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Combined Variation — Definition, Formula & Examples

Combined variation is a relationship where one variable depends on two or more other variables through a mix of direct and inverse variation. For example, yy might vary directly with xx and inversely with zz at the same time.

A variable yy exhibits combined variation if it can be expressed as y=k⋅x1a1x2a2⋯w1b1w2b2⋯y = k \cdot \dfrac{x_1^{a_1} x_2^{a_2} \cdots}{w_1^{b_1} w_2^{b_2} \cdots}, where kk is a nonzero constant of proportionality, the numerator variables represent direct (or joint) variation, and the denominator variables represent inverse variation.

Key Formula

y=k⋅xzy = k \cdot \frac{x}{z}
Where:
  • yy = Dependent variable
  • kk = Constant of variation (nonzero)
  • xx = Variable that y varies directly with
  • zz = Variable that y varies inversely with

How It Works

Start by translating the verbal description into an equation. Variables that yy varies directly with go in the numerator; variables that yy varies inversely with go in the denominator. Multiply by a constant kk. Then substitute a known set of values to solve for kk. Once you know kk, use the equation to find unknown values.

Worked Example

Problem: Suppose y varies directly with x and inversely with z. If y = 12 when x = 8 and z = 2, find y when x = 5 and z = 4.
Write the equation: Because y varies directly with x and inversely with z:
y=k⋅xzy = k \cdot \frac{x}{z}
Solve for k: Substitute y = 12, x = 8, z = 2:
12=k⋅82=4k  ⟹  k=312 = k \cdot \frac{8}{2} = 4k \implies k = 3
Find the new y: Substitute k = 3, x = 5, z = 4:
y=3⋅54=154=3.75y = 3 \cdot \frac{5}{4} = \frac{15}{4} = 3.75
Answer: y=3.75y = 3.75

Why It Matters

Many real-world formulas are combined variations. The ideal gas law PV=nRTPV = nRT combines direct and inverse relationships among pressure, volume, and temperature. Recognizing combined variation lets you set up these equations quickly in physics and chemistry courses.

Common Mistakes

Mistake: Placing an inversely related variable in the numerator instead of the denominator.
Correction: "Varies inversely with z" means z belongs in the denominator. Re-read the problem carefully: direct → numerator, inverse → denominator.

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