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Understanding Direct Proportion
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Today, we're going to learn about direct proportion. Can anyone tell me what happens to the cost if I buy more kilograms of sugar?
The cost increases as we buy more sugar!
"Exactly! If 1 kg costs
Examples of Direct Proportion
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Let's look at another example involving petrol and distance. If 4 litres of petrol takes a car 60 km, how far will it go with 12 litres?
It should travel 180 km.
Exactly! The distance is three times because the petrol consumed is three times more. Can anyone explain why we can say there is a direct proportion here?
Because both the petrol and distance increase in the same ratio.
Right! Let's make a quick table together to visualize this information.
What do we put in the table?
We'll fill in the litres of petrol and corresponding distances! This will help us see the pattern. Remember, for direct proportions, you can check if the ratios stay the same!
Proportion Activities in Class
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Now, letβs do an activity with a clock's minute hand. When we record the time and angle, what will we find out?
Weβll see if the angle changes the same with every minute?
Exactly! Make sure to observe how they relate to each other. If they keep the same ratio, then they are in direct proportion. Let's start recording!
In this table, as time increases, the angle turns too. Looks like they do stay proportional!
Great job! This is a classic example of direct proportion in motion.
Practical Applications of Direct Proportion
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Can anyone think of a situation where direct proportion applies? Perhaps in cooking or shopping?
In recipes! If you double the servings, you double the ingredients!
Exactly! And what about when we buy certain items? Let's brainstorm some other examples.
When we buy fruits, if I buy more apples, I pay more money but the price per apple stays the same!
Well said! Itβs so important to understand how these quantities affect each other in life.
Understanding Non-Direct Proportions
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Finally, let's understand that not all relationships are direct proportion. Can anyone provide an example?
Changes in age between friends?
Correct! Age changes, but not in a constant ratio. It's critical to identify these cases too.
So we have to be careful about how we classify relationships in math?
Yes! Always evaluate if the ratios hold true to ensure they are in direct proportion.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, direct proportion is explored through various examples such as the relationship between cost and quantity of sugar, petrol consumption and distance traveled, and more. The concept is illustrated with interactive activities and real-life applications, reinforcing the idea that two quantities in direct proportion change consistently in relation to one another.
Detailed
Direct Proportion
Direct proportion is a fundamental concept in mathematics that describes a relationship between two variables, where an increase in one variable results in a proportional increase in the other. This section outlines the basic principles of direct proportion through several relatable examples and exercises. If the cost of 1 kg of sugar is
36, then the cost of 3 kg is
108, demonstrating that as the quantity of sugar increases, the cost also increases in a consistent ratio.
Key Concepts
- Definition: Two quantities are said to be in direct proportion when their ratio remains constant.
- Examples provided include the use of petrol and the distance traveled by a car, and the relationship between the length of cloth and its cost.
Applications
This section also includes practical activities such as measuring the angle turned by a clock's minute hand and drawing direct comparisons between ages and their ratios. It concludes with exercises that require the application of the direct proportion concept in various scenarios, reinforcing the understanding of how different variables are interconnected.
Example
The scale of a map is given as 1:500000. Two cities are 5 cm apart on the map. Find the actual distance between them.
Solution: Let the map distance be \( x \) cm and actual distance be \( y \) cm, then:
\[ 1:500000 = \frac{x}{y} \]
or
\[ 500000 \cdot x = y \]
Since \( x = 5 \):
\[ 500000 \cdot 5 = y \]
Thus, two cities, which are 5 cm apart on the map, are actually \( 500000 \cdot 5 = 2500000 \) m or 2500 km away from each other.
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Audio Book
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Understanding Direct Proportion
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If the cost of 1 kg of sugar is βΉ36, then what would be the cost of 3 kg sugar? It is βΉ108. Similarly, we can find the cost of 5 kg or 8 kg of sugar. Study the following table. Observe that as the weight of sugar increases, cost also increases in such a manner that their ratio remains constant.
Detailed Explanation
In direct proportion, two quantities increase or decrease together at a constant rate. In this example, when the weight of sugar increases from 1 kg to 3 kg, the cost increases from βΉ36 to βΉ108. The ratio of cost to weight remains the same, which is 36/1 = 108/3. This constant ratio is the key feature of direct proportion.
Examples & Analogies
Imagine you are buying fruits. If 1 kg of apples costs βΉ50, then 2 kg will cost βΉ100. The cost scales directly with the weight of apples. This means if you buy more apples, you always pay more, and the ratio of cost to quantity remains the same.
Example with Petrol Consumption
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Take one more example. Suppose a car uses 4 litres of petrol to travel a distance of 60 km. How far will it travel using 12 litres? The answer is 180 km. How did we calculate it? Since petrol consumed in the second instance is 12 litres, i.e., three times of 4 litres, the distance travelled will also be three times of 60 km.
