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Mathematics · Class 10 · Chapter 18: Variations

8 min

Key points

Direct Variation: When one quantity increases, the other quantity also increases in the same ratio
Inverse Variation: When one quantity increases, the other quantity decreases in the same ratio
Joint Variation: When a quantity varies directly as the product of two or more quantities
Combined Variation: When a quantity varies directly with some quantities and inversely with others
Constant of Variation (k): The constant value that relates the varying quantities
Graph of direct variation is a straight line passing through origin
Graph of inverse variation is a rectangular hyperbola
In direct variation, y = kx where k is constant of variation
In inverse variation, xy = k or y = k/x where k is constant
Always find the constant first, then solve for unknown values

Definitions

Direct Variation: Two quantities are said to be in direct variation if when one quantity increases the other quantity also increases in the same ratio, or when one quantity decreases the other quantity also decreases in the same ratio.
Inverse Variation: Two quantities are said to be in inverse variation if when one quantity increases the other quantity decreases in the same ratio, or when one quantity decreases the other quantity increases in the same ratio.
Joint Variation: A quantity is said to vary jointly as two or more quantities when it varies directly as their product.
Constant of Variation: The constant value k which relates the varying quantities is called the constant of variation or constant of proportionality.

Worked example

Define variation and explain the difference between continuous and discontinuous variations with examples. (6 marks)

Variation is the differences that exist between individuals of the same species. Continuous variation shows a range of phenotypes between two extremes with no distinct categories, controlled by multiple genes and greatly influenced by environment. Example: human height ranges from very short to very tall. Discontinuous variation shows distinct categories with no intermediates, controlled by one or few genes with little environmental influence. Example: ABO blood groups show only A, B, AB, O types with no intermediates.

Common mistakes

Students confuse direct and inverse variation - remember direct means both increase/decrease together
Forgetting to find the constant of variation first before solving for unknown values
Writing wrong mathematical forms - direct is y = kx, inverse is y = k/x
Not showing step-by-step calculation of constant k in exam answers
Mixing up joint variation (multiply) with combined variation (both direct and inverse)

Quick recap

Variation means differences between individuals of same species. Two main types: continuous (range of phenotypes, multiple genes, environment important) like height and weight; discontinuous (distinct categories, few genes, less environmental influence) like blood groups and eye color. Variations can be inherited (genetic) or acquired (environmental). Environmental factors modify genetic potential especially in continuous traits.

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