?/24>1/4>4/?

Question

What makes this a true statement ?/24 > 1/4 > 4/?

Answer 1

Split this into 2 inequalities: (1) ?/24>1/4 and (2) 1/4>4/?

Multiply both sides by 4: (1) ?/6>1 so ?>6 (2) 1>16/?. ? is positive, because it has to be bigger than 6, so ?>16. 

If ? has to be the same in each case then, since 16>6, ? must be greater than 16.

Let’s check by putting ?=17:

17/24>¼>4/17. Multiply through by 408: 289>102>96 which is true.

Now we try ?=16 and we get ⅔>¼>¼ (8>3>3) which is false, as we would expect, because we haven’t satisfied (2).

So the answer is confirmed: ?>16.