Detailed Explanation
In this case, the relationship between petrol consumption and distance travelled is directly proportional. We notice that if the car uses three times more petrol (12 litres instead of 4 litres), it will also travel three times further (180 km instead of 60 km). This reinforces the principle that in direct proportion, if one value increases multiplicatively, the other does too in the same ratio.
Examples & Analogies
Think of filling a swimming pool. If it takes 1 hour to fill the pool with a hose, with a double-sized hose, it will take half the timeβ30 minutes. The volume of water flowing in directly affects how quickly the pool fills up. If you double the flow, the time to fill is halved.
Ratios in Direct Proportion
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We say that x and y are in direct proportion, if \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \) or \( x = ky \). The consumption of petrol and the distance travelled by a car is a case of direct proportion. Similarly, the total amount spent and the number of articles purchased is also an example of direct proportion.
Detailed Explanation
This statement describes how we express the relationship between two variables in direct proportion. The equation \( \frac{x_1}{y_1} = \frac{x_2}{y_2} \) signifies that the ratios remain consistent. Therefore, doubling one quantity will double the other. For example, if you buy twice as many items, the total amount spent also doubles.
Examples & Analogies
Consider going to a pizza shop. If one pizza costs βΉ200, then twice the number of pizzas (2 pizzas) will cost βΉ400. The total cost is directly proportional to the number of pizzas you buy.
Activities for Understanding
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Observe the angle turned by the minute hand of a clock relative to the time that has passed. Is the angle turned through by the minute hand directly proportional to the time that has passed? Yes! From the above table, you can also see T : T = A : A, because T1 : T2 = 15 : 30 = 1:2.
Detailed Explanation
This activity demonstrates the concept of direct proportion through practical observation. By measuring the angle the minute hand moves as time passes, we found that as time doubles, the angle also doubles, confirming that they are in direct proportion.
Examples & Analogies
If you think about cooking, if a recipe calls for 30 minutes of simmering, and you decide to double the amount of soup, you likely need to simmer it for 60 minutes. The time required to cook increases directly with the amount of soup.
Solving Problems with Direct Proportion
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Let's consider some solved examples where we would use the concept of direct proportion. Example 1: The cost of 5 metres of a particular quality of cloth is βΉ210. Tabulate the cost of 2, 4, 10 and 13 metres of cloth of the same type.
Detailed Explanation
In this example, we evaluate the cost of different lengths of cloth knowing the price for 5 metres. With direct proportion, the cost is calculated proportionally based on the length. If 5 metres cost βΉ210, 2 metres would logically cost less, and we can determine the costs of 4, 10, and 13 metres similarly.
Examples & Analogies
Imagine filling up gas in your car. If βΉ100 gives you 5 litres of petrol, you can quickly find out how much you would pay for 2 litres (which would be less) or 10 litres (which would be more) based on direct proportion.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Definition: Two quantities are said to be in direct proportion when their ratio remains constant.
-
Examples provided include the use of petrol and the distance traveled by a car, and the relationship between the length of cloth and its cost.
-
Applications
-
This section also includes practical activities such as measuring the angle turned by a clock's minute hand and drawing direct comparisons between ages and their ratios. It concludes with exercises that require the application of the direct proportion concept in various scenarios, reinforcing the understanding of how different variables are interconnected.
-
Example
-
The scale of a map is given as 1:500000. Two cities are 5 cm apart on the map. Find the actual distance between them.
-
Solution: Let the map distance be \( x \) cm and actual distance be \( y \) cm, then:
-
\[ 1:500000 = \frac{x}{y} \]
-
or
-
\[ 500000 \cdot x = y \]
-
Since \( x = 5 \):
-
\[ 500000 \cdot 5 = y \]
-
Thus, two cities, which are 5 cm apart on the map, are actually \( 500000 \cdot 5 = 2500000 \) m or 2500 km away from each other.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example: If the cost of 1 kg of sugar is 36, then the cost of 3 kg is 108.
-
Example: If a car uses 4 litres of petrol to cover 60 km, it will cover 180 km using 12 litres.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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When one goes up, the other stays true, that's direct proportion just for you!
π Fascinating Stories
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Imagine two friends, Sam and Tim. When Sam eats more, Tim eats the same. If Sam eats two sandwiches, Tim has two too. This shows how they are in direct proportion!
π§ Other Memory Gems
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Remember: DRβ Direct & Ratio. Direct proportion is all about maintaining a ratio.
π― Super Acronyms
P.A.R. - Proportion Always Remains! Keep this in mind when assessing direct proportions.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Direct Proportion
Definition:
A relationship between two quantities where an increase in one leads to a proportionate increase in the other.
-
Term: Ratio
Definition:
A relationship between two numbers indicating how many times the first number contains the second.
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Term: Constant (k)
Definition:
A fixed value that maintains the ratio between two quantities in direct proportion.
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Term: Proportionality
Definition:
The quality of a relationship where two variables change in a consistent ratio.