What is the Constant of Proportionality?

S3-SA1-0219

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Constant of Proportionality is a fixed, unchanging number that links two quantities that are directly proportional to each other. It tells us how much one quantity changes for every unit change in the other quantity, represented as k in the equation y = kx.

Simple Example

Quick Example

Imagine you are buying samosas. If one samosa costs ₹10, then two samosas cost ₹20, and three cost ₹30. Here, the total cost is directly proportional to the number of samosas. The constant of proportionality is ₹10 per samosa, because for every additional samosa, the cost increases by ₹10.

Worked Example

Step-by-Step

PROBLEM: An auto-rickshaw charges ₹12 for every kilometer travelled. What is the constant of proportionality? How much will it cost to travel 5 km?

Step 1: Understand the relationship. The total fare (y) is directly proportional to the distance travelled (x).
---Step 2: Write the direct proportion equation: y = kx, where k is the constant of proportionality.
---Step 3: Use the given information: The auto charges ₹12 for 1 km. So, when x = 1 km, y = ₹12.
---Step 4: Substitute these values into the equation: 12 = k * 1.
---Step 5: Solve for k: k = 12.
---Step 6: The constant of proportionality is 12 (₹ per km).
---Step 7: Now, to find the cost for 5 km, use y = kx with k=12 and x=5. y = 12 * 5.
---Step 8: Calculate y: y = ₹60.

ANSWER: The constant of proportionality is 12. It will cost ₹60 to travel 5 km.

Why It Matters

Understanding the constant of proportionality is crucial for solving problems in science and technology. Data scientists use it to model relationships in data, physicists use it in laws like Ohm's Law (V=IR), and economists use it to understand how prices or quantities change. It helps engineers design systems and predict outcomes.

Common Mistakes

MISTAKE: Confusing direct proportion with inverse proportion. | CORRECTION: Direct proportion means as one quantity increases, the other increases (y=kx). Inverse proportion means as one quantity increases, the other decreases (y=k/x). Always check if the quantities move in the same direction.

MISTAKE: Not stating the units for the constant of proportionality. | CORRECTION: The constant 'k' always has units, which are the units of 'y' divided by the units of 'x'. For example, if y is in Rupees and x is in kilograms, k will be in Rupees/kilogram.

MISTAKE: Assuming a relationship is proportional without checking. | CORRECTION: Always check if the ratio y/x is constant for all given pairs of values. If y/x changes, then it's not a direct proportion, and there's no single constant of proportionality.

Practice Questions

Try It Yourself

QUESTION: If 3 litres of petrol cost ₹300, what is the constant of proportionality (cost per litre)? | ANSWER: ₹100 per litre

QUESTION: A factory produces 25 toys in 5 hours. If the production rate is constant, what is the constant of proportionality (toys per hour)? How many toys will be produced in 8 hours? | ANSWER: 5 toys per hour; 40 toys

QUESTION: The electricity bill (in ₹) is directly proportional to the number of units consumed. If 150 units cost ₹1200, find the constant of proportionality. Then, calculate the cost for 220 units. | ANSWER: ₹8 per unit; ₹1760

MCQ

Quick Quiz

If y is directly proportional to x, and y = 40 when x = 8, what is the constant of proportionality?

320

0.2

The Correct Answer Is:

Since y is directly proportional to x, we have y = kx. Substituting y=40 and x=8, we get 40 = k * 8. Dividing both sides by 8 gives k = 5. Options A, C, D are incorrect calculations.

Real World Connection

In the Real World

In cricket, a batsman's strike rate is a constant of proportionality. If a batsman scores 60 runs in 30 balls, their strike rate (runs per ball) is 2. This helps coaches and analysts predict performance and make strategic decisions. Similarly, UPI transaction fees (if any) could be seen as a constant of proportionality for the amount transferred.

Key Vocabulary

Key Terms

PROPORTION: A relationship between quantities where their ratio is constant | DIRECT PROPORTION: A relationship where two quantities increase or decrease at the same rate (y=kx) | INVERSE PROPORTION: A relationship where as one quantity increases, the other decreases (y=k/x) | RATIO: A comparison of two quantities by division | VARIABLE: A symbol (like x or y) that represents a quantity that can change

What's Next

What to Learn Next

Great job learning about the constant of proportionality! Next, explore 'Inverse Proportionality'. You'll see how the constant of proportionality works differently when two quantities move in opposite directions, which is super useful in real-world problems like speed and time.