OK, following up on our previous post, we have another fraction question. This one is not new, has appeared in the comments of a previous post, and many readers will have heard us bang on about it. Nonetheless, given the discussion on the previous question, and given the possibility that some new readers might not have yet read Marty’s Collected Sermons, it seems worthwhile giving the question its own post. Here it is:
That is, are the two quantities equal or unequal? Loyal readers of the Sermons might hold off commenting for a little while, to allow others time to ponder.
Thanks to everyone who has commented. The answer is that the two quantities are equal, as best explained by Sir Humphrey:* since 2/6 equals 1/3, whatever quantity we make from 2/6 will equal the corresponding quantity made from 1/3. That’s it. That’s all there is.
The trouble is, plenty of teachers and students treat “index laws” as if they are commandments from God, and trick themselves into all manner of inequality. This question was born, a decade ago, with a textbook that got it wrong.
*) SIr Humphrey’s perfectly blunt response reminded us of the greatest line of dialogue from the great writer Ring Lardner:
“Shut up”, he explained.