Directly Proportional and Inversely Proportional

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🎬 Math Angel Video: Direct Proportion vs Inverse Proportion

What Does “Directly Proportional” Mean?

Direct proportionality between the number of watermelons and their total cost, with calculations.

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🛎️ Directly Proportionality Definition:

When two quantities are directly proportional, as one increases or decreases, the other increases or decreases by the same factor.

🛎️ Key Features of Direct Proportionality:

  • If you double one quantity, the other will also double.
  • If you triple one quantity, the other will triple too.
  • If you halve one quantity, the other will also halve.

     

🛎️ Example of Direct Proportionality:

Imagine you are buying watermelons at a shop. Each watermelon costs £4.

The total cost is directly proportional to the number of watermelons you buy, because each one costs the same amount.

Number of WatermelonsTotal Cost
1£4 = 1 x £4
3£12 = 3 x £4
5£20 = 5 x £4
8£32 = 8 x £4
10£40 = 10 x £4


You can calculate the total cost using this formula:

$$\text{Total Cost} = \text{Number of Watermelons} \times £4$$

What Is Inversely Proportional?

Inverse proportion chart showing number of workers and hours needed, demonstrating that more workers reduce the time required to complete a task.

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🛎️ Inverse Proportionality Definition:

When two quantities are inversely proportional, it means that as one increases, the other decreases in such a way that their product always stays the same.

 

🛎️ Key Features of Inverse Proportionality:

  • If you double one quantity, the other is halved.
  • If you triple one, the other becomes one third as much.
  • The product (when you multiply the two quantities together) is always the same.

 

🛎️ Example of Inverse Proportionality:

Imagine you are working on a job.

  • If 5 people work together, it takes 6 hours to finish.
  • If only 3 people work, it takes 10 hours to finish.
  • If 30 people work, it only takes 1 hour to finish.

Number of Workers Hours Needed Product (Workers × Hours)
5 6 30
3 10 30
30 1 30
15 2 30


Notice that if you multiply the number of workers by the number of hours, you always get the same result: 30.

Therefore, the number of workers and the number of hours needed to finish this task is inversely proportional

You can calculate the number of workers or hours needed using this formula:

$$ \text{Number of Workers} \times \text{Hours Needed} = \text{Constant}$$


❇️ Exam Tip:
If the product of the two quantities always stays the same, they are inversely proportional.

Direct vs Inverse Proportion

Direct and inverse proportionality concepts with real-life examples showing the relationship between quantities and costs or time.

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It’s important to learn how to spot whether a relationship is directly or inversely proportional. This skill often comes up in math exams.


🛎️ How to Spot Direct Proportion:

  • If you divide one quantity by the other, you always get the same answer.

  • Example: Divide total cost by number of watermelons → always £4.

  • This means the total cost is directly proportional to the number of watermelons.

 

🛎️ How to Spot Inverse Proportion:

  • If you multiply one quantity by the other, you always get the same answer.

  • Example: Multiply number of workers by hours needed → always 30.

  • This means the time needed is inversely proportional to the number of workers.

🍪 Quiz: Direct and Inverse Proportion Practice

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