How The Mathematical Ninja Divides By 49

“… which works out to be $\frac{13}{49}$,” said the student, carefully avoiding any calculator use.

“Which is $0.265306122…$”, said the Mathematical Ninja, with the briefest of pauses after the 5.

“I presume you could go on?”

“$…448979591…$”

“All right, all right, all right. I suppose you’re going to tell me the trick?”

“But of course! It’s one I picked up from Vedic Mathematics. Dreadful book. Take the top, as two digits, and double it.¹”

“$26$.”

“That’s the first two decimal places. Double it again - and (here’s the tricky bit) - if it’s more than $50$, add one.”

“$53$?”

“Yep! Then double it again - same rules, except you ignore any hundreds as you double.”

“Is that because the adding one thing was a carry?”

“Precisely!”

“So… can I try a different one?”

“Please do. How about $\frac{34}{49}$?

“OK. That’ll be $0. 68…$ wait, no, add one… $0.6938775510204081…$ I could go on!”

“You most certainly could! It always repeats after $42$ digits, because $49$ is a factor of $10^{42} - 1$.”

“I’ll take your word for it. Why does the trick work?”

“Oh! It’s because you can express $\frac{1}{49}$ as a geometric series with $a = \frac{1}{50}$ and $r = \frac{1}{50}$. Each next term is two-hundredths of the previous one, so you double what you’ve got and move it two places to the right.”

“And then if you get something over $50$, you know you’ll be carrying next time, so you can add one to save you time. Nice trick.”

“Thank you,” said the Mathematical Ninja. “I have others.”