Crux the Two-Directions

The second way he called trading off, and it was the one his well had taught him in his shoulders. Put twice as many workers on a ditch and it takes half the time. Double the speed of a journey and it takes half as long. Here, when one quantity doubles, the other halves — they move against each other, one rope down as the other climbs.

The Advanced trap, Crux warned, is to reach for the same method as before and scale both the same way. It comes out backwards. "In inverse proportion," he would say, "the ratio does not stay fixed — it changes with every pair. The thing that holds steady is the product. Six workers times four days is twenty-four worker-days of digging, and that ditch needs twenty-four worker-days no matter how you split them. Twelve workers? The product must still be twenty-four, so the time is two days. Three workers? Still twenty-four, so eight days. Multiply the two quantities and you always land on the same number. That number, k, is what the two are really guarding — not their ratio, their product. x y = k."

He drilled the distinction until the children could feel which was which. Direct proportion, graphed, is a straight line climbing from the corner: every step right is the same step up, because the ratio is fixed. Inverse proportion, graphed, is a curve that swoops down and flattens — a hyperbola — falling steeply then levelling, because as one quantity grows huge the other must shrink toward nothing to keep the product constant. "Two shapes," Crux said, tapping each. "A straight climb, and a falling curve. If you know which shape your problem is before you compute, you will never solve it backwards."

The hinge, he told them, was a single question — the crux, the point a whole problem turns on. "Ask it first, always. Do these two rise together, or do they trade off? Rise together: fix the ratio, use y = k x. Trade off: fix the product, use x y = k. Get the crux right and the arithmetic is easy. Get it wrong and every number after it is wrong, however careful your sums."

A boy named Fenn, who was quick with figures but had failed the same problem three times, stayed after one afternoon and turned the little model wheel Crux kept on his desk — one rope sinking as the other rose. His trouble, it turned out, was never the arithmetic. He had been treating a trade-off as a rise-together. He had doubled the workers and doubled the time, and been baffled that the ditch was never finished.

Crux sat beside him and let him work the wheel. "You did the sum perfectly," he said. "You just answered a different question than the one the ditch was asking. The workers and the days are on opposite ropes. When you pulled one up, you should have let the other down." Fenn ran it again — twelve workers, product still twenty-four, so two days — and the answer came out clean.