What is Variation?

S6-SA5-0006

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Variation describes how one quantity changes in relation to another quantity. It helps us understand if quantities increase or decrease together, or if one increases while the other decreases.

Simple Example

Quick Example

Imagine you are buying samosas. If one samosa costs Rs. 10, then two samosas will cost Rs. 20, and three will cost Rs. 30. Here, the total cost varies directly with the number of samosas you buy.

Worked Example

Step-by-Step

Problem: If a car travels at a constant speed, the distance covered varies directly with the time taken. If the car covers 120 km in 2 hours, how much distance will it cover in 5 hours?

Step 1: Understand direct variation. This means Distance = k * Time, where 'k' is a constant.
---Step 2: Use the given information to find 'k'. Distance = 120 km, Time = 2 hours. So, 120 = k * 2.
---Step 3: Solve for 'k'. k = 120 / 2 = 60. This 'k' is the speed of the car (60 km/hour).
---Step 4: Now use 'k' to find the distance for the new time. Time = 5 hours.
---Step 5: Distance = k * Time = 60 * 5.
---Step 6: Calculate the distance. Distance = 300 km.

Answer: The car will cover 300 km in 5 hours.

Why It Matters

Understanding variation is crucial in many fields. Engineers use it to design structures, scientists in AI/ML use it to model data relationships, and doctors use it to understand how medicine dosage affects health outcomes. It's the basic building block for predicting and controlling real-world systems.

Common Mistakes

MISTAKE: Confusing direct variation with inverse variation. Students often assume all related quantities increase or decrease together. | CORRECTION: Remember that in inverse variation, as one quantity increases, the other decreases (e.g., more workers, less time to finish a job).

MISTAKE: Forgetting to find the constant of proportionality ('k'). Students might try to directly calculate without establishing the relationship first. | CORRECTION: Always find the constant 'k' using the given set of values before solving for an unknown quantity.

MISTAKE: Incorrectly setting up the variation equation. For example, writing x = y/k for direct variation. | CORRECTION: For direct variation, it's y = kx or y/x = k. For inverse variation, it's y = k/x or xy = k.

Practice Questions

Try It Yourself

QUESTION: If the cost of 5 pens is Rs. 75, what is the cost of 8 pens, assuming direct variation? | ANSWER: Rs. 120

QUESTION: If 3 workers can complete a task in 10 days, how many days will 5 workers take to complete the same task, assuming inverse variation? | ANSWER: 6 days

QUESTION: The volume (V) of a sphere varies directly as the cube of its radius (r). If a sphere with radius 2 cm has a volume of 33.5 cm^3 (approximately), what is the volume of a sphere with radius 4 cm? | ANSWER: Approximately 268 cm^3

MCQ

Quick Quiz

Which of the following is an example of inverse variation?

More hours worked, more salary earned

More petrol in a car, more distance it can travel

More speed of a bike, less time to reach the destination

More samosas bought, more money spent

The Correct Answer Is:

Option C describes inverse variation because as the speed increases, the time taken to cover a fixed distance decreases. Options A, B, and D are examples of direct variation.

Real World Connection

In the Real World

When you use navigation apps like Google Maps or Ola/Uber, the estimated time of arrival (ETA) varies inversely with the average speed of the vehicle. If traffic increases (meaning lower speed), the ETA increases. Similarly, the price of vegetables in a mandi varies based on supply and demand, showing complex variations.

Key Vocabulary

Key Terms

DIRECT VARIATION: When two quantities increase or decrease together in the same ratio. | INVERSE VARIATION: When one quantity increases as the other decreases, and vice-versa. | CONSTANT OF PROPORTIONALITY: The fixed ratio or product between two quantities in a variation relationship, often denoted by 'k'. | JOINT VARIATION: When one quantity varies directly with two or more other quantities. | COMBINED VARIATION: When a quantity varies directly with some quantities and inversely with others.

What's Next

What to Learn Next

Great job understanding variation! Next, you can explore 'Proportion and Ratios' to see how these concepts are deeply connected. You'll then be ready to tackle more complex problems involving percentages and real-world financial calculations.