Rank

When Rank was small, comparing fractions made his head ache.

His family ran the village relay races, and every single race seemed to end in the same argument: whose runner had actually covered more of the track? One runner would have managed three-fifths of the way round. Another would have managed two-thirds. The children would shout numbers at one another across the field, and not a soul could say for certain who had truly won.

Little Rank tried to settle these disputes the thorough way — by converting everything, cutting the whole track into tiny matching pieces, fifteenths, and counting them all up one by one. It worked, every time. But it was painfully slow, and by the time he had finished his careful sums, half the crowd had wandered off home and stopped caring who'd won at all.

His grandfather, who judged the races, took him quietly aside one afternoon. "You don't always need the entire calculation," he said. "Look here. Is three-fifths more than half the track, or less than half?"

Rank thought it through. Half of five-fifths would be two-and-a-half fifths. Three-fifths was past that. "More than half."

"And two-thirds?"

Half of three thirds would be one-and-a-half thirds. Two-thirds was past that as well. "Also more than half." Rank frowned, disappointed. "But that doesn't settle which one's bigger."

"No," his grandfather agreed, unbothered, "but now you've only got two close ones left to think hard about, instead of a whole jumbled field of them. You've already thrown out all the easy cases in a single stroke. Half did most of the work for you, before you'd even properly started."

Rank stared out at the track. Half. The middle marker. All at once, comparing fractions stopped feeling like a wall of arithmetic he had to climb — and started feeling like sorting, which was a thing he found he rather enjoyed.

Rank came to the academy years later carrying a deck of fraction cards he could shuffle and order faster than anyone alive.

Slice, the old tortoise, fanned out four cards on the table to test him: *2/5, 4/4, 1/6, 7/8*. "Order them," he said. "Out loud. And tell me how you're doing it as you go."

Rank didn't so much as reach for a pencil. "Four-fourths is a whole one — a complete thing, nothing missing — so that's the biggest of the lot. One-sixth is a tiny sliver, way under half, so that's the smallest. Two-fifths is under half too, but it's a fair bit bigger than that little sliver. And seven-eighths is almost a whole, just barely short of four-fourths." He laid them out, left to right. "One-sixth, two-fifths, seven-eighths, four-fourths."

"And if two of them were both sitting close to half?" Slice pressed.

"Then I'd think harder, and earn it," Rank said. "Same top number on both? The one with the smaller bottom is the bigger — fewer pieces, but fatter ones. Same bottom number on both? The one with the bigger top wins — more pieces, all the same size. And if I'm genuinely stuck between two of them, I'll do it Stretch's way and give them both a common bottom. But most of the time, plain reasoning gets me there long before I'd need to."

Slice looked over the tidy row of cards. He had taught comparison as always convert first, then look for sixty years. He had never once watched a student sort an entire hand by simply thinking out loud. "You may stay," he said.

In his corner of the academy, Rank met a worried vole named Wisp, clutching two cards as though they might bite.

"I have to work out which of these is bigger," Wisp said, "three-fourths or two-thirds, and I get them muddled every single time."

"Middle marker first, always," Rank said. "Are they over half, or under half?"

Wisp thought hard. "Three-fourths... half of four is two, and three is past two, so — over half. Two-thirds... half of three is one-and-a-half, and two is past that, so — also over half." Wisp's shoulders sagged. "They're both over half. So now I'm stuck all over again."

"Good — that means you've earned the harder look," Rank said warmly. "Now think about it the other way round. Think about how far short of a whole each one falls. Three-fourths is missing only one-fourth to make a whole. Two-thirds is missing one-third to make a whole." He held two fingers up, a little apart. "So which gap is the bigger one — a fourth, or a third?"

Wisp pictured the two pieces side by side. "A third is bigger than a fourth. So two-thirds is missing the bigger chunk."

"And so which of them is closer to being a whole?"

"Three-fourths! It's only missing a little fourth, so it's nearly there. So three-fourths is the bigger one!" Wisp very nearly dropped both cards in excitement. "And I worked it out by thinking about what was missing, not what was there."

"That's the magpie's favourite trick," Rank grinned. "Sometimes the quickest way to see which fraction is bigger is to see which one has less missing from it."