Direct vs. Inverse Proportion¶
A direct proportionality relationship and an inverse proportionality relationship describe how two quantities are related to each other:
1. Direct Proportion:¶
In a proportional (or directly proportional) relationship, two quantities increase or decrease at the same rate. This means that as one quantity increases, the other increases by the same factor, and vice versa.
- Mathematical Form:
If \( y \) is directly proportional to \( x \), we write:
where \( k \) is a constant of proportionality.
- Example:
If you buy apples, the cost (\( y \)) is proportional to the number of apples (\( x \)) you buy. If the price of each apple is \( k \), the total cost is \( y = kx \).
- Graphical Representation: A graph of a proportional relationship is a straight line that passes through the origin (0, 0).
2. Inverse Proportion:¶
In an inversely proportional relationship, as one quantity increases, the other decreases. This means that if one quantity is doubled, the other is halved, and so on.
- Mathematical Form:
If \( y \) is inversely proportional to \( x \), we write:
where \( k \) is a constant.
- Example:
The time it takes to complete a task (\( y \)) is inversely proportional to the number of people (\( x \)) working on it. If more people work on the task, the time to finish decreases.
- Graphical Representation:
A graph of an inverse relationship is a hyperbola, where one axis represents \( y \) and the other represents \( x \), and it curves toward the axes but never touches them.
Summary:
- Proportional (Direct): As \( x \) increases, \( y \) increases (or decreases together).
- Inversely Proportional: As \( x \) increases, \( y \) decreases.
Exercises - Direct vs Inverse Prooportionality Relationship¶
Here are a few exercises to help understand proportional and inversely proportional relationships:
Exercise 1: Identifying Direct Proportionality Relationships¶
For each of the following scenarios, state whether the relationship is proportional or inversely proportional. Explain your reasoning.
- The cost of buying multiple loaves of bread if each loaf costs $2.
- The speed of a car and the time it takes to travel a fixed distance.
- The length of a shadow and the height of the object casting the shadow (assuming the light source remains constant).
- The time required to fill a swimming pool with a fixed capacity if you increase the rate at which water flows into it.
Exercise 2: Proportional Relationship Table¶
Complete the table to show how \( y \) changes with \( x \) when \( y \) is directly proportional to \( x \), with a proportionality constant of 3.
| \( x \) | \( y \) |
|---|---|
| 1 | ? |
| 2 | ? |
| 4 | ? |
| 5 | ? |
| 10 | ? |
Exercise 3: Inversely Proportional Relationship Table¶
Complete the table for \( y \) and \( x \), where \( y \) is inversely proportional to \( x \), and \( y = \frac{12}{x} \).
| \( x \) | \( y \) |
|---|---|
| 1 | ? |
| 2 | ? |
| 3 | ? |
| 6 | ? |
| 12 | ? |
Exercise 4: Word Problem (Proportional Relationship)¶
A car rental company charges $50 per day to rent a car. Write an equation to represent the total cost \( C \) as a function of the number of days \( d \). How much will it cost to rent the car for 7 days?
Exercise 5: Word Problem (Inversely Proportional Relationship)¶
The amount of time \( T \) it takes to complete a project is inversely proportional to the number of people \( p \) working on it. If it takes 5 people 10 hours to complete the project, how long would it take 2 people?
These exercises encourage understanding of how proportional and inversely proportional relationships work in various scenarios.
Solutions¶
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