# Representing Proportional Relationships by Equations

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Representing Proportional Relationships by Equations
CCSS.MATH.CONTENT.7.RP.A.2.C

## Representing Proportional Relationships by Equations – Introduction

When we talk about relationships in mathematics, we're often referring to how one quantity relates to another. Proportional relationships are an essential concept in this context, much like a blueprint for understanding how different variables interact. In this introduction, we'll start exploring how to represent these proportional relationships using equations, a fundamental skill in both math and real-life applications.

## Understanding Proportional Relationships – Definition

A proportional relationship exists between two quantities when they increase or decrease at the same rate. This means that the ratio between these quantities remains constant.

For instance, if you have a situation where the more hours you work, the more money you earn at a constant rate, there's a proportional relationship between your work hours and your earnings.

Let’s look at an example!

Example: If you have 4 bags of flour weighing 8 kilograms in total, are the number of bags and the total weight proportional?

Yes, they are proportional. The constant ratio (weight per bag) is $2$ kilograms, since $8$ kilograms divided by $4$ bags is $2$ kilograms per bag. This means for every bag of flour, the weight increases uniformly by $2$ kilograms.

Try the following on your own!

If you buy 5 notebooks for $15, are the cost and the number of notebooks proportional? Are the two quantities proportional if a car travels 100 miles in 2 hours and 150 miles in 3 hours? ## Representing Proportional Relationships – Example Let's consider the following scenario: a taxi company charges a flat rate of$5 plus $2 per mile driven. How would we represent this proportional relationship by an equation? • Identify the constant of proportionality: The price per mile is$2.
• Set up the equation: The total cost ($C$) is equal to the flat rate ($\$$5) plus the cost per mile (\$$2) times the number of miles ($m$). Thus, the equation representing this scenario is:$C = 5 + 2m$Practice by using the information above to answer the following question. What would be the total cost for a 15-mile taxi ride? A fruit seller charges$\1.50 per pound of apples.

Write the equation to calculate the total cost ($T$) for buying p pounds of apples.

## Representing Proportional Relationships – Summary

Key Learnings from this Text:

• A proportional relationship is when two quantities increase or decrease at the same rate.
• The constant of proportionality is the constant ratio between two proportional quantities.
• You can represent proportional relationships using linear equations.
• These concepts are widely used in real-life situations like calculating expenses, distances travel, and more.

Explore other content on our platform for interactive practice problems, videos, and printable worksheets to enhance your understanding of proportional relationships and equations.

## Representing Proportional Relationships by Equations – Frequently Asked Questions

What is a proportional relationship?
How do you know if two quantities are in a proportional relationship?
What is the constant of proportionality?
How do you represent a proportional relationship with an equation?
Can a proportional relationship be represented by a graph?
What is the formula for the constant of proportionality?
Is the equation y = mx + b always a representation of a proportional relationship?
Can a table of values be used to determine a proportional relationship?
How does the constant of proportionality affect the graph of a proportional relationship?
What happens to the equation of a proportional relationship if the constant of proportionality is doubled?

## Representing Proportional Relationships by Equations exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Representing Proportional Relationships by Equations .
• ### What is the unit rate (k) of the bananas?

Hints

Remember: The unit rate (k) is the cost of fruit per 1 pound.

For example, if Penny was selling melons for $2.15 per pound, the unit rate of the melons would be 2.15. Solution The unit rate (k) in a proportional relationship is the cost of fruit per 1 pound. Penny is selling the bananas for$1.25 per pound.

Therefore, the unit rate of the bananas is 1.25.

• ### Write the equation that represents the proportional relationship of pounds of apples to cost.

Hints

The unit rate of cost per pound for the apples is 2.65.

k = 2.65

To write the equation that represents the proportional relationship, the unit rate is substituted for k in:

y = kx

Solution

The equation that represents a proportional relationship is y = kx.

The unit rate (k) of cost per pound for the apples is 2.65.

Therefore, to find the equation that represents the proportional relationship of pounds of apples to cost, we substitute 2.65 for k in y = kx.

The answer is: y = 2.65x.

• ### Assign the pairs.

Hints

The unit rate is the cost of fruit per 1 pound. Which variable can the unit rate be represented with?

$y$ is affected by the value of $x$.

For example, the total cost of apples is affected by (dependent on) the pounds of apples bought.

Solution

Dependent variable: $y$

The equation that represents a proportional relationship: $y = kx$

Unit rate: $k$

Independent variable: $x$

• ### Penny begins selling vegetables.

Hints

The dependent variable ($y$) is affected by the independent variable ($x$). The total cost of vegetables is affected by the pounds of vegetables bought. Therefore, what would $x$ = and $y$ =?

The equation to represent the proportional relationship between two variables is $y = kx$, where $k$ = the unit rate (the cost per 1 pound).

Solution

Carrots:

• $x$ = pounds of carrots and $y$ = total cost of carrots.
• The unit rate $k$ is 1.9.
• The equation to represent the proportional relationship of pounds of carrots to cost is $y = 1.9x$.

Broccoli:

• $x$ = pounds of broccoli and $y$ = total cost of broccoli.
• The unit rate $(k)$ is $1.15$.
• The equation to represent the proportional relationship of pounds of broccoli to cost is $y = 1.15x$.

Onions:

• $x$ = pounds of onions and $y$ = total cost of onions.
• The unit rate $(k)$ is $2.4$.
• The equation to represent the proportional relationship of pounds of onions to cost is $y = 2.4x$.

Explanation:

• The dependent variable ($y$) is affected by the independent variable ($x$). The total cost of vegetables is affected by the pounds of vegetables bought. Therefore, the independent variable ($x$) would be pounds of vegetables, and the dependent variable ($y$) would be total cost of vegetables.