Problem 3 - Intermediate Algebra

Question

In \(\frac{50}{3}=\frac{x}{9},\) the terms 50 and 9 are called the _________ and the terms 3 and \(x\) are called the _________ of the proportion. In a the product of the _________ is equal to the product of the _________.

Step-by-Step Solution

Answer

The terms 50 and 9 are called the extremes, while 3 and \(x\) are called the means. The product of the means is equal to the product of the extremes.

Step 1: Identify terms of a proportion

In a proportion \(\frac{a}{b} = \frac{c}{d}\), the terms \(a\) and \(d\) are called the extremes, while \(b\) and \(c\) are called the means. In our given proportion \(\frac{50}{3} = \frac{x}{9}\), \(50\) and \(9\) are the extremes, and \(3\) and \(x\) are the means.

Step 2: Describe the product of means and extremes

In a proportion, the product of the means is equal to the product of the extremes. This principle can be written as \(b \times c = a \times d\). Applied to our proportion \(\frac{50}{3} = \frac{x}{9}\), it implies \(3 \times x = 50 \times 9\).

Key Concepts

Understanding Extremes in a Proportion

In the context of a proportion, such as \( \frac{a}{b} = \frac{c}{d} \), the terms known as the extremes play a crucial role. The extremes are the numbers located at the far ends of the proportion. Here, they are \(a\) and \(d\). This placement is indispensable when working with proportions as it helps in various calculations such as discovering unknowns or equating products. In our specific example, \( \frac{50}{3} = \frac{x}{9} \), the extremes are \(50\) and \(9\). They sit at opposite ends, emphasizing their special positions. Recognizing extremes is essential for understanding the proportion's balance and performing manipulations such as cross-multiplication.

The Role of Means in Proportions

Means in a proportion like \( \frac{a}{b} = \frac{c}{d} \) are the terms \(b\) and \(c\). Unlike extremes, means are situated in the middle of the equation. They work together to maintain the proportion's balance. In the example \(\frac{50}{3} = \frac{x}{9}\), the means are the numbers \(3\) and \(x\). Understanding that the means are placed between the extremes helps you see how related numbers interact in a proportion. A proportion maintains its harmony through the equality of products: the product of the means equals the product of the extremes. This characteristic is a key component in solving for unknown values within a proportion.

Cross-Multiplication Magic

Cross-multiplication is a powerful technique derived from the algebraic property of proportions. The technique is based on the equality of the product of means and the product of extremes. Here's how to use it: in a proportion \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying the extremes (\(a\times d\)) and the means (\(b\times c\)). You equate the results: \( a\times d = b\times c \). This method simplifies solving for unknowns.

For instance, in the proportion \(\frac{50}{3} = \frac{x}{9}\), cross-multiplication results in \(3 \times x = 50 \times 9\). Here, you can solve for \(x\) by calculating the products and then dividing by one of the remaining known terms. Cross-multiplication is simple, direct, and cuts down the steps to find unknown values efficiently, making it a favorite tool among students and mathematicians alike.