**False Analogy**:

Scientific discoveries are sometimes made by using analogy. It certain features of two objects are similar, perhaps other features are also similar. Analogy, however, is only a tool for good guesses. The guesses have to be tested.

Analogy has its place in math also, but alas, so does false analogy:

“By how much is 40 larger than 32?”

“By 8.”

“By how much is 32 smaller than 40?”

“By 8.”

“By what percentage is 40 larger than 32?”

“By 25 percent.”

“By what percentage is 32 smaller than 40?”

“By 25 percent.”

But, it is only 20 percent smaller.

(A) Suppose your monthly income increases 30 percent. By what percentage does your purchasing power increase?

(B) Suppose that your monthly income does not change, but prices go down 30 percent. By what percentage does your purchasing power increase?

(C) When a secondhand bookstore holds a sale with a 10 percent reduction in prices, it makes an 8 percent profit on each book sold. What was the profit before the sale?

(D) If a metal worker reduces the time per part by p percent, how much does he increase his productivity?

Cheers,

Nalin Pithwa

Reference:

The Moscow Puzzles by Boris A. Kordemsky, Dover Publications.

PS: One of the best quote about analogies in math was made by Stefan Banach, a famous Polish mathematician, with whom Stanislaw Ulam had collaborated. The quote is as follows:

One can imagine that the ultimate mathematician is one who can see analogies between analogies. A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories.

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