The rule
result × (ratio of each quantity, inverted where the relationship is inverse)
Why it works
A compound proportion multiplies one ratio per quantity. For a direct relationship the ratio is after over before; for an inverse one it is before over after. Getting each direction right is the whole task — the arithmetic afterwards is trivial.
How to do it by hand
- Decide for each quantity whether it is direct or inverse
- Direct: divide after by before. Inverse: divide before by after
- Multiply all the ratios together
- Multiply the known value by that product
What is worth knowing
The classic worked example is workers, days and output, and it is a good test of whether you have the directions right: more workers means fewer days for the same job, so workers against days is inverse; more days means more output, so days against output is direct. Reason it out one quantity at a time rather than trying to see the whole thing at once. A quick sanity check at the end saves most errors: decide whether the answer should be bigger or smaller than the starting value, and see whether it is.