Question

Explain what is meant by the phrase ' \(a\) is inversely proportional to \(b\) '.

Short answer

Answer: When we say that 'a is inversely proportional to b', it means that one variable (a) is directly proportional to the reciprocal of the other variable (b). In other words, when one variable increases, the other variable decreases, and vice versa. This can be represented as a ∝ 1/b or ab = k, where k is a constant.

Step-by-step solution

Define inversely proportional

In mathematics, two variables are said to be inversely proportional if one of the variables is directly proportional to the reciprocal of the other variable. It means that when one variable increases, the other variable decreases, and vice versa. In other words, the product of the two variables remains constant. This can be represented as: a ∝ 1/b or ab = k, where k is a constant.

Provide an example to illustrate the concept

Let's take an example to better understand the concept of inverse proportionality. Suppose we have a fixed amount of money, \(60, and we want to buy chocolates that cost \)3 per piece. The number of chocolates we can buy (a) is inversely proportional to the price of each chocolate (b). As the price of each chocolate (b) increases, the number of chocolates (a) we can buy with the fixed amount of money decreases, and vice versa. This can be represented as: a ∝ 1/b or ab = 60 If the price per chocolate (b) is $3, then the number of chocolates (a) we can buy is 20: a = 60 ÷ 3 = 20 If the price per chocolate (b) is $6, the number of chocolates (a) we can buy is 10: a = 60 ÷ 6 = 10 This example demonstrates how an increase in one variable (price per chocolate) leads to a decrease in the other variable (number of chocolates) and vice versa, illustrating the concept of inverse proportionality.

Key concepts

Variables

Variables are fundamental elements in mathematics and science that hold various values. When we discuss relationships between variables, like inverse proportionality, it involves understanding how the change in one affects the other. In an inversely proportional relationship, suppose we have a variable \(a\) and another variable \(b\). Here, as \(a\) increases, \(b\) decreases, and the opposite is also true. This connection is crucial, as it shows how dependent factors work together.

• Independent Variable: This is the variable you change to see its effect on another variable.
• Dependent Variable: This changes in response to the independent variable.

Think of it like this: if you increase the number of workers on a job (variable \(a\)), the time it will take to complete the job (variable \(b\)) should decrease. This helps identify cause and effect between factors.

Constant Product

The constant product is a central feature of inverse proportionality. If two variables, \(a\) and \(b\), are inversely proportional, their multiplied value is always constant. This means no matter how much one variable increases or decreases, the product of the two will remain the same. Mathematically, this is shown as \(ab = k\), where \(k\) is the constant.
Using our chocolate example: If the overall spend is fixed at \(60\) dollars, then \(\text{number of chocolates} (a) \times \text{price per chocolate} (b) = 60\). This remains true regardless of the separate values of \(a\) and \(b\).

• If \(a = 20\), \(b\) should be \(3\), since \(20 \times 3 = 60\).
• Similarly, if \(b = 6\), then \(a\) must be \(10\) because \(10 \times 6 = 60\).

The constant product concept ensures that changes in variables maintain a balance in real-world applications, like budgeting or resource management.

Reciprocal Relationship

A reciprocal relationship is another way to understand inverse proportionality. It means that one variable is proportional to the reciprocal of the other variable. So, if \(a \propto \frac{1}{b}\), this indicates that every change in one variable creates a reciprocal effect in the other.
Reciprocal relationships are easy to visualize:

• If \(b\) becomes twice as large, \(a\) becomes half as much.
• In simpler terms, as one goes up, the other comes down to maintain the consistent product \(k\).

This relationship helps in deriving formulas for inverse proportional scenarios and solving practical everyday problems. For example, if you divide the responsibility to ten workers, each has a small workload. But if only two workers are available, the workload increases reciprocally for each worker. Such understanding can be beneficial in time management, resource allocation, and optimizing processes.