Real-World Examples of Proportionality (Middle School)

📚 Quick Study Guide

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• Definition: Proportionality shows the relationship between two quantities where their ratio is constant. If one quantity changes, the other changes by a consistent factor.
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• Direct Proportionality: As one quantity increases, the other increases. Think: More pizza slices, more calories! The formula is $y = kx$, where $k$ is the constant of proportionality.
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• Inverse Proportionality: As one quantity increases, the other decreases. Think: More people sharing a chocolate bar, less chocolate per person! The formula is $y = \frac{k}{x}$, where $k$ is the constant of proportionality.
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• Constant of Proportionality: This is the constant value ($k$) that relates the two quantities in a proportional relationship. It's the key to solving proportionality problems!
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• Tips: Always identify whether the relationship is direct or inverse first. Set up a proportion or use the appropriate formula to solve for the unknown quantity. Practice makes perfect!

📝 Practice Quiz

1. Which of the following is an example of direct proportionality?
   1. The number of workers and the time it takes to complete a job.
   2. The speed of a car and the time it takes to travel a certain distance.
   3. The number of books purchased and the total cost.
   4. The temperature outside and the amount of clothing you wear.
2. If 3 apples cost $2.25, how much would 7 apples cost, assuming direct proportionality?
   1. $4.50
   2. $5.25
   3. $6.75
   4. $7.50
3. Which of the following is an example of inverse proportionality?
   1. The number of hours worked and the amount earned.
   2. The size of a room and the amount of paint needed to cover it.
   3. The number of guests at a party and the amount of pizza each guest gets.
   4. The distance traveled and the amount of fuel consumed.
4. If 4 workers can complete a task in 6 hours, how long would it take 8 workers to complete the same task, assuming inverse proportionality?
   1. 3 hours
   2. 4 hours
   3. 9 hours
   4. 12 hours
5. A recipe calls for 2 cups of flour to make 24 cookies. How many cups of flour are needed to make 60 cookies, assuming direct proportionality?
   1. 3 cups
   2. 4 cups
   3. 5 cups
   4. 6 cups
6. The area of a rectangle is constant. If the length is doubled, what happens to the width, assuming inverse proportionality?
   1. The width doubles.
   2. The width is halved.
   3. The width remains the same.
   4. The width is tripled.
7. A cyclist travels 30 miles in 2 hours. Assuming direct proportionality, how far will they travel in 5 hours?
   1. 50 miles
   2. 60 miles
   3. 75 miles
   4. 90 miles