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.”