Problem 8 - 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. $$\frac{0.3}{1.2}=\frac{1}{4}$$

Step-by-Step Solution

Answer

The means are 1.2 and 1; the extremes are 0.3 and 4; both products equal 1.2.

Step 1: Identify the Means and Extremes

In a proportion \( \frac{a}{b} = \frac{c}{d} \), the means are the inner terms \(b\) and \(c\) while the extremes are the outer terms \(a\) and \(d\). Here, the proportion is \( \frac{0.3}{1.2} = \frac{1}{4} \), so the means are 1.2 and 1, and the extremes are 0.3 and 4.

Step 2: Calculate the Product of the Means

To find the product of the means, multiply the means from Step 1: \(1.2 \times 1 = 1.2\).

Step 3: Calculate the Product of the Extremes

To find the product of the extremes, multiply the extremes from Step 1: \(0.3 \times 4 = 1.2\).

Step 4: Compare the Products

The product of the means is 1.2, and the product of the extremes is also 1.2. Since these products are equal, the condition that the product of the means is equal to the product of the extremes is satisfied.

Key Concepts

Understanding Means and Extremes

When dealing with proportions, it's essential to grasp the concepts of means and extremes. A proportion represents two equivalent fractions, such as \(\frac{a}{b} = \frac{c}{d}\). Here, the terms \(b\) and \(c\) are known as the means, while \(a\) and \(d\) are the extremes. Understanding these classifications will help you solve proportion problems easily.

• Means: Inner terms of the proportion (in this example, 1.2 and 1)
• Extremes: Outer terms of the proportion (here, 0.3 and 4)

Recognizing and naming these terms proves valuable, especially when you need to check equality by calculating products.

Decoding the Product of Proportions

The product of proportions is a key concept that shows the relationship between the means and extremes. In any valid proportion, the product of the means is always equal to the product of the extremes. Calculating these products helps confirm the equality and validity of the proportion.To calculate:

• Multiply the means: For our example \(1.2 \times 1 = 1.2\).
• Multiply the extremes: Similarly, calculate \(0.3 \times 4 = 1.2\).

This step confirms that the values satisfy the condition of a proportion when both products are equal, as shown in our example.

Guided Step-by-Step Solution

Breaking down a problem into step-by-step solutions makes it easier to understand. Let's consider how to approach a solution using the steps given in the problem. Understanding each step individually helps solidify the concepts. **Step 1:** Identify the means and extremes. Recognize the inner and outer terms in the fraction set. **Step 2:** Calculate the product of the means. Multiply the inner terms (means) to verify part of the proportion's equality. **Step 3:** Calculate the product of the extremes. Repeat this multiplication process for the outer terms. **Step 4:** Compare the products. Confirm that both products are equal, ensuring the proportion holds true. This approach, when followed carefully, is a reliable way to handle similar exercises in proportion problems.