Chapter 2: Problem 33

a. State the general equation for quantities, x and y, that are inversely proportional. b. For two inversely proportional quantities, what happens to one variable when the other increases?

Short Answer

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a) \( y = \frac{k}{x} \)b) One variable decreases when the other increases.

Step by step solution

Understanding Inverse Proportionality

Inverse proportionality implies that as one quantity increases, the other decreases in such a way that their product is a constant. This relationship is stated mathematically as follows: If quantities \( x \) and \( y \) are inversely proportional, then \[ x \times y = k \]where \( k \) is a constant.

General Equation of Inverse Proportionality

The general equation for quantities \( x \) and \( y \) that are inversely proportional is \[ y = \frac{k}{x} \]where \( k \) is a constant.

Impact of Increasing One Variable

For two inversely proportional quantities, if one variable increases, the other variable decreases so that their product remains constant. This means \( x \times y = k \) is maintained. For instance, if \( x \) is doubled, \( y \) must be halved to keep their product constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inverse relationship

Inverse proportionality is a unique kind of relationship between two quantities. Imagine two values, say x and y, which are tied together by a special rule. This rule states that if one value increases, the other decreases in a way that their product always remains the same.
For example, if x and y are inversely proportional, you can say that x times y equals some constant number, k. This is written as:

mathematical constants

In mathematics, constants are numbers that remain fixed and do not change. When dealing with inverse proportionality, we see the appearance of a constant in the equation which is denoted by k.
This k is crucial because no matter how x or y changes, their product always equals k. For instance, if x increases, y must decrease in such a manner that their product k is unchanged. Conversely, if x decreases, y increases.
Understanding constants helps us make sense of how variables in inverse relationships behave.

proportional variables

Proportional variables can be either directly or inversely proportional. In direct proportionality, as one variable increases, the other does as well. In our case of inverse proportionality, as one variable increases, the other decreases. This is mathematically represented by the equation: Understanding these relationships helps in predicting and calculating the behaviour of quantities based on their proportional variables.
For instance, if you know that speed and time for a fixed distance are inversely proportional, doubling the speed will halve the time taken.

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a. State the general equation for quantities, x and y, that are inversely proportional. b. For two inversely proportional quantities, what happens to one variable when the other increases?

Short Answer

Step by step solution

Understanding Inverse Proportionality

General Equation of Inverse Proportionality

Impact of Increasing One Variable

Key Concepts

inverse relationship

mathematical constants

proportional variables

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