In essence, if a and b form a proportional relationship, the ratio of their values is considered normal. As the value of b increases, a also increases; conversely, as the value of b decreases, a also decreases.
While the term "proportional" is typically used to describe relationships where one variable directly impacts another, it can be challenging to define the exact nature of a proportional relationship when dealing with specific numerical variables like those found in mathematics. For example, let's consider the relationship between height (h) and weight (w) in human body.
A proportion could take several forms depending on how weight and height are measured, but if we assume that the mass (m) and height (h) are measured accurately using the same unit of measurement, such as kilograms (kg), then a linear relationship can be established. In this case, h = m / k, where m represents the weight and h represents the height.
If h represents the height, then for every kilogram increase in weight, the corresponding height increase is directly proportional. This means that as the weight (in kg) increases by say 10%, the corresponding height (in meters) would also increase by approximately 10% since height is often measured in meters.
On the other hand, if h represents weight (in kg) and w represents height (in meters), then the relationship might not necessarily be proportional due to the difference in units of measurement. If we use meters as the base unit, and change the measurement system from kilograms to meters per square meter, a non-linear relationship might emerge. The conversion factor between kg and m/s² could affect the ratio, causing a decrease in the ratio with an increasing weight, as long as it remains within a certain range.
To better illustrate these concepts, here is an example: Suppose someone weighs 70 kg and measures their height at 1.8 meters, which converts to 96 inches. With a linear relationship, the height would grow as weight increases by 10%:
- Initial height (height without considering weight): 1.8 meters (or 96 inches)
- Height after a weight increase of 10% (h_{new}):
Since we are using meters as the base unit, 1.8 meters (or 96 inches) divided by meters per square meter equals approximately 1.47 meters or 63.6 inches.
- Weight after the weight increase of 10% (w_{new}):
Since the weight now increases by 10%, from 70 kg to 70 kg + 0.10 * 70 kg = 87 kg.
As the weight continues to increase, the height would still follow the same growth pattern with an increased linear rate due to the linear equation h = m / k.
In conclusion, while "proportional" can provide a general framework for understanding proportional relationships in mathematical settings, the specific nature of a proportional relationship in everyday life may involve differences in units of measurement, the presence of other factors like gravity or external forces, or deviations from a linear trend. By recognizing and accounting for these variations, we can effectively apply proportional relationships to various contexts, including physical proportions and non-linear relationships encountered in real-world situations.