What is Combined Variation?

S3-SA1-0218

Grade Level:

Class 8

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, and some of these relationships are direct variations while others are inverse variations. It's like mixing different types of dependencies into one problem.

Simple Example

Quick Example

Imagine the number of samosas you can make. It directly varies with the amount of dough you have (more dough, more samosas) and inversely varies with the time each samosa takes to make (more time per samosa, fewer samosas in total). This is a combined variation.

Worked Example

Step-by-Step

PROBLEM: The cost (C) of painting a wall varies directly with its area (A) and inversely with the number of painters (N). If it costs Rs 1200 to paint a 60 sq meter wall with 3 painters, how much will it cost to paint a 90 sq meter wall with 4 painters?

STEP 1: Write the combined variation equation. C = k * (A/N), where 'k' is the constant of variation.
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STEP 2: Use the given values to find 'k'. 1200 = k * (60/3).
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STEP 3: Simplify and solve for 'k'. 1200 = k * 20 => k = 1200 / 20 => k = 60.
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STEP 4: Now we have the full equation: C = 60 * (A/N).
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STEP 5: Use this equation to find the new cost. C = 60 * (90/4).
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STEP 6: Calculate the new cost. C = 60 * 22.5 => C = 1350.
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ANSWER: It will cost Rs 1350 to paint a 90 sq meter wall with 4 painters.

Why It Matters

Understanding combined variation is super important for jobs in Data Science and Engineering, where you predict outcomes based on multiple factors. For example, AI/ML models use similar logic to understand how different inputs affect an output, like predicting cricket match scores or stock prices.

Common Mistakes

MISTAKE: Confusing direct and inverse relationships in the equation. | CORRECTION: Remember, direct variation means the quantity is in the numerator, and inverse variation means it's in the denominator.

MISTAKE: Forgetting to find the constant of variation (k) first. | CORRECTION: Always use the initial set of values to calculate 'k' before solving for the unknown in the second scenario.

MISTAKE: Incorrectly performing algebraic operations when solving for 'k' or the final answer. | CORRECTION: Double-check your multiplication and division steps, especially when dealing with fractions.

Practice Questions

Try It Yourself

QUESTION: The speed (S) of a car varies directly with the distance (D) it travels and inversely with the time (T) taken. If a car travels 150 km in 3 hours at a speed of 50 km/h, what is the constant of variation? | ANSWER: k = 1

QUESTION: The number of rotis (R) a cook can make varies directly with the amount of flour (F) available and inversely with the number of rotis per minute (M) the cook makes. If a cook makes 20 rotis with 500g flour at a rate of 2 rotis/minute, how many rotis can they make with 750g flour at a rate of 3 rotis/minute? | ANSWER: 30 rotis

QUESTION: The pressure (P) of a gas varies directly with its temperature (T) and inversely with its volume (V). If a gas has a pressure of 10 Pascals at 300 Kelvin and a volume of 5 liters, what will be its pressure if the temperature is 360 Kelvin and the volume is 6 liters? | ANSWER: 12 Pascals

MCQ

Quick Quiz

Which of the following equations represents combined variation where 'y' varies directly with 'x' and inversely with the square of 'z'?

y = kxz^2

y = kx/z^2

y = kx^2/z

y = kx + z^2

The Correct Answer Is:

Direct variation means the variable is in the numerator, so 'x' is on top. Inverse variation means the variable is in the denominator, so 'z^2' is at the bottom. Option B correctly shows y = k * (x / z^2).

Real World Connection

In the Real World

Combined variation is used in logistics companies like Delhivery or Zepto to calculate delivery times. The time taken varies directly with distance and number of packages, and inversely with the number of delivery riders available. This helps them optimize routes and delivery schedules.

Key Vocabulary

Key Terms

DIRECT VARIATION: When one quantity increases, the other also increases proportionally. | INVERSE VARIATION: When one quantity increases, the other decreases proportionally. | CONSTANT OF VARIATION (k): The constant value that relates two or more varying quantities. | QUANTITY: A measurable amount or number.

What's Next

What to Learn Next

Once you've mastered combined variation, you're ready to explore more complex algebraic relationships and functions. This will help you understand graphs and how real-world data can be modeled, opening doors to advanced topics in science and technology!