Problem 1 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

For the proportion a, 3 and 12 are the extremes while 4 and 9 are the means. For the proportion b, 9 and 4 are the extremes while 12 and 3 are the means.

Step 1: Identification of extremes and means for proportion a

In the proportion \(\frac{3}{4}= \frac{9}{12}\), 3 and 12 are the extremes while 4 and 9 are the means.

Step 2: Identification of extremes and means for proportion b

In the proportion \(\frac{9}{12}= \frac{3}{4}\), 9 and 4 are the extremes while 12 and 3 are the means.

Key Concepts

Extremes in Proportions

When dealing with proportions, it's crucial to identify the extremes because they play a key role in understanding the relationships between two ratios. The extremes are simply the outermost numbers in a proportion. For example, in the proportion \( \frac{3}{4} = \frac{9}{12} \), the first and last numbers (3 and 12) are the extremes.
Understanding which numbers are the extremes can help verify if the proportion is set up correctly. In any proportion \( \frac{a}{b} = \frac{c}{d} \), the extremes are \(a\) and \(d\).

• They are situated diagonally from each other in the written proportion.
• Reversing the order of the numbers in the ratio changes the means but not the extremes.

Means in Proportions

Just as important as identifying the extremes, recognizing the means in a proportion is essential. The means are the middle numbers when a proportion is set out.
For example, in \( \frac{9}{12} = \frac{3}{4} \), the numbers 12 and 3 are the means of the proportion.
The general setup of a proportion is \( \frac{a}{b} = \frac{c}{d} \), where \(b\) and \(c\) are the means.

• They are located nicely in the middle, connecting the two ratios.
• If you switch the terms in the proportions, these numbers (means) will change positions with the extremes.

This interchangeability of means can be helpful in cross multiplication or when verifying the validity of a proportion.

Understanding Proportional Relationships

Proportional relationships involve two ratios or fractions set to be equal. They help to maintain a balance between quantities.
For example, the proportion \( \frac{3}{4} = \frac{9}{12} \) tells us that three is to four as nine is to twelve.
Here's how you can truly grasp proportional relationships:

• Utilize cross-multiplication to verify proportions. For \( \frac{a}{b} = \frac{c}{d} \), the cross product \( ad = bc \) helps confirm equality.
• Proportions are especially handy in scaling problems, like when altering the size of a recipe.
• Understanding them aids in solving everyday real-world problems, like determining distances on maps using scale factors.

With a solid grasp of proportional relationships, you can efficiently solve problems that involve direct equivalence between different sets of numbers.