What is Joint Variation (y = kxz)? | Simple Explanation for Class 7

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What is the Joint Variation (y = kxz)?

Grade Level:

Class 7

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

Definition

What is it?

Joint Variation describes a situation where one quantity depends directly on two or more other quantities. If 'y' varies jointly as 'x' and 'z', it means 'y' is directly proportional to both 'x' and 'z' at the same time. This relationship is written as y = kxz, where 'k' is a constant of proportionality.

Simple Example

Quick Example

Imagine the total money you earn (y) from selling samosas. This money depends on how many samosas you sell (x) AND the price of each samosa (z). If you sell more samosas or sell them at a higher price, you earn more money. This is a joint variation because your earnings depend on both factors.

Worked Example

Step-by-Step

PROBLEM: The cost of making a certain type of fabric (C) varies jointly with its length (L) and its width (W). If a fabric of length 5 meters and width 2 meters costs Rs 400, find the cost of a fabric of length 8 meters and width 3 meters.

STEP 1: Write the joint variation equation. C = kLW
---STEP 2: Use the given values to find the constant 'k'. We know C = 400 when L = 5 and W = 2. So, 400 = k * 5 * 2.
---STEP 3: Solve for 'k'. 400 = 10k. Dividing both sides by 10 gives k = 40.
---STEP 4: Now we have the complete relationship: C = 40LW.
---STEP 5: Use this equation to find the new cost. We need to find C when L = 8 and W = 3. So, C = 40 * 8 * 3.
---STEP 6: Calculate the cost. C = 40 * 24. C = 960.
---ANSWER: The cost of the fabric of length 8 meters and width 3 meters is Rs 960.

Why It Matters

Understanding joint variation is crucial in fields like Physics to calculate forces or energy, and in Economics to model how prices depend on multiple factors. Data Scientists and Engineers use these relationships to build predictive models and design systems, helping them understand complex real-world problems.

Common Mistakes

MISTAKE: Writing y = kx + z instead of y = kxz. | CORRECTION: Remember that 'jointly' means multiplication between the varying quantities, not addition. The constant 'k' multiplies all the variables.

MISTAKE: Forgetting to find the constant 'k' first. | CORRECTION: Always use the initial set of values given in the problem to calculate 'k'. This 'k' is unique for that specific problem and must be found before solving for any unknown quantity.

MISTAKE: Confusing joint variation with direct variation. | CORRECTION: Direct variation involves one quantity varying directly with only ONE other quantity (y = kx). Joint variation involves one quantity varying directly with TWO OR MORE other quantities (y = kxz).

Practice Questions

Try It Yourself

QUESTION: The volume (V) of a rectangular box varies jointly as its length (L), width (W), and height (H). If a box with L=10 cm, W=5 cm, H=2 cm has a volume of 100 cubic cm, what is the constant of proportionality 'k'? | ANSWER: k = 1

QUESTION: The number of rotis (R) a cook can make varies jointly with the number of hours worked (H) and the number of cooks (C). If 2 cooks can make 120 rotis in 3 hours, how many rotis can 3 cooks make in 4 hours? | ANSWER: 240 rotis

QUESTION: The simple interest (I) earned on an investment varies jointly with the principal amount (P), the rate of interest (R), and the time in years (T). If Rs 5000 invested for 2 years at 5% interest earns Rs 500, how much interest will Rs 8000 earn in 3 years at 4% interest? (Hint: Convert percentage to decimal for R). | ANSWER: Rs 960

MCQ

Quick Quiz

If the strength of a signal (S) varies jointly with its power (P) and inversely with the square of the distance (D), which of the following equations represents this relationship?

S = kPD^2

S = kP/D^2

S = k(P + D^2)

S = kD^2/P

The Correct Answer Is:

Joint variation means multiplication (kP) and inverse variation means division (by D^2). So, S is proportional to P and inversely proportional to D^2, making the equation S = kP/D^2.

Real World Connection

In the Real World

In cricket analytics, a player's 'impact score' might jointly vary with the number of runs scored and the number of wickets taken. Or, the cost of an Uber/Ola ride often varies jointly with the distance traveled and the peak-hour multiplier, helping the app calculate your fare.

Key Vocabulary

Key Terms

JOINT VARIATION: A relationship where one quantity depends directly on two or more other quantities by multiplication. | CONSTANT OF PROPORTIONALITY (k): A fixed, non-zero number that relates the quantities in a variation. | DIRECTLY PROPORTIONAL: When two quantities increase or decrease together at a constant rate. | EQUATION: A mathematical statement that shows two expressions are equal.

What's Next

What to Learn Next

Great job understanding joint variation! Next, you can explore 'Combined Variation', which mixes joint and inverse variations. This will help you solve even more complex real-world problems where quantities depend on many factors in different ways.