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What is the Inverse Variation (y = k/x)?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Inverse variation describes a relationship where if one quantity increases, the other quantity decreases proportionally, and vice-versa. When two quantities vary inversely, their product remains constant. The formula for inverse variation is y = k/x, where 'k' is a constant.
Simple Example
Quick Example
Imagine you have 12 ladoos to share equally among your friends. If you have 2 friends, each gets 6 ladoos. If you invite 4 friends, each gets 3 ladoos. As the number of friends (x) increases, the number of ladoos each friend gets (y) decreases. The total number of ladoos (12) remains constant.
Worked Example
Step-by-Step
PROBLEM: If 4 workers can paint a house in 12 days, how many days will it take 6 workers to paint the same house, assuming they work at the same rate?
---Step 1: Identify the relationship. More workers mean less time to paint the house. This is an inverse variation.
---Step 2: Write the formula for inverse variation: y = k/x. Here, 'x' is the number of workers and 'y' is the number of days. 'k' is the constant.
---Step 3: Use the given information (4 workers, 12 days) to find the constant 'k'. We know y = 12 when x = 4. So, 12 = k/4.
---Step 4: Solve for 'k'. Multiply both sides by 4: k = 12 * 4 = 48.
---Step 5: Now we know the constant k = 48. We need to find 'y' (days) when x = 6 workers. Use the formula again: y = k/x.
---Step 6: Substitute the values: y = 48/6.
---Step 7: Calculate y: y = 8.
---Answer: It will take 6 workers 8 days to paint the house.
Why It Matters
Inverse variation is crucial in fields like Physics to understand how pressure and volume of gas relate. In Engineering, it helps design systems where resources are distributed. Data scientists use it to analyze how different factors impact outcomes, helping them build smarter AI models.
Common Mistakes
MISTAKE: Assuming it's direct variation and setting up y = kx. | CORRECTION: Remember that in inverse variation, as one quantity increases, the other decreases. Always use y = k/x.
MISTAKE: Forgetting to find the constant 'k' first. | CORRECTION: Always use the initial pair of values given to calculate the constant 'k' before solving for the unknown quantity.
MISTAKE: Dividing 'k' by the known 'y' value to find 'x' when 'x' is in the denominator. | CORRECTION: If y = k/x, then x = k/y. Always rearrange the formula correctly to isolate the variable you need to find.
Practice Questions
Try It Yourself
QUESTION: If 3 taps can fill a tank in 10 hours, how long will 5 taps take to fill the same tank? | ANSWER: 6 hours
QUESTION: The speed of a car and the time taken to cover a certain distance are inversely proportional. If a car travels at 60 km/h and takes 4 hours, what speed is needed to cover the same distance in 3 hours? | ANSWER: 80 km/h
QUESTION: A group of friends decided to buy a gift worth Rs. 1200. If 4 friends contribute, how much does each pay? If 2 more friends join, how much less does each friend pay compared to the first situation? | ANSWER: Each of the 4 friends pays Rs. 300. With 6 friends, each pays Rs. 200. So, each friend pays Rs. 100 less.
MCQ
Quick Quiz
Which of the following scenarios represents an inverse variation?
The cost of pens and the number of pens bought.
The amount of petrol in a car and the distance it can travel.
The number of workers building a wall and the time taken to build it.
The height of a plant and the number of days it grows.
The Correct Answer Is:
C
In option C, as the number of workers increases, the time taken to build the wall decreases. This is the definition of inverse variation. Other options show direct variation or no direct relationship.
Real World Connection
In the Real World
Think about mobile data usage. If you have a fixed data pack (like 10 GB), the more videos (large files) you download, the fewer videos you can store on your phone before running out of space. Or consider a delivery service like Zepto: more delivery riders mean faster delivery times for a fixed number of orders.
Key Vocabulary
Key Terms
INVERSE: Opposite in effect or relationship | VARY: To change or differ | PROPORTIONAL: Having a constant ratio or relationship | CONSTANT: A value that does not change | RECIPROCAL: The inverse of a number (1/x)
What's Next
What to Learn Next
Great job understanding inverse variation! Next, you can explore 'Direct and Inverse Proportions' together to compare and contrast them. This will help you identify different types of relationships in problems more easily and prepare you for advanced algebra.
