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Chapter 11: Problem 1

Write the extremes and the means of the proportion \(\frac{3}{4}=\frac{9}{12}\).

Short Answer

Expert verified
The extremes of the proportion are 3 and 12, while the means are 4 and 9.

Step by step solution

01

Identify the Extremes

In a proportion, the first and the fourth terms are referred to as the extremes. In this case, these are the numbers 3 and 12.
02

Identify the Means

The second and the third terms of a proportion are called the means. Here, these are 4 and 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Extremes in Proportions
In the world of proportions, the concept of "extremes" refers to the outer numbers in a fraction comparison. When we look at a proportion, it forms a statement where two fractions are equal. For example, in the given proportion \(\frac{3}{4} = \frac{9}{12}\), the numbers 3 and 12 are referred to as the extremes.
The first step to identify extremes is to recognize them as the first and fourth numbers in this arrangement:
  • 3 (first number in the first fraction)
  • 12 (second number in the second fraction)
Identifying extremes is essential for understanding the bigger picture of how proportions are structured.
The Role of Means in Proportions
Proportions are composed of both means and extremes. While the extremes sit on the outside, means are the two numbers that find their place in the middle of this "equal fractions" setup. Using our proportion example \(\frac{3}{4} = \frac{9}{12}\), the means are the numbers 4 and 9.
To find the means, you simply look at the pair that isn鈥檛 the extremes:
  • 4 (second number of the first fraction)
  • 9 (first number of the second fraction)
Understanding the means is crucial in scaling problems and maintaining balance within equations.
Working with Fractions in Proportions
Fractions play a pivotal role in proportions, as proportions essentially compare two equal fractions. In the case of \(\frac{3}{4} = \frac{9}{12}\), we're actually comparing the relationship between these two fractions.
When dealing with fractions:
  • Recognize the numerator (the top number of the fraction) and the denominator (the bottom number of the fraction).
  • Understand how equivalent fractions work, often involving scaling both the numerator and the denominator by the same number.
Proportions are a direct result of this scaling and equivalence process in fractions.
Applying Mathematical Reasoning in Proportions
When working with proportions, mathematical reasoning helps you decipher why certain numbers are paired as means and extremes. It's important to see how these relationships hold true across various problems. Let鈥檚 enhance our reasoning skills through this approach:
  • Verify if the cross-products of the means and extremes equal each other. This checks proportion validity: \(3 \times 12 = 36\) and \(4 \times 9 = 36\).
  • Use reasoning to potentially solve for missing values in similar setups by maintaining the equality of cross-products.
Developing strong mathematical reasoning enables you to solve complex equations and understand how different parts of proportions interact.

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