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

Step-by-Step Solution

Answer

Means: \(\frac{1}{4}, 4\) ; Extremes: \(2, \frac{1}{2}\); Products: Both equal \(1\).

Step 1: Identify the Proportion

The given proportion is \( \frac{2}{\frac{1}{4}} = \frac{4}{\frac{1}{2}} \). In this proportion, we need to identify the means and the extremes.

Step 2: Identify the Means

In a proportion \( \frac{a}{b} = \frac{c}{d} \), the means are \(b\) and \(c\). For the given proportion, the means are \(\frac{1}{4}\) and \(4\).

Step 3: Identify the Extremes

In the same proportion \( \frac{a}{b} = \frac{c}{d} \), the extremes are \(a\) and \(d\). For the given proportion, the extremes are \(2\) and \(\frac{1}{2}\).

Step 4: Calculate Product of Means

Calculate the product of the means: \( \frac{1}{4} \times 4 \). Simplifying yields \(1\).

Step 5: Calculate Product of Extremes

Calculate the product of the extremes: \( 2 \times \frac{1}{2} \). Simplifying also yields \(1\).

Step 6: Confirm Products are Equal

Since the product of the means (\(1\)) is equal to the product of the extremes (\(1\)), the property of proportions holds true for this equation.

Key Concepts

Means and Extremes in Proportions

When dealing with proportions, it's important to understand the terms 'means' and 'extremes'. A proportion is an equation that states two ratios are equal, written in the form \( \frac{a}{b} = \frac{c}{d} \). In this setup:

• The 'means' are the middle terms of the proportion, which are \(b\) and \(c\).
• The 'extremes' are the outer terms, which are \(a\) and \(d\).

Understanding how to find these pairs helps you solve and verify proportions correctly. In the exercise you're working on, we identified the means as \( \frac{1}{4} \) and \(4\), and the extremes as \(2\) and \( \frac{1}{2} \). Knowing these roles, you'll better understand and balance proportions effectively.

Products of Fractions in Proportions

Proportions aren't just about comparing sizes; they're also about ensuring balance. This balance comes from the products of the means and extremes. To see this in action:

• Find the product of the means: \( \frac{1}{4} \times 4 \). When you multiply these, you get \(1\).
• Similarly, find the product of the extremes: \( 2 \times \frac{1}{2} \), which also results in \(1\).

In every proportion, the products of the means and extremes must be equal for the proportion to hold true. This equality confirms that the two ratios are indeed proportional. By practicing finding these products, you will develop a strong grasp on balance in fractional equations.

Prealgebra - Understanding Proportions

Proportions are a fundamental concept in prealgebra. They're pivotal in understanding how different parts of an equation relate to each other. Here’s why mastering them is crucial:

• Proportions allow you to solve for unknowns by setting up an equation with equal ratios.
• They teach the importance of maintaining balance, which is a core concept in all equations and manipulations you'll encounter.
• Understanding proportions helps in real-world applications like scaling recipes or converting units.

As seen in the exercise, identifying and working with means and extremes shows how proportional relationships function. Continue practicing proportions to build a solid mathematical foundation, ensuring you can tackle more complex problems with confidence.