Learning Objectives
- explain joint variation;
- translate joint-variation statements into equations;
- find the constant k;
- solve problems involving three or more quantities;
- recognise powers in joint variation.
Introduction
This lesson develops Joint Variation from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Visual Exploration
Use the diagrams, tables or examples in this lesson to identify what changes, what stays fixed and which relationship connects the values.
What Is Joint Variation?
A quantity varies jointly when it is directly proportional to the product of two or more other quantities.
Visualising the Relationship
If x doubles while y stays fixed, z doubles. If both x and y double, z becomes four times as large.
Worked Example 1
z varies jointly as x and y. When x = 2, y = 5 and z = 30, find z when x = 4 and y = 3.
Write z = kxy.
30 = k(2)(5), so k = 3.
z = 3xy.
z = 3(4)(3) = 36.
Joint Variation with Powers
A quantity may vary jointly as one variable and the square or cube of another.
P varies jointly as a and b². If P = 72 when a = 2 and b = 3, find k.
P = kab².
72 = k(2)(3²) = 18k.
k = 4.
🔍 How Do I Recognise Joint Variation?
- “varies jointly as x and y”
- “is proportional to the product of...”
- several variables multiplied together;
- phrases involving area, volume or combined factors.
⚠️ Common Mistakes
✅ Quick Check
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Write equations for:
- q varies jointly as r and s;
- V varies jointly as l, w and h;
- P varies jointly as x and y².
- z = 5xy. Find z when x = 4 and y = 6.
- z varies jointly as x and y. If z = 48 when x = 3 and y = 4, find z when x = 5 and y = 2.
- P varies jointly as a and b². If P = 100 when a = 4 and b = 5, find P when a = 6 and b = 2.
- Q varies jointly as x² and y. When x = 2 and y = 3, Q = 72. Find y when Q = 300 and x = 5.
✅ Check Your Work
Show practice answers
Foundation
-
- q = krs
- V = klwh
- P = kxy²
- 120
Developing
-
40.
48 = k(3)(4), so k = 4.
z = 4(5)(2) = 40. -
24.
100 = k(4)(25), so k = 1.
P = 1(6)(2²) = 24.
Challenge
-
y = 2.
Q = kx²y
72 = k(4)(3), so k = 6.
300 = 6(25)y
y = 2.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Joint variation involves a product of variables.
- Translate z ∝ xy into z = kxy.
- Powers must be copied carefully.
- Find k before calculating new values.