#math trivia for #February20: #51 is 33 base 16. Approximately what fraction is 33 base 16 of 100 base 16?

— Burt Kaliski Jr. (@modulomathy) February 20, 2012

33 is just under one third of 100 base 10 — but the question is asking about base 16.

There are two approaches to solve this one. First, we could convert both 33 and 100 base 16 to base 10, then divide. 33 base 16 is the day’s number, 51 base 10; 100 base 16 is 256 base 10 (16*16). 51 is just under one fifth of 256 (= 5*51+1), so the answer is one fifth.

The other way is to look at the digits. 100 base 16 is one more than FF base 16 (the F represents the digit with decimal value 15 in usual hexadecimal notation with symbols 0-9 and A-F). A 3 is one fifth of an F, so 33 is one fifth of FF and just under one fifth of 100 base 16. (Just as a 3 is one third of a 9 in the base-10 version.)

Conclusion: The ratio of 100 to 33 depends on the base — 33 is a third in one case, a fifth in another, or even a half if we were working in base 7!

See also the solution to #46 for more about bases.