Warp and Portion

Portion reached for her method first. "Count the shares," she said. "Three parts and five parts — eight shares in all. One share is the whole over eight — forty over eight — five yards. So one house gets three shares, fifteen yards, and the other gets five shares, twenty-five." She measured, and it was correct, and it summed back to forty. But a student watching asked why the eight, and Portion found she could only say "because you add the parts." She knew the step, but not the reason the step was allowed.

Warp, watching, saw the reason plainly, because it was her whole craft. "The eight is the whole," she said. "The ratio three to five is part-to-part — it does not yet mention the whole at all. But the cloth is a whole, and to take a share of a whole you first have to name the whole. Add the parts — three and five — and you have named it: eight. Now each part becomes a fraction of the whole. The first house is three-eighths of the cloth; the second is five-eighths. Portion's five-yard share is just one-eighth of forty, wearing a different hat."

Portion stared. "So my counting of shares and your fractions of the whole are the same move."

"They are the same move seen from two sides," Warp said. "You divide by eight because I have already turned the ratio into eighths. I could name the eighths all day, but I could never cut the cloth. And you could cut it, but you were cutting on faith."

They taught it together after that, and it became the academy's favourite lesson on why the two skills are one. Warp goes first now, always. "Never take a share of a whole until you have named the whole," she tells the children. "A part-to-part ratio hides the whole. Add the parts to reveal it. Three to five hides an eight." Then Portion steps in with the cloth. "Now that the whole is named, taking the shares is easy — three-eighths of forty is fifteen, five-eighths is twenty-five, and they sum back to forty, which is how you know you were fair."

The Advanced point they drive home is the sequence. "You cannot share in a ratio directly," Warp says. "You must pass through the whole. Ratio, to fractions-of-the-whole, to shares of the quantity. Skip the middle step — try to take a fifth of the cloth because the ratio said five — and you will hand out too much and never know why."