What is Combined Variation? | Simple Explanation for Class 7

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What is the Combined Variation?

Grade Level:

Class 7

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

Definition

What is it?

Combined Variation happens when one quantity depends on two or more other quantities, where some are directly proportional and others are inversely proportional. It's like mixing both direct and inverse variations in one problem. So, if A varies directly with B and inversely with C, that's a combined variation.

Simple Example

Quick Example

Imagine the time it takes to deliver food. It varies directly with the distance to be covered (more distance, more time) and inversely with the number of delivery riders working (more riders, less time). This is a combined variation problem.

Worked Example

Step-by-Step

Let's say the cost of painting a wall (C) varies directly with its area (A) and inversely with the number of painters (P). If it costs Rs. 1200 to paint a 60 sq meter wall with 2 painters, how much will it cost to paint a 90 sq meter wall with 3 painters?

Step 1: Write the combined variation formula. C = k * (A/P), where k is the constant of proportionality.
---Step 2: Use the given values to find k. 1200 = k * (60/2).
---Step 3: Simplify and solve for k. 1200 = k * 30. So, k = 1200 / 30 = 40.
---Step 4: Now that we have k = 40, write the specific formula for this problem: C = 40 * (A/P).
---Step 5: Use the new values (A = 90, P = 3) to find the new cost. C = 40 * (90/3).
---Step 6: Calculate the new cost. C = 40 * 30 = 1200.
---Answer: It will cost Rs. 1200 to paint a 90 sq meter wall with 3 painters.

Why It Matters

Understanding combined variation helps scientists predict outcomes, engineers design efficient systems, and economists model market behaviour. From designing rockets at ISRO to predicting stock prices, this concept is a fundamental tool for problem-solving in many exciting careers.

Common Mistakes

MISTAKE: Mixing up direct and inverse relationships, for example, treating an inverse variation as direct. | CORRECTION: Always clearly identify which quantities are directly related (numerator) and which are inversely related (denominator) in the variation equation.

MISTAKE: Forgetting to find the constant of proportionality (k) first. | CORRECTION: The first step in solving any variation problem is to use the given initial conditions to calculate the value of 'k'. Without 'k', you cannot solve for new values.

MISTAKE: Incorrectly setting up the equation, like writing A = k * B * C when A varies directly with B and inversely with C. | CORRECTION: Remember, direct variations go in the numerator, and inverse variations go in the denominator. The correct form would be A = k * (B/C).

Practice Questions

Try It Yourself

QUESTION: The number of samosas made (S) varies directly with the amount of flour (F) and inversely with the time taken (T). If 5 kg of flour makes 100 samosas in 2 hours, how many samosas can be made with 7 kg of flour in 1 hour? | ANSWER: 280 samosas

QUESTION: The electricity consumption of a fan (E) varies directly with its speed (S) and the hours it runs (H). If a fan at speed 3 consumes 12 units in 4 hours, how many units will it consume at speed 4 for 6 hours? | ANSWER: 24 units

QUESTION: The cost of building a wall (C) varies directly with its length (L) and the square of its height (H^2), and inversely with the number of workers (W). If a 10-meter long, 2-meter high wall costs Rs. 8000 with 4 workers, what will be the cost of a 15-meter long, 3-meter high wall with 6 workers? | ANSWER: Rs. 18000

MCQ

Quick Quiz

If 'P' varies directly with 'Q' and inversely with the square of 'R', which of the following equations represents this relationship?

P = k * Q * R^2

P = k * (Q / R^2)

P = k * (R^2 / Q)

P = k * Q / R

The Correct Answer Is:

Direct variation means the quantity is in the numerator, and inverse variation means it's in the denominator. So, 'Q' is in the numerator, and 'R^2' is in the denominator, making P = k * (Q / R^2) the correct form.

Real World Connection

In the Real World

Delivery apps like Zomato or Swiggy use combined variation to estimate delivery times. The time depends directly on the distance to the customer and the number of orders in the queue, but inversely on the number of available delivery partners. This helps them give you accurate estimates and manage their fleet efficiently.

Key Vocabulary

Key Terms

DIRECT VARIATION: When one quantity increases, the other increases proportionally. | INVERSE VARIATION: When one quantity increases, the other decreases proportionally. | CONSTANT OF PROPORTIONALITY (k): A fixed, non-zero number that relates two or more quantities in a variation. | QUANTITY: A measurable property, like distance, time, or cost.

What's Next

What to Learn Next

Great job understanding Combined Variation! Next, you can explore 'Joint Variation,' which is a special type of direct variation involving multiple quantities. This will further strengthen your understanding of how different factors influence each other in real-world situations.