Mathematics
Grade 8
15 min
Find the constant of proportionality: word problems
Find the constant of proportionality: word problems
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1
Introduction & Learning Objectives
Learning Objectives
Identify proportional relationships within word problems.
Correctly identify the independent (x) and dependent (y) variables in a proportional word problem.
Calculate the constant of proportionality (k) from given information in word problems.
Interpret the meaning of the constant of proportionality in the context of a word problem.
Write an equation in the form y = kx to represent a proportional relationship described in a word problem.
Use the constant of proportionality to solve related problems.
Ever wonder how much you'd earn for babysitting for a certain number of hours, or how many cookies you can bake with a specific amount of flour? 🤔 These are all examples of proportional relationships!
In this lesson, you'll learn how to find the &#...
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Key Concepts & Vocabulary
TermDefinitionExample
Proportional RelationshipA relationship between two quantities where their ratio is always constant. When one quantity changes, the other quantity changes by a constant factor.If you earn $10 per hour, your total earnings are proportional to the number of hours you work. The ratio of earnings to hours is always 10/1.
Constant of Proportionality (k)The constant ratio between two quantities in a proportional relationship. It represents the unit rate or the value of y when x is 1.In the earnings example, $10 per hour, the constant of proportionality (k) is 10.
Independent Variable (x)The quantity that can be changed or controlled in an experiment or situation. Its value determines the value of the dependent variable.In the earnings example, the 'number of hours wor...
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Core Formulas
Formula for Constant of Proportionality
$k = \frac{y}{x}$
To find the constant of proportionality (k), divide the dependent variable (y) by the independent variable (x). This ratio must be constant for all pairs of data in a proportional relationship.
Equation of a Proportional Relationship
$y = kx$
Once you find the constant of proportionality (k), you can write an equation that describes the entire proportional relationship. This equation allows you to find any 'y' value for a given 'x' value, or vice versa.
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Challenging
A graphic designer is mixing a shade of orange. She uses 5 parts red paint for every 2 parts yellow paint. To create a large batch for a 300 square foot mural, she uses a total of 140 fluid ounces of paint. What is the constant of proportionality (k) for the relationship between the amount of red paint (y) and the amount of yellow paint (x)?
A.0.4 parts red per part yellow
B.2.5 parts red per part yellow
C.7 parts red per part yellow
D.20 parts red per part yellow
Challenging
A recipe states that for every 7 cups of flour used, you should add 3 cups of sugar. If you want to find the amount of flour needed (y) based on the amount of sugar you have (x), what is the constant of proportionality for this specific relationship?
A.3/7
B.7/3
C.10
D.4
Challenging
The circumference (C) of a circle is directly proportional to its radius (r). A circle with a radius of 5 inches has a circumference of approximately 31.4 inches. What is the constant of proportionality that relates the circumference to the radius, and what is the value of this constant multiplied by the circle's diameter?
A.k = 3.14; value = 31.4
B.k = 6.28; value = 62.8
C.k = 3.14; value = 62.8
D.k = 6.28; value = 31.4
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