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.
See how directly varying quantities and inversely varying quantities are placed in the numerator and denominator of the equation.
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:
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 (\(k \neq 0\)).
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 \(z^3\):
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?
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.
Complete all parts first, then press the check answer button.
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, \(k\)?
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 \(A =\) .
Drag each variation statement to the correct relation. On a phone, tap the answer choice first, then tap the matching box.
There is something wrong with this question.
