Problem 39 - Introductory Chemistry Atoms First

Question

"The older I get, the fewer hairs I have on my head." What kind of relationship (proportion or inverse proportion) exists between this gentleman's age and his hair? Explain your answer.

Step-by-Step Solution

Answer

The relationship between the gentleman's age and the number of hairs on his head is an inverse proportion, as his age increases, the number of hairs decreases at a constant rate.

Step 1: Understanding the relationship

In general, a direct proportion (or proportion) means that as one variable increases, the other variable also increases at a constant rate. On the other hand, an inverse proportion means that as one variable increases, the other variable decreases at a constant rate. In this case, the gentleman's age is increasing, and the number of hairs on his head is decreasing. Therefore, we can conclude that this relationship is an inverse proportion.

Key Concepts

Direct Proportion

In the realm of mathematics, a direct proportion describes a scenario where two variables change in relation to each other in a consistent manner. This means, when one variable increases or decreases, the other does the same at a constant ratio. For instance, if you were buying apples and the price per apple stayed the same, the total cost would be directly proportional to the number of apples you buy. If you double the number of apples, your cost would also double. This relationship can be expressed mathematically as: \[ y = kx \] where \( y \) and \( x \) are the variables in question, and \( k \) is the constant of proportionality, remaining the same irrespective of the values \( y \) and \( x \) take.

Variable Relationship

Variables are key components in any mathematical relationship, representing quantities that can change or vary. Understanding variable relationships helps us predict how a change in one quantity might affect another. In simple terms, if you have two variables, the way they affect each other is their "relationship". This could be directly proportional, inversely proportional, or have no set pattern at all. For example, if studying time increases, test scores may also increase, showing a direct relationship. Conversely, as the age of a vehicle increases, its value might decrease, exemplifying an inverse relationship. Recognizing these patterns is vital in fields ranging from economics to the natural sciences.

Constant Rate

A constant rate signifies uniform change between variables. When something changes at a constant rate, there is a steady, predictable pattern involved. For instance, if a car travels at a constant speed, say 60 kilometers per hour, it covers the same distance each hour continuously. This concept is fundamental in equations involving direct and inverse proportions. With direct proportion, changes between the variables follow a constant multiplier. Similarly, in inverse proportion, there is a consistent factor dictating how much one variable decreases as the other increases. Recognizing a constant rate can simplify solving or predicting outcomes in a variety of practical situations.

Age and Hair Relationship

The relationship between age and the amount of hair, as mentioned in the exercise, illustrates an inverse proportion. In essence, as the gentleman's age increases, the number of hairs on his head decreases. This relationship can be represented by the inverse proportion formula: \[ xy = k \] where \( x \) could be the age, \( y \) the number of hairs, and \( k \) is the constant of proportionality. Such relationships are common in everyday life. They help us understand how one factor might inversely affect another. In this context, it becomes clear that as time progresses, fewer hairs remain, mapping a classic real-life example of inverse proportion.