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Understanding Inverse Proportion
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Today, we are going to explore the concept of inverse proportion. Can anyone tell me what happens to one quantity when the other increases?
I think it decreases, right?
Exactly! That's correct. If one quantity increases, the other decreases. This relationship is very important in math and real life. For example, if we have a fixed amount of work, adding more workers will decrease the time taken. Letβs think of a formula. Can anyone remember what correlates their relationship?
Is it like xy = k?
Yes! Well done! So, if x is the number of workers and y is the time taken, their product remains constant. Remember this as we continue!
Examples of Inverse Proportion
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Letβs explore some real-life examples. For instance, if Zaheeda travels faster, how does her travel time change?
Her travel time would decrease!
Perfect! As we look at various cases, when speed increases, time decreases. Remember, if we double the speed, the time taken becomes half. What if we examine how it relates to buying books?
If the price of each book increases, the number of books you can buy goes down!
Exactly! So we can see this is again an example where one quantity's increase leads to the otherβs decrease. Letβs think of a table we can create to show this.
Practical Applications
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Now, letβs put this concept into practice. If I have six pipes that can fill a tank in 80 minutes, how long would five pipes take?
We can set up the equation 80 * 6 = x * 5, right?
Yes, that's right! So we calculate. What are we solving for?
Weβre looking for x, the time it would take with 5 pipes!
Correct! Letβs perform that multiplication, and what do we get?
We get 96 minutes!
Well done! Now you understand how to apply inverse proportions effectively.
Exploring More Examples
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Let's analyze more scenarios! If we keep a fixed amount of food for students and add more, what happens?
The food would run out faster!
Great observation! So we expect to see another case of inverse proportion here. Let's write this out. For 100 students, it lasts 20 days. If we have 125 students, what's the new number of days?
It would last only 16 days!
Excellent! You are really grasping the concept of how one quantity can affect another inversely.
Recap and Quiz
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To wrap up, can someone summarize what we learned about inverse proportions?
If one quantity increases, the other decreases, and their product remains constant.
That's right! Now, letβs have a quick quiz. If 15 workers can complete a task in 48 hours, how many are needed to finish in 30 hours?
That will be 24 workers!
Excellent job! Remember to keep practicing this concept as it comes up in various scenarios!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
This section introduces the concept of inverse proportion, highlighting how two quantities vary together in opposite directions. Examples include how more workers reduce the time to complete a task, or how increasing the speed of a vehicle decreases the time taken for a journey. The key equation xy = k illustrates this relationship.
Detailed
Inverse Proportion
Inverse proportion describes a fundamental relationship between two quantities: when one quantity increases, the other decreases in such a way that the product of the two quantities remains constant. For example, if we consider the time taken to complete a job with respect to the number of workers, as more workers join a task, the time to finish decreases correspondingly. The inverse relationship can be expressed through the equation xy = k, where k is a constant.
Key Points:
- Examples of inverse proportions are highlighted through practical scenarios such as Zaheeda's travel speeds, book purchases based on price, and resource allocation.
- A table illustrates how increasing the price of books results in fewer books purchased with a fixed budget, showing an inverse relationship.
- Activities and thought experiments encourage students to identify and understand further examples of inverse proportion in everyday life.
This section builds upon the principles of direct proportion by contrasting them and enabling students to grasp the versatile nature of mathematical relationships.
Example :
There are 80 students in a dormitory. Food provisions for them is for 25 days. How long will these provisions last if 15 more students join the group?
Solution: Suppose the provisions last for \( x \) days when the number of students is 95. We have the following table:
\[ \text{Number of students} = 80 \quad \text{Number of days} = 25 \]
Note that the more the number of students, the sooner would the provisions exhaust. Therefore, this is a case of inverse proportion.
\[ 80 \times 25 = 95 \times y \]
So, \[ 2000 = 95y \quad \Rightarrow \quad y = \frac{2000}{95} \approx 21.05 \]
Thus, the provisions will last for approximately 21 days if 15 more students join the dormitory.
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Understanding Inverse Proportion
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Two quantities may change in such a manner that if one quantity increases, the other quantity decreases and vice versa. For example, as the number of workers increases, time taken to finish the job decreases. Similarly, if we increase the speed, the time taken to cover a given distance decreases.
Detailed Explanation
Inverse proportion describes a relationship between two quantities where an increase in one quantity results in a decrease in the other. For instance, if more workers are assigned to a job, they can complete it in less time. Conversely, if the speed of travel increases, the time taken to reach a destination decreases. This means that the product of the two quantities remains constant; if one goes up, the other goes down.
Examples & Analogies
Consider a pizza delivery scenario: If a delivery person is on a motorbike (fast), they reach the customer quickly and take less time. However, if the same person were to walk (slow), it would take much longer to deliver the pizza. The faster the delivery method, the less time it takes to reach the same customer.
