Problem 5 - 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. $$\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{4}{6}$$

Step-by-Step Solution

Answer

The means are \(\frac{1}{2}\) and \(4\); the extremes are \(\frac{1}{3}\) and \(6\); both products equal 2.

Step 1: Identify Elements of the Proportion

The given proportion is \( \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{4}{6} \). A proportion represents two ratios that are equal. For this proportion, \( \frac{1}{3} \) and \( 6 \) are the extremes, while \( \frac{1}{2} \) and \( 4 \) are the means.

Step 2: Verify Product of Means

Calculate the product of the means \( \frac{1}{2} \) and \( 4 \) by multiplying them: \( \frac{1}{2} \times 4 = 2 \).

Step 3: Verify Product of Extremes

Calculate the product of the extremes \( \frac{1}{3} \) and \( 6 \) by multiplying them: \( \frac{1}{3} \times 6 = 2 \).

Step 4: Conclusion About Product Equality

Since the product of the means (\(2\)) is equal to the product of the extremes (\(2\)), this confirms that the given proportion is valid and satisfies the property that the product of the means is equal to the product of the extremes.

Key Concepts

Understanding Ratios

Ratios are a fundamental concept in mathematics that help us compare two numbers or amounts. They tell us how much of one thing there is compared to another. For example, in the ratio \( \frac{1}{3} : \frac{1}{2} \), it tells us how \( \frac{1}{3} \) parts compare to \( \frac{1}{2} \) parts. Ratios can be expressed in different forms such as fractions, decimal numbers, or even with a colon between terms, like \( 3 : 4 \). They are great for comparing quantities and making sense of data by showing the relative size of two values.

Whenever you're dealing with ratios, it's important to ensure that both sides of the comparison are in the same units. This maintains accuracy and consistency in comparison. Converting ratios into fractions can often make calculations easier, especially when solving problems involving proportions.

Exploring Means and Extremes

In a proportion, there are two types of terms called means and extremes. A proportion is an equation that states two ratios are equal. In this context, given a proportion \( \frac{a}{b} = \frac{c}{d} \), \(b\) and \(c\) are the means, while \(a\) and \(d\) are called the extremes.

• Means: These are the middle terms in the equal proportion. In our exercise \( \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{4}{6} \), \( \frac{1}{2} \) and \(4\) are identified as the means.
• Extremes: These are the outer terms in the proportion. Similarly, \( \frac{1}{3} \) and \(6\) are known as the extremes in the given example.

Understanding the difference between means and extremes is essential because their relationship helps us verify proportions. The concept of means and extremes is significant as it simplifies the process of checking whether two ratios form a valid proportion.

Verification of Proportions

The verification of proportions involves checking if the product of the means is equal to the product of the extremes. This is a crucial property that can be used to validate a given proportion. If both these products are equal, it confirms that the proportion is correct.

To verify a proportion, like \( \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{4}{6} \), we calculate:

• The product of means: \( \frac{1}{2} \times 4 = 2 \)
  
• The product of extremes: \( \frac{1}{3} \times 6 = 2 \)

As both products are equal to \(2\), this confirms that the given proportion is valid. Therefore, using this method of verification is both reliable and straightforward, offering quick validation for any proportion you encounter. Ensuring this balance of products not only ascertains the correctness of the proportion but also deepens your understanding of ratios and proportion relationships.