Mathematics
Grade 7
15 min
Solve problems involving proportional relationships
Solve problems involving proportional relationships
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1
Introduction & Learning Objectives
Learning Objectives
Identify proportional relationships from tables, graphs, and equations.
Calculate the constant of proportionality from given data.
Write equations to represent proportional relationships.
Use proportional relationships to solve real-world problems involving rates and ratios.
Distinguish between proportional and non-proportional relationships.
Apply unit rates to solve problems involving proportional relationships.
Have you ever needed to double a recipe or figure out how much gas you'll need for a road trip? 🚗 These everyday situations often involve proportional relationships!
In this lesson, you'll learn how to recognize, represent, and solve problems involving proportional relationships. Understanding these relationships will help you make s...
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Key Concepts & Vocabulary
TermDefinitionExample
Proportional RelationshipA relationship between two quantities where their ratio is constant. This means that as one quantity changes, the other quantity changes by a constant multiplier.If 2 apples cost $1, then 4 apples cost $2. The ratio of cost to apples (1/2) is constant.
Constant of Proportionality (k)The constant ratio between two proportional quantities, often represented by 'k'. It's the value you multiply one quantity by to get the other. In the equation y = kx, 'k' is the constant of proportionality.If a car travels 60 miles in 1 hour, the constant of proportionality (speed) is 60 miles per hour. For every hour (x), the distance (y) is 60 times x.
Unit RateA rate in which the second quantity in the comparison is one unit. It's...
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Core Formulas
Equation of a Proportional Relationship
$y = kx$
This equation represents a proportional relationship where 'y' is the dependent variable, 'x' is the independent variable, and 'k' is the constant of proportionality. It shows that 'y' is always 'k' times 'x'.
Calculating the Constant of Proportionality
$k = \frac{y}{x}$
To find the constant of proportionality 'k', divide any 'y' value by its corresponding 'x' value in a proportional relationship. This 'k' value will be the same for all pairs (x, y).
Solving Proportions (Cross-Multiplication)
If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$
When two ratios are equal (a proportion), their cross products are also equal. This...
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Challenging
Two rectangles are proportional. Rectangle A has a width of 8 cm and a length of 12 cm. Rectangle B has a width of 10 cm. What is the area of Rectangle B?
A.150 cm²
B.120 cm²
C.160 cm²
D.130 cm²
Challenging
Company A charges $60 for 4 hours of kayak rental. Company B's rental cost is represented by the equation C = 16h, where C is the cost and h is the hours. If you need to rent a kayak for 5 hours, which company is cheaper and by how much?
A.Company A is cheaper by $5.
B.Company B is cheaper by $5.
C.Company A is cheaper by $10.
D.Company B is cheaper by $10.
Challenging
A baker makes a large batch of muffins where the amount of blueberries (b, in cups) is proportional to the amount of flour (f, in cups). The equation is b = 0.4f. If the baker used 10 cups of blueberries in total, and each batch requires 2.5 cups of flour, how many batches of muffins did he make?
A.8 batches
B.25 batches
C.4 batches
D.10 batches
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