Speed and Time Example
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To understand this, let us look into the following situation. Zaheeda can go to her school in four different ways. She can walk, run, cycle or go by car. Observe that as the speed increases, time taken to cover the same distance decreases.
Detailed Explanation
Zaheeda demonstrates the concept of inverse proportion through her choice of travel speed. When she chooses to run, she doubles her speed, which halves her time to reach school. Likewise, if she cycles and increases her speed even further, the time needed decreases even more. This situation perfectly illustrates how speed and time are inversely proportional: as speed increases, time decreases.
Examples & Analogies
Imagine a race where two athletes are competing. The faster athlete (let's say they run at 10 mph) will finish the race in a shorter time compared to the slower athlete (who runs at 5 mph). If the race distance remains the same, the relationship between their speeds and the time taken to finish embodies the principle of inverse proportion.
Cost and Quantity Example
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A school wants to spend ο 6000 on mathematics textbooks. How many books could be bought at ο 40 each? Clearly 150 books can be bought. If the price of a textbook is more than ο 40, then the number of books which could be purchased with the same amount of money would be less than 150.
Detailed Explanation
When the price per textbook increases, the number of textbooks that can be purchased decreases. For example, if one textbook costs βΉ40, then 150 books can be bought for βΉ6000. However, if the price rises to βΉ50, fewer books can be obtained. This relationship shows that price and quantity purchased are in inverse proportion: as the price increases, the quantity decreases.
Examples & Analogies
Think about going to a grocery store with βΉ1000 to spend. If apples cost βΉ100 per kg, you can buy 10 kg. But if the price per kg rises to βΉ200, you would only be able to buy 5 kg. This illustrates how changes in price affect the quantity you can afford, exemplifying inverse proportion.
The Inverse Relationship Explained
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Ratio by which the price of books increases when going from 40 to 50 is 4 : 5, and the ratio by which the corresponding number of books decreases from 150 to 120 is 5 : 4. This means that the two ratios are inverses of each other. Notice that the product of the corresponding values of the two quantities is constant.
Detailed Explanation
The relationships in cost and quantity exemplify inverse proportion through the concept of constant product. When the price of a book increases, the number of books purchased decreases, yet if we multiply the number of books by their price, the total remains constant (βΉ6000). This reflects the definition of inverse proportion where the product of the quantities (price and quantity) is constant.
Examples & Analogies
Consider filling a tank: If you use one large pipe, it fills quickly. If you switch to a smaller pipe, it takes longer, but the total amount of water (the volume of the tank) remains unchanged. Regardless of which pipe is used, the relationship illustrates that as one variable (pipe size) changes, the other (time taken to fill the tank) adjusts in a way that keeps the total volume constant.
Inverse Proportion in Action
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If we represent the price of one book as x and the number of books bought as y, then as x increases y decreases and vice-versa. It is important to note that the product xy remains constant. We say that x varies inversely with y and y varies inversely with x.
Detailed Explanation
Mathematically, inverse proportion can be represented as xy = k, where k is a constant. Thus, an increase in one variable requires a decrease in the other in order to keep their product constant. This concept allows us to unravel many real-life situations involving resource management and budgeting.
Examples & Analogies
Think about sharing food among friends. If you have a single pizza to share among four friends, each person gets a larger slice. If five friends show up, the size of each slice gets smaller. This relationship between the number of friends and the size of their pizza slices reflects inverse proportion: as the guest count increases, the pizza slice size decreases.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Inverse Proportion: A relationship where one quantity increases while the other decreases.
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Constant Product: Inverse proportionality implies a fixed product of two quantities.
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Reciprocal Relationship: The inverse of a relationship where increasing one quantity affects the other negatively.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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If a car travels at a speed of 60 km/h and takes 2 hours to reach a destination, traveling at 80 km/h will reduce the time taken.
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Buying books with a fixed budget of $600 means if the price per book goes up, the total number of books you can purchase decreases.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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If you hire more crew, the days will reduce, thatβs inverse proportion, good to deduce.
π§ Other Memory Gems
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RAP - Remember: As one rises, Another Plummets.
π Fascinating Stories
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Imagine a baker with just one oven. He can bake 12 pies in 3 hours. If he buys another oven, now he can bake 24 pies in the same time, making baking even faster!
π― Super Acronyms
IP = Inverse Equals Increase and Decrease
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Inverse Proportion
Definition:
A relationship between two quantities where an increase in one results in a decrease in the other, with their product remaining constant.
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Term: Constant (k)
Definition:
A fixed value in the equation xy = k that represents the relationship between two inversely proportional quantities.
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Term: Reciprocal
Definition:
The multiplicative inverse of a number; for a number x, its reciprocal is 1/x.