What is the Constant of Proportionality (k)?

S3-SA1-0619

Grade Level:

Class 7

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

Definition

What is it?

The Constant of Proportionality (k) is a fixed number that relates 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, always staying the same.

Simple Example

Quick Example

Imagine you buy samosas. If one samosa costs Rs 10, then two samosas cost Rs 20, and three samosas cost Rs 30. Here, the cost is directly proportional to the number of samosas. The constant of proportionality (k) is Rs 10, because for every one samosa, the cost increases by Rs 10.

Worked Example

Step-by-Step

PROBLEM: An auto-rickshaw charges Rs 15 for every kilometer travelled. Find the constant of proportionality (k) and use it to calculate the fare for 5 kilometers.

STEP 1: Identify the two quantities that are directly proportional. Here, it's 'Distance travelled (in km)' and 'Fare (in Rs)'.
---STEP 2: Write the relationship as a direct proportion: Fare = k * Distance.
---STEP 3: Use the given information to find k. We know for 1 km, the fare is Rs 15. So, 15 = k * 1.
---STEP 4: Solve for k. k = 15 / 1 = 15.
---STEP 5: State the constant of proportionality. The constant of proportionality (k) is 15.
---STEP 6: Use k to find the fare for 5 km. Fare = k * Distance = 15 * 5.
---STEP 7: Calculate the fare. Fare = 75.
---ANSWER: The constant of proportionality (k) is 15. The fare for 5 kilometers is Rs 75.

Why It Matters

Understanding 'k' is crucial for predicting outcomes and making smart decisions. Engineers use it to design bridges, data scientists use it to understand trends in customer behavior, and even economists use it to model market changes. It's a fundamental concept for careers in AI/ML, Data Science, and Computer Science.

Common Mistakes

MISTAKE: Confusing direct proportion with inverse proportion. | CORRECTION: In direct proportion, if one quantity increases, the other increases. For inverse proportion, if one increases, the other decreases.

MISTAKE: Calculating 'k' incorrectly by dividing the wrong way (e.g., dividing x by y instead of y by x when y = kx). | CORRECTION: Always remember the formula y = kx, so k = y/x. Make sure to put the dependent quantity (y) in the numerator.

MISTAKE: Thinking 'k' changes for different values in the same problem. | CORRECTION: The 'constant' in constant of proportionality means it stays the same throughout that particular relationship.

Practice Questions

Try It Yourself

QUESTION: If 3 pens cost Rs 45, what is the constant of proportionality (k) for the cost per pen? | ANSWER: k = 15

QUESTION: A car travels 120 km in 2 hours. Assuming constant speed, find the constant of proportionality (k) representing speed in km/hr. How far will it travel in 5 hours? | ANSWER: k = 60 km/hr; Distance = 300 km

QUESTION: The amount of sugar needed for a recipe is directly proportional to the number of servings. If 2 cups of sugar are needed for 8 servings, how many cups of sugar are needed for 12 servings? First find k. | ANSWER: k = 0.25 cups/serving; Sugar needed = 3 cups

MCQ

Quick Quiz

If the cost of mobile data is Rs 200 for 2 GB, and it's directly proportional, what is the constant of proportionality (k) in Rs/GB?

100

200

400

The Correct Answer Is:

The cost (y) is Rs 200 and data (x) is 2 GB. k = y/x = 200/2 = 100. So, k is Rs 100 per GB.

Real World Connection

In the Real World

Many apps like food delivery (Swiggy, Zomato) or ride-sharing (Ola, Uber) use proportionality. The fare or delivery charge often depends directly on the distance or time. The constant of proportionality here would be the per-kilometer or per-minute rate they charge.

Key Vocabulary

Key Terms

PROPORTION: A statement that two ratios are equal. | DIRECT PROPORTION: A relationship where two quantities increase or decrease at the same rate. | RATIO: A comparison of two quantities by division. | CONSTANT: A value that does not change.

What's Next

What to Learn Next

Now that you understand the constant of proportionality, you're ready to explore inverse proportionality and its constant. This will show you how quantities relate when one increases while the other decreases, which is common in many real-world situations!