In mathematics, the terms or elements in proportion are known as extremes and mean terms. These specific labels apply to the four elements involved when two ratios are declared equal.
Understanding Proportion
A proportion is a statement that two ratios are equal. As per the definition, if two ratios are equal, they are said to be in proportion. This fundamental concept is expressed by equating two fractions. For instance, if a, b, c, d are four elements such that the ratio of a to b is equal to the ratio of c to d, then they are in proportion, written as:
a/b = c/d
This can also be written as a : b :: c : d, which reads "a is to b as c is to d."
Identifying the Terms in Proportion
The four elements (a, b, c, d) in a proportion a/b = c/d are classified into two distinct types: extremes and mean terms. Understanding these terms is crucial for working with proportions.
Extremes
The elements a and d are called the extremes because they are at the "extremes" or outer positions of the proportion when written in the a : b :: c : d format.
Mean Terms
The elements b and c are called the mean terms (or simply "means") because they are in the "middle" or central positions of the proportion.
Here's a breakdown of the elements:
| Element | Position in a/b = c/d |
Term Name |
|---|---|---|
| a | Numerator of first ratio | Extreme |
| b | Denominator of first ratio | Mean Term |
| c | Numerator of second ratio | Mean Term |
| d | Denominator of second ratio | Extreme |
Key Property of Proportions
An important property derived from the definition of proportion is that the product of the means equals the product of the extremes. For a/b = c/d, this property is expressed as:
a * d = b * c
This property is widely used to solve for an unknown term in a proportion, verify if ratios are proportional, or understand the relationship between the quantities involved.
Example:
Consider the proportion 2/4 = 5/10.
- Extremes: The elements a and d are 2 and 10, respectively.
- Mean Terms: The elements b and c are 4 and 5, respectively.
Let's verify the product of means and extremes:
- Product of Extremes:
2 * 10 = 20 - Product of Mean Terms:
4 * 5 = 20
Since 20 = 20, the ratios are indeed in proportion, confirming the labels for the terms.
For more detailed information on ratios and proportions, you might find resources on basic algebra helpful.
