Inverse proportion: why doubling gives half

You must process 120 pages. At 20 pages per hour this takes 6 hours; at 40, it takes 3. Why does doubling productivity halve the time? The work stays fixed: productivity multiplied by time does not change.

Use rational expressions and functions. This is hypothetical: constant pace, equally difficult pages, no preparation, breaks or rework.

A constant product first, a fraction second

Let $p>0$ be numerical productivity in pages/hour and $t>0$ numerical time in hours. From $pt=120$ :

t=\frac{120}{p}.

The 120 represents pages: $(\text{pages}/\text{hour})\cdot\text{hour}=\text{pages}$ . At $p=30$ , $t=4$ hours. Tripling productivity gives one third of the time while the same assumptions hold.

The general rule of inverse proportion is $y=k/x$ for a nonzero constant $k$ . Its equivalent form $xy=k$ reveals what stays fixed. This says more than “one increases while the other decreases.” For instance, $y=10-x$ decreases too, but $xy$ is not constant. Inverse variation, OpenStax §7.5.

One area, different shapes

A rectangle of area 12 cm² has numerical side lengths in centimeters satisfying $xy=12$ , hence $y=12/x$ . Each pair $(2,6)$ , $(3,4)$ and $(6,2)$ gives the same area. Double one side and the other must halve.

One area: 12 cm²

Positive branch y=12/x

Rectangles use the same schematic scale. The graph shows part of the positive branch, continuing beyond the frame. Points (2,6), (3,4), (6,2); both axes use uniform scales.

Rectangles model positive lengths only. In pure algebra, $y=12/x$ also allows negative $x$ , with negative $y$ . The full graph is a hyperbola with two branches. For $k>0$ , they occupy quadrants I and III; for $k<0$ , II and IV.

Why the axes describe approaching behavior

$x=0$ is excluded by the denominator. For positive $x$ approaching zero, $12/x$ grows without bound: at 0.1 it is 120, at 0.01 it is 1200. As $x$ grows, the output approaches zero: at 120 it is 0.1, at 1200 it is 0.01.

Thus $x=0$ and $y=0$ are this graph’s asymptotes: they describe approaching behavior, not attainable zero values. For finite nonzero $x$ , $12/x$ never equals zero. This is a property of this particular function; other graphs can cross horizontal asymptotes. Hyperbolas and asymptotes, OpenStax §5.6.

Two meanings of “inverse”

Inverse proportion describes a constant product. An inverse function recovers the original input from the output. For $f(x)=2x$ , the reverse rule is $f^{-1}(y)=y/2$ , not $1/(2y)$ . Here $f^{-1}$ does not mean the reciprocal $1/f$ .

The time model does not promise zero hours in actual work either. Add a fixed 1-hour preparation and total time becomes $T=1+120/p$ ; now $pT=p+120$ is not constant. Before applying “double means half,” identify what remains fixed and whether any term does not scale.