Certainly! To understand proportional relationships and unit rates, here’s a quick overview and example problems that might be relevant to your math 7A curriculum.
Proportional Relationships
A proportional relationship between two quantities can be described by the equation \( y = kx \), where \( k \) is the constant of proportionality. This means that as one quantity increases or decreases, the other does so at a constant rate.
Unit Rate
The unit rate is a way to express how much of one quantity corresponds to one unit of another quantity. For example, if you traveled 300 miles in 5 hours, the unit rate of speed would be: \[ \text{Unit rate} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{300 \text{ miles}}{5 \text{ hours}} = 60 \text{ miles/hour} \]
Quick Check Questions
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Identify the Proportional Relationship: A recipe calls for 3 cups of flour to make 24 cookies. What would be the amount of flour needed to make 48 cookies?
Solution: Find the unit rate (flour per cookie): \[ \text{Flour per cookie} = \frac{3 \text{ cups}}{24 \text{ cookies}} = \frac{1}{8} \text{ cups per cookie} \] For 48 cookies, you multiply the unit rate by 48: \[ \text{Flour for 48 cookies} = 48 \times \frac{1}{8} = 6 \text{ cups} \]
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Finding Unit Rates: If a car uses 12 gallons of gas to travel 360 miles, what is the unit rate in miles per gallon?
Solution: \[ \text{Unit rate (miles per gallon)} = \frac{360 \text{ miles}}{12 \text{ gallons}} = 30 \text{ miles per gallon} \]
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Determine if Relationships are Proportional: Graph the following points: (1, 2), (2, 4), (3, 6), (4, 8). Are these points proportional?
Solution: Check if \( \frac{y}{x} \) is constant: \[ \frac{2}{1} = 2, \quad \frac{4}{2} = 2, \quad \frac{6}{3} = 2, \quad \frac{8}{4} = 2 \] Since the ratio remains constant, the relationship is proportional.
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Use Cross-Multiplication: If \( \frac{3}{x} = \frac{6}{12} \), find the value of \( x \).
Solution: Cross-multiply: \[ 3 \times 12 = 6 \times x \implies 36 = 6x \implies x = 6 \]
Summary
Understanding proportions and unit rates is critical in solving many real-world problems. You can assess whether a relationship is proportional by checking the constant ratio, and you can find unit rates by dividing the quantities involved. Practice these concepts by working through problems in your textbook or additional math resources.
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