inversely proportional
Summary
Two quantities are inversely proportional when their product stays constant, so multiplying one by a factor divides the other by that same factor. The MCAT tests this by scaling one variable and asking for the other, such as doubling pressure at constant temperature to halve gas volume.
Detail
Meaning: If y is inversely proportional to x, then y = k/x, where k is a constant while the other conditions are held fixed. Doubling x halves y, tripling x cuts y to one third, and halving x doubles y. A graph of y against x is a hyperbola, while y against 1/x is a straight line through the origin. A related case is inverse-square dependence, y = k/x², where doubling x cuts y to one quarter. Light intensity versus distance from a point source is an example.
Equation: y = k/x, so xy = k and x₁y₁ = x₂y₂. Here x is the independent variable, y is the dependent variable, and k is the constant of proportionality, with SI units of y multiplied by units of x.
On the MCAT: Expect a scenario in which one variable changes by a stated factor and you must find the new value of the other without a calculator. Typical cases are Boyle's law (P₁V₁ = P₂V₂ at constant n and T), λ = v/f in a given medium, a = F/m, and I = V/R at fixed voltage. You may also be asked to recognize a hyperbola or to pick the axis that linearizes the data.
Common confusion: A variable that falls as another rises is not automatically inversely proportional. A linear relation such as y = 10 − x decreases, but doubling x does not halve y. The test is whether the product xy stays constant.
Sources
- Physics (Giancoli)
- College Physics (Serway and Vuille)
- Physics for Scientists and Engineers (Knight)
Written for the MCAT and checked before it was published. For study reference only. Spotted an error? Let us know.