This chapter introduces two important concepts of Direct Proportion and Inverse Proportion. These concepts help to determine how two quantities are related to each other mathematically. The chapter also covers solving problems using proportionality.
Direct Proportion
In direct proportion, two quantities are said to be directly proportional when an increase in one quantity leads to a proportional increase in the other, and a decrease leads to a proportional decrease.
Formula for Direct Proportion
If two quantities x and y are in direct proportion, they are related as follows:
x / y = k (where k is a constant)
This can also be written as:
x = k × y or y = x / k
Real-life Example of Direct Proportion
Example 1: The cost of apples is directly proportional to the number of apples purchased. If the cost of 5 apples is ₹20, then the cost of 10 apples would be proportional.
Process:
Use the direct proportion formula:
5 / 20 = 10 / x
Cross multiply to solve for x:
5 × x = 200 → x = 40
Answer: The cost of 10 apples would be ₹40.
Inverse Proportion
In inverse proportion, two quantities are said to be inversely proportional if an increase in one quantity leads to a proportional decrease in the other and vice versa.
Formula for Inverse Proportion
If two quantities x and y are in inverse proportion, they are related as follows:
x × y = k (where k is a constant)
This can also be written as:
x = k / y or y = k / x
Real-life Example of Inverse Proportion
Example 2: The time taken to complete a task is inversely proportional to the number of people working. If 6 workers can complete a task in 12 hours, how many hours would it take for 9 workers to complete the same task?
Process:
Use the inverse proportion formula:
6 × 12 = 9 × x
Solve for x:
72 = 9 × x → x = 72 / 9 = 8
Answer: It would take 9 workers 8 hours to complete the task.
Solving Direct and Inverse Proportion Problems
The chapter includes solving a variety of problems based on direct and inverse proportions. Let’s look at some solved examples.
Example 3 – Direct Proportion
Problem: If 3 meters of fabric costs ₹180, find the cost of 7 meters of fabric.
Process:
Let the cost of 7 meters of fabric be x.
Since the cost is directly proportional to the length, use the formula:
3 / 180 = 7 / x
Cross-multiply and solve for x:
3 × x = 7 × 180 → x = 1260 / 3 = 420
Answer: The cost of 7 meters of fabric is ₹420.
Example 4 – Inverse Proportion
Problem: If 10 people can paint a house in 15 days, how many days would it take for 5 people to paint the same house?
Process:
Let the number of days taken by 5 people be x.
Since the number of people and days are inversely proportional, use the formula:
10 × 15 = 5 × x
Solve for x:
150 = 5 × x → x = 150 / 5 = 30
Answer: It would take 5 people 30 days to paint the house.
Key Terms and Concepts
Direct Proportion: A relationship where one quantity increases or decreases with the other.
Inverse Proportion: A relationship where one quantity increases while the other decreases.
Proportionality Constant (k): The constant that relates two proportional quantities.
Cross-Multiplication: A method used to solve problems involving direct and inverse proportions.
Real-life Applications of Direct and Inverse Proportions
Direct Proportion in Shopping: If the price of 1 liter of milk is ₹20, then the price of 5 liters will be proportional.
Inverse Proportion in Speed and Time: If the speed of a car increases, the time taken to reach the destination decreases, and vice versa.
Common Mistakes to Avoid
Confusing direct and inverse proportions: Always remember that in direct proportion, both quantities move in the same direction, while in inverse proportion, they move in opposite directions.
Not cross-multiplying correctly: Ensure that cross-multiplication is applied properly when solving equations.
Practice Problems
Problem 1: If 4 workers can complete a task in 8 hours, how many workers would be required to complete the task in 6 hours? (Inverse Proportion)
Problem 2: The cost of 9 kg of sugar is ₹315. Find the cost of 12 kg of sugar. (Direct Proportion)
FAQs on Chapter 11: Direct and Inverse Proportions
1. What is direct proportion?
Direct proportion is when two quantities increase or decrease together in the same ratio. For example, if the quantity of apples increases, the cost also increases proportionally.
2. What is inverse proportion?
Inverse proportion is when one quantity increases while the other decreases. For example, if the number of workers increases, the time taken to complete a task decreases proportionally.
3. How do you recognize a direct proportion problem?
A direct proportion problem involves two quantities that increase or decrease together. If dividing one by the other gives a constant value, the problem involves direct proportion.
4. How do you recognize an inverse proportion problem?
An inverse proportion problem involves two quantities where one increases as the other decreases. If multiplying them together gives a constant value, it involves inverse proportion.
5. What is the formula for direct proportion?
The formula for direct proportion is x / y = k, where k is a constant.
