Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S3-SA1-0615
What is the Direct Variation (y = kx)?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Direct Variation describes a relationship between two quantities where if one quantity increases, the other quantity also increases proportionally, and if one decreases, the other decreases proportionally. This means their ratio always remains constant. We write it as y = kx, where 'k' is a constant value.
Simple Example
Quick Example
Imagine you buy samosas. If one samosa costs ₹10, then two samosas will cost ₹20, and three will cost ₹30. Here, the number of samosas and the total cost are in direct variation. As the number of samosas increases, the total cost increases proportionally.
Worked Example
Step-by-Step
QUESTION: An auto-rickshaw travels 5 km in 15 minutes. How long will it take to travel 12 km if it maintains the same speed? Assume distance and time are in direct variation.
STEP 1: Identify the two quantities: Distance (d) and Time (t). We know they are in direct variation, so t = kd.
---STEP 2: Use the given information to find the constant 'k'. We have d = 5 km and t = 15 minutes. So, 15 = k * 5.
---STEP 3: Solve for 'k'. Divide both sides by 5: k = 15 / 5 = 3. This means for every km, it takes 3 minutes.
---STEP 4: Now use the value of k to find the time for 12 km. The equation is t = 3d. Substitute d = 12 km.
---STEP 5: Calculate the time: t = 3 * 12 = 36 minutes.
---ANSWER: It will take the auto-rickshaw 36 minutes to travel 12 km.
Why It Matters
Direct variation helps us predict how one thing changes when another changes, which is super useful! Engineers use it to design bridges, physicists use it to understand forces, and even data scientists use it to find patterns in information. Learning this helps you think like a scientist!
Common Mistakes
MISTAKE: Confusing direct variation with just any increasing relationship. For example, 'As I get older, I get taller' isn't direct variation because the ratio of height to age isn't constant. | CORRECTION: Remember, for direct variation, the ratio y/x must always be the same constant 'k'. If y/x changes, it's not direct variation.
MISTAKE: Not finding the constant 'k' first. Students sometimes jump directly to finding the unknown value without establishing the relationship. | CORRECTION: Always calculate 'k' using the given pair of values (x1, y1) before using it to find another unknown value (x2 or y2).
MISTAKE: Thinking that if one quantity increases, the other *must* increase. This is true for direct variation, but sometimes students forget the 'proportional' part. | CORRECTION: The increase (or decrease) must be at a steady, constant rate. If x doubles, y must also double. If x halves, y must also halve.
Practice Questions
Try It Yourself
QUESTION: The cost of milk is in direct variation with its quantity. If 2 litres of milk cost ₹100, how much will 5 litres of milk cost? | ANSWER: ₹250
QUESTION: A painter can paint 3 walls in 6 hours. If the number of walls painted is directly proportional to the time taken, how many hours will it take to paint 7 walls? | ANSWER: 14 hours
QUESTION: If y varies directly as x, and y = 24 when x = 8. Find the value of x when y = 39. | ANSWER: x = 13
MCQ
Quick Quiz
Which of the following scenarios represents direct variation?
The number of workers and the time taken to complete a job.
The speed of a car and the time taken to cover a fixed distance.
The number of items bought and the total cost.
The age of a person and their shoe size.
The Correct Answer Is:
C
In option C, as the number of items bought increases, the total cost increases proportionally (assuming a constant price per item). Options A and B are inverse variations, and D is not a direct proportional relationship.
Real World Connection
In the Real World
When you use a mobile data pack, the amount of data you use (in MB or GB) directly varies with the total cost you pay for that usage. Similarly, in cricket, if a player scores runs at a constant rate, the total runs scored directly vary with the number of balls faced. Even apps like Swiggy or Zomato use this concept to calculate delivery charges based on distance.
Key Vocabulary
Key Terms
VARIATION: How one quantity changes in relation to another | CONSTANT OF PROPORTIONALITY (k): The fixed ratio between two quantities in direct variation | PROPORTIONAL: Changing at a constant rate relative to something else | RATIO: A comparison of two numbers by division
What's Next
What to Learn Next
Great job understanding direct variation! Next, you should explore 'Inverse Variation'. It's another crucial way quantities can relate to each other, but in the opposite way, and knowing both will give you a powerful tool for solving many real-world problems.
