1.3 Combined Variation
Combined variation is a mathematical relationship that involves a combination of direct variation (or joint variation) and inverse variation. In everyday situations and scientific applications, a quantity often depends directly on one or more variables while simultaneously depending inversely on other variables. Understanding combined variation allows us to connect multiple variables into a single algebraic relationship and solve practical real-world problems.
Learning Objectives
- Determine the relation between three or more variables for a combined variation.
- Express a combined variation in the form of a variation relation using the symbol \(\propto\) and in the form of an algebraic equation with a constant of variation, \(k\).
- Solve problems involving combined variation using systematic mathematical procedures.
Structure of a Combined Variation Equation
See how directly varying quantities and inversely varying quantities are placed in the numerator and denominator of the equation.
Brief Explanation
What is Combined Variation?
Combined variation involves a combination of direct variation or joint variation, and inverse variation. If a variable \(y\) varies directly as \(x\) and varies inversely as \(z\), then \(y\) varies directly as \(x\) and inversely as \(z\), written as:
General Representation
For a combined variation where \(y\) varies directly as \(x^m\) and inversely as \(z^n\):
where \(m = 1, 2, 3, \frac{1}{2}, \frac{1}{3}\), \(n = 1, 2, 3, \frac{1}{2}, \frac{1}{3}\), and \(k\) is a constant (\(k \neq 0\)).
Smart Tips on Combined Variation Relations
If \(y\) varies directly as \(w\) and the cube of \(x\), and inversely as the cube root of \(z\):
If \(y\) varies directly as the product of \(w^2\) and \(x\), and inversely as \(z^3\):
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Write in the form of variation and equation: \(w\) varies directly as the cube root of \(v\) and inversely as the square of \(x\).
It is given that \(y\) varies directly as \(x^2\) and inversely as \(\sqrt{z}\). If \(y = 8\) when \(x = 4\) and \(z = 36\), find the value of \(k\) and express \(y\) in terms of \(x\) and \(z\).
The pressure, \(p\text{ N m}^{-2}\), on a tyre varies directly as mass, \(m\text{ kg}\), and inversely as contact surface area, \(l\text{ m}^2\). If \(m\) is fixed, what happens to the pressure \(p\) when the contact area \(l\) increases?
Common Mistakes
- Placing inversely proportional variables in the numerator alongside directly proportional variables instead of in the denominator.
- Forgetting to apply powers or roots, for example writing \(z\) instead of \(\sqrt{z}\) or \(x\) instead of \(x^2\).
- Forgetting to write the constant \(k\) when converting from variation relation (\(\propto\)) to equation form (\(=\)).
- Incorrect algebraic manipulation when solving for a variable located in the denominator or under a radical.
Misconceptions
Combined variation is identical to joint variation.
Joint variation involves direct variation of two or more variables only, for example \(y \propto xz\). Combined variation must include a combination of direct/joint variation and inverse variation, for example \(y \propto \frac{x}{z}\).
The constant \(k\) changes when the values of the variables change.
The value of \(k\) is fixed for a given relationship; it remains constant across all sets of corresponding values.
Summary
- Combined variation is the combination of direct variation (or joint variation) and inverse variation.
- General algebraic form: \(y = \frac{k x^m}{z^n}\), where \(k\) is a constant.
- Directly proportional variables are placed in the numerator, and inversely proportional variables are placed in the denominator.
- Problems are solved by identifying the relationship, determining the constant \(k\) from known data, and substituting the target values into the resulting equation.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which of the following represents the statement: “\(F\) varies directly as \(G\) and \(H^3\), and inversely as \(\sqrt{t}\)”?
2 It is given that \(P\) varies directly as \(Q^3\) and inversely as \(R\). When \(Q = 2\) and \(R = 0.6\), the value of \(P\) is \(4\). What is the value of the constant of variation, \(k\)?
B. Fill in the Blanks
3 Combined variation involves a combination of direct variation or joint variation, and variation.
4 If the acceleration of an object, \(A\), varies directly as distance travelled, \(s\), and inversely as the square of time taken, \(t\), the equation relating them with constant \(k\) is \(A =\) .
C. Drag and Drop: Match Each Statement to Its Variation Relation
Drag each variation statement to the correct relation. On a phone, tap the answer choice first, then tap the matching box.
