How would you approach this problem in a timely manner regarding different way to multiply

Question

If you had no more than 1-2 minutes, how would you approach this Q:

We have four numbers (a, b, c, d), that will create the following pairs

a*b = 5

a*c = 8

a*d = 10

b*c = 16

b*d = 32

c*d = x

Find x

EDIT

Original question pic:

Answer 1

You can calculate $abcd$ in three ways from the given numbers:

$$abcd = ad\times bc = ac\times bd = ab\times cd $$

Every number is used exactly once in the above equations, so we must be able to find two pairs of numbers with the same product. In fact, we can do it in two ways:

$$5\times 16=10\times 8=32\times cd\\ 5\times 32=10\times 16=8\times cd$$

This gives us the two solutions $cd= 5/2$ and $cd=20$.

There seems to be no reason to prefer one solution more than the other. In fact, with some more time you can find out the possibilities for the four variables themselves to confirm that both solutions are valid. In each case there are two possibilities.

The set $\{2, 5/2, 4, 8\}$ gives products $\{5, 8, 10, 16, 20, 32\}$. An alternative set that gives the same products is $\{ \sqrt{10}/2, \sqrt{10}, 8\sqrt{10}/5, 2\sqrt{10} \}$.

The set $\{ 5/4, 2, 4, 8\}$ gives products $\{5/2, 5, 8, 10, 16, 32\}$. An alternative set that gives the same products is $\{ \sqrt{5}/2, \sqrt{5}, 2\sqrt{5}, 16\sqrt{5}/5 \}$.