our example, the means are 5 and 1, and the extremes are 0.5 and 10. To solve or check a proportion, you need to be able to identify these terms properly so that you can verify the equality of their products.

Equivalent Ratios

A proportion represents equivalent ratios. This means that the two fractions or ratios, \( \frac{0.5}{5} \) and \( \frac{1}{10} \), represent the same relationship between numbers.

• \( \frac{0.5}{5} \) indicates that for every 5 units of the second quantity, there are 0.5 units of the first quantity.
• \( \frac{1}{10} \) tells you that for every 10 units of the second, there's 1 unit of the first.

Despite being written differently, these two ratios are the same because their cross products (means and extremes) are equal, confirming their equivalence.

Product Equality in Proportions

A key property of proportions is that the product of the means equals the product of the extremes. This property allows you to confirm if two ratios form a proportion. For the proportion \( \frac{0.5}{5} = \frac{1}{10} \):- Calculate the means product: \( 5 \times 1 = 5 \)- Calculate the extremes product: \( 0.5 \times 10 = 5 \)Both products equal 5, which checks out, showing that the proportion is valid. This concept of "product equality" is crucial, not only for verifying proportions but also for solving problems involving ratios in real-world applications, such as determining how changes in one variable affect another.