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MathidiaMathematics Academy
Ratio, variation and rates
Lesson 31 of 75
Introduction

Combined variation

Objective. Model quantities directly and inversely proportional to several variables.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Combined variation describes a quantity depending on several other quantities at once, some directly and some inversely proportional. Write the relationship using a single constant of proportionality k, such as y = kx/z² when y varies directly with x and inversely with the square of z, then use given values to find k before predicting new outcomes. For instance, if y = 20 when x = 5 and z = 2, then k = 16, giving y = 16x/z².

Worked example

The combined variation p varies directly with q and inversely with r² is given by p = kq/r². When q = 4 and r = 2, p = 10. Find p when q = 9 and r = 3.

4 … 2
Illustrated card showing 4 … 2
  1. 1Substitute known values: 10 = k(4)/2² = 4k/4 = k, so k = 10
  2. 2Write the formula: p = 10q/r²
  3. 3Substitute q = 9 and r = 3: p = 10(9)/3² = 90/9
  4. 4Simplify the fraction: p = 10

Answer: p = 10

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Common mistake. Students often multiply by z instead of dividing when a quantity varies inversely with z, muddling direct and inverse parts of the formula. Always write y ∝ x for direct and y ∝ 1/z for inverse separately before combining them into one constant equation.

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