6. What is the formula for inverse proportion?
The formula for inverse proportion is x × y = k, where k is a constant.
7. Can you give a real-life example of direct proportion?
A real-life example of direct proportion is the relationship between distance and fuel consumption. The more distance you drive, the more fuel you consume, assuming fuel consumption remains constant.
8. Can you give a real-life example of inverse proportion?
A real-life example of inverse proportion is the relationship between the number of workers and the time required to complete a task. More workers mean less time is needed to finish the task.
9. What happens if the proportionality constant (k) is zero?
If the proportionality constant (k) is zero, then the quantities are not related through direct or inverse proportion. In direct proportion, both quantities would be zero, and in inverse proportion, the relationship wouldn’t hold.
10. How can you solve direct proportion problems using cross multiplication?
To solve direct proportion problems, you can set up a proportion equation (x1 / y1 = x2 / y2), then cross-multiply and solve for the unknown variable.
11. How can you solve inverse proportion problems using cross multiplication?
To solve inverse proportion problems, use the equation x1 × y1 = x2 × y2, and then solve for the unknown variable by rearranging the equation.
12. Can direct and inverse proportions occur together?
Yes, in some complex problems, certain parts of the system may be directly proportional, while other parts may be inversely proportional.
13. What is the difference between direct and inverse proportion?
In direct proportion, both quantities move in the same direction, either increasing or decreasing together. In inverse proportion, one quantity increases as the other decreases.
14. How is direct proportion used in real-world applications?
Direct proportion is used in various real-world applications, such as shopping (more quantity means more cost) and scaling models (if dimensions increase, the area or volume increases proportionally).
15. How is inverse proportion used in real-world applications?
Inverse proportion is used in applications like speed and time (increasing speed decreases travel time) and work-related tasks (more workers mean less time is needed).
16. What are the key signs that two quantities are in direct proportion?
Key signs include that as one quantity increases, the other increases proportionally, and when you divide one by the other, the result is a constant value.
17. What are the key signs that two quantities are in inverse proportion?
In inverse proportion, one quantity increases while the other decreases proportionally, and when you multiply the two, the result is a constant value.
18. Is the relationship between speed and distance direct or inverse proportion?
The relationship between speed and distance is direct proportion if time is constant. As speed increases, the distance covered also increases proportionally.
19. Is the relationship between speed and time direct or inverse proportion?
The relationship between speed and time is an inverse proportion if the distance is constant. As speed increases, the time taken decreases.
20. How do you solve direct proportion problems in shopping?
In shopping, if the price per unit is constant, you can solve direct proportion problems by dividing the cost by the quantity, or using cross-multiplication for unknowns.
21. What is the role of the constant in direct proportion?
In direct proportion, the constant is the value that remains the same for all proportional relationships. It represents how one quantity is related to another.
22. What is the role of the constant in inverse proportion?
In inverse proportion, the constant represents the product of two variables. It shows how an increase in one quantity results in a decrease in another.
23. Can proportions be applied to geometry?
Yes, proportions are often used in geometry, especially when dealing with similar shapes, where the ratio of corresponding sides is in direct proportion.
24. How does cross-multiplication help in solving proportion problems?
Cross-multiplication allows you to solve for an unknown in proportion problems by multiplying across the equals sign and isolating the unknown variable.
25. What are common mistakes to avoid when solving proportion problems?
Common mistakes include mixing up direct and inverse proportions, failing to cross-multiply correctly, and misinterpreting the relationships between variables.
MCQs on Chapter 11: Direct and Inverse Proportions
MCQs on Chapter 11: Direct and Inverse Proportions
1. What is direct proportion?
2. What is inverse proportion?
3. What is the formula for direct proportion?
4. What is the formula for inverse proportion?
5. In a direct proportion, if x increases, what happens to y?
6. In an inverse proportion, if x increases, what happens to y?
7. If 6 workers can finish a task in 12 days, how many days will it take for 9 workers to finish the same task?
8. The cost of 3 meters of cloth is ₹180. What will be the cost of 7 meters of the same cloth?
9. If 10 people can paint a house in 15 days, how long will it take 5 people to paint the house?
10. If 4 workers can complete a task in 8 hours, how many workers would be required to complete the task in 6 hours?
11. If the speed of a car is increased, what happens to the time taken to reach the destination?
12. In direct proportion, what is the relationship between the two quantities?
13. In inverse proportion, what is the relationship between the two quantities?
14. If a car travels 100 km in 2 hours, how far will it travel in 5 hours at the same speed?
15. What is the unit of speed in the SI system?
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