Representing Proportional Relationships by Equations
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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 maths 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!
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.
A fruit seller charges £1.50 per pound 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
Transcript Representing Proportional Relationships by Equations
June is buying fruit from Penny to enter a worlds biggest fruit salad competition. To find out how much money she will spend, June will be representing proportional relationships by equations. June needs to identify the 'unit rate' in cost of fruit per kilogram, which can be represented with the variable
Representing Proportional Relationships by Equations exercise
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What is the unit rate (k) of the bananas?
HintsRemember, the unit rate (k) is the cost of fruit per 1 kilogram.
For example, if Penny was selling melons for £2.15 per kilogram, the unit rate of the melons would be 2.15.
SolutionThe unit rate (k) in a proportional relationship is the cost of fruit per 1 kilogram.
Penny is selling the bananas for £1.25 per kilogram.
Therefore, the unit rate of the bananas is 1.25.
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Write the equation that represents the proportional relationship of kilograms of apples to cost.
HintsThe unit rate of cost per kilogram 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
SolutionThe equation that represents a proportional relationship is y = kx.
The unit rate (k) of cost per kilogram for the apples is 2.65.
Therefore, to find the equation that represents the proportional relationship of kilograms of apples to cost, we substitute 2.65 for k in y = kx.
The answer is: y = 2.65x.
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Assign the pairs.
HintsThe unit rate is the cost of fruit per 1 kilogram. 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 kilograms of apples bought.
SolutionDependent variable: $y$
Unit rate: $k$
Independent variable: $x$
The equation that represents a proportional relationship: $y = kx$
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Penny begins selling vegetables.
HintsThe dependent variable ($y$) is affected by the independent variable ($x$). The total cost of vegetables is affected by the kilograms 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 kilogram).
SolutionCarrots:
- $x$ = kilograms of carrots and $y$ = total cost of carrots.
- The unit rate $k$ is 1.9.
- The equation to represent the proportional relationship of kilograms of carrots to cost is $y = 1.9x$.
Broccoli:
- $x$ = kilograms of broccoli and $y$ = total cost of broccoli.
- The unit rate $(k)$ is $1.15$.
- The equation to represent the proportional relationship of kilograms of broccoli to cost is $y = 1.15x$.
Onions:
- $x$ = kilograms of onions and $y$ = total cost of onions.
- The unit rate $(k)$ is $2.4$.
- The equation to represent the proportional relationship of kilograms 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 kilograms of vegetables bought. Therefore, the independent variable ($x$) would be kilograms of vegetables and the dependent variable ($y$) would be total cost of vegetables.
- The unit rate ($k$) is the cost per 1 kilogram of vegetables. For example: the carrots cost £1.90 per kilogram. Therefore, the unit rate is 1.9.
- The equation to represent the proportional relationship between two variables is $y = kx$, where $k$ = the unit rate. For example, to find the equation for the carrots, we substitute the 1.9 for $k$: $y = 1.9x$.
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What is the equation that represents the proportional relationship of kilograms of raspberries to cost?
HintsThe unit rate of cost per kilogram for the raspberries is 1.85.
k = 1.85
To write the equation that represents the proportional relationship, the unit rate is substituted for k in:
y = kx
SolutionThe equation that represents a proportional relationship is y = kx.
The unit rate (k) of cost per kilogram for the raspberries is 1.85.
Therefore, to find the equation that represents the proportional relationship of kilograms of raspberries to cost, we substitute 1.85 for k in y = kx.
The answer is: y = 1.85x.
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How much money does June spend on blueberries?
HintsThe equation that represents the proportional relationship of kilograms of blueberries to cost is:
$y = 3.75x$
where $x$ = kilograms of blueberries and $y$ = total cost of blueberries.
You can use this equation to calculate how much June spends on blueberries (the total cost of blueberries $y$).
To calculate how much money June spends on blueberries, substitute the number of kilograms June buys for $x$.
$y = 3.75(4)$
SolutionThe equation that represents the proportional relationship of kilograms of blueberries to cost is:
$y = 3.75x$
where $x$ = kilograms of blueberries and $y$ = total cost of blueberries.
You can use this equation to calculate how much June spends on blueberries (the total cost of blueberries $y$).
Substitute the number of kilograms June buys for $x$: $y = 3.75(4) = 15$.
The answer is £15.
This was very easy-_-