Determining which number pair has greater product - without multiplying.

Question

This is a 5x5 multiplication table.

    1   2   3   4   5
  ===================
1 | 1   2   3   4   5
2 | 2   4   6   8   10
3 | 3   6   9   12  15
4 | 4   8   12  16  20
5 | 5   10  15  20  25

Ordering all unique products by magnitude, you get:

5*5, 5*4, 4*4, 5*3, 4*3, 5*2, 3*3, 4*2, 3*2, 5*1, 4*1, 3*1, 2*1, 1*1
 25,  20,  16,  15,  12,  10,   9,   8,   6,   5,   4,   3,   2,   1

Given any two product pairs, could you predict which would have the greater product without performing the multiplication?

(3,2) or (5,1) ?

My first guess, whichever pair has a greater sum, is proven incorrect by the above example. Is there another test that can be done?

Answer 1

You are asking to compare $ab$ with $cd$. You can ask whether $\frac ac$ is greater or less than $\frac db$, which lets you divide instead of multiply. You can also compare $\log a + \log b$ with $\log c + \log d$, which is easy if you have log tables around.