In honour of @teakayb’s birthday this week, here’s a post with a vaguely Douglas-Adams-related theme.

The student looked at the Mathematical Ninja and decided this was a moment where reaching for the calculator would be appropriate.

”$\frac{29}{42}$…” she said aloud.

“0.69,” said the Mathematical Ninja.

She threw the calculator down in disgust. “How do you DO that?” she asked, an elementary error: now he’d just go and explain.

“It’s simple enough,” said the Mathematical Ninja. “$\frac{30}{42}$ is five-sevenths, which is 0.714.”

“And you just know that?! Of course, you just know that.”

“Well yes, I just know that. If I had to work it out, I’d multiply top and bottom by 14 to get $\frac{70}{98}$ - and 0.7, plus 2%, is 0.714.”

The student rolled her eyes. “Oh, wait - wasn’t there something about sevenths being in a pattern? Like 14, 28, 56 or something?”

“That’s good! The pattern is 142857, and you just need to find the start number. The 5th biggest number in there is 7, so $\frac{5}{7}$ is actually $0.\dot{7}1428 \dot{5}$.”

“And from there, I suppose you know what $\frac{1}{42}$ is and you take it away?”

The Mathematical Ninja nodded. “But of course! It’s 0.024, give or take. So, to two decimal places, $\frac{29}{42} = 0.69$.”

“So, $\frac{13}{42}$ would be a shade over… two-sevenths, which is 0.2857… so 0.31?”

The Mathematical Ninja nodded. “You have learned well.”

“Now, why would you ever need to divide by 42 in your head?” asked the student.

The Mathematical Ninja looked crestfallen ((One possible scenario: you want to know what percentage of a marathon you’ve run.)).

* Edited 2021-03-08 to fix formatting.