What is Direct Variation?

S3-SA1-0215

Grade Level:

Class 8

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 quantity decreases, the other quantity also decreases proportionally. This means their ratio always remains constant.

Simple Example

Quick Example

Imagine you are buying ladoos from a sweet shop. If one ladoo costs ₹20, then two ladoos will cost ₹40, and three ladoos will cost ₹60. As the number of ladoos increases, the total cost also increases in the same proportion. This is a direct variation.

Worked Example

Step-by-Step

PROBLEM: An auto-rickshaw charges ₹15 for every kilometer travelled. How much will it cost to travel 8 kilometers?

STEP 1: Identify the relationship. The cost increases as the distance increases, so it's a direct variation.
---STEP 2: Write the direct variation formula: y = kx, where y is the total cost, x is the distance, and k is the constant of variation (cost per km).
---STEP 3: Find the constant of variation (k). We know 1 km costs ₹15, so k = 15/1 = 15.
---STEP 4: Now use the formula with the given distance. We need to find y when x = 8 km.
---STEP 5: Substitute the values: y = 15 * 8.
---STEP 6: Calculate: y = 120.
---ANSWER: It will cost ₹120 to travel 8 kilometers.

Why It Matters

Understanding direct variation is crucial in fields like Physics to calculate speed or force, in Economics to understand supply and demand, and in Data Science to model relationships between variables. Engineers use it to design structures, and it's even fundamental to how AI algorithms make predictions.

Common Mistakes

MISTAKE: Confusing direct variation with addition or subtraction (e.g., thinking if x increases by 2, y also increases by 2). | CORRECTION: Remember that in direct variation, quantities change by multiplication or division, maintaining a constant ratio, not a constant difference.

MISTAKE: Forgetting to find the constant of variation (k) or miscalculating it. | CORRECTION: Always calculate 'k' first using a known pair of values (y/x). This 'k' is the unique multiplier that connects the two variables.

MISTAKE: Assuming any increasing relationship is direct variation. | CORRECTION: For direct variation, the graph must be a straight line passing through the origin (0,0). If it doesn't pass through the origin, or isn't a straight line, it's not direct variation.

Practice Questions

Try It Yourself

QUESTION: If 5 pens cost ₹75, how much will 9 pens cost? | ANSWER: ₹135

QUESTION: A car travels 120 km in 2 hours. Assuming constant speed (direct variation between distance and time), how far will it travel in 5 hours? | ANSWER: 300 km

QUESTION: The amount of flour needed for a recipe varies directly with the number of rotis you want to make. If 3 cups of flour make 18 rotis, how many cups of flour are needed to make 30 rotis? | ANSWER: 5 cups

MCQ

Quick Quiz

Which of the following equations represents a direct variation?

y = x + 5

y = 3x

y = x - 2

y = 6/x

The Correct Answer Is:

Direct variation is represented by the equation y = kx, where k is a non-zero constant. Option B, y = 3x, fits this form with k=3. The other options are not direct variations.

Real World Connection

In the Real World

In India, when you pay for mobile data, the cost often varies directly with the amount of data you use (e.g., ₹10 per GB). Similarly, the amount of fuel your car consumes is directly proportional to the distance you travel, a concept used by GPS apps like Google Maps to estimate fuel costs for your trip.

Key Vocabulary

Key Terms

PROPORTIONAL: Changing in size or amount in relation to something else, maintaining a constant ratio. | CONSTANT OF VARIATION (k): The fixed, non-zero number that relates two variables in a direct variation. | RATIO: A comparison of two numbers or quantities by division. | ORIGIN: The point (0,0) on a coordinate plane where the x and y axes intersect.

What's Next

What to Learn Next

Great job understanding direct variation! Next, you should explore 'Inverse Variation'. It's another fundamental relationship between quantities, but they behave in opposite ways, which is equally important in many real-world situations.