Coke Competition Math

Long time no post!

I recently had a Coke Zero and noticed that they had a competition on (https://www.coke.co.nz/cash) :

Check under specially marked labels on 'Coca-Cola', 'Coca-Cola Zero' and 'Diet Coca-Cola' 600mL bottles to see if you are a winner of $20, $50 or $100.

There is $200k in prizes to be won, so I asked myself: how many of each price is there?

x = number of $20 prizes
y = number of $50 prizes
z = number of $100 prizes
so 20x + 50y + 100z = 200,000
update: I forgot to mention that there are 4,600 prizes, so x + y + z = 4600

With some headscratching I rejigged the problem to:

 x = 5n
y = 8(650-n)
z = 3(n-200)
200 <= n &<= 650 if we want positive results

I then explored that range with excel to see what my best guess is for the distribution of cash:

$20	$50	$100	n
1000	3600	0	200
2000	2000	600	400	Min where x <= y <= z
2005	1992	603	401	Min where x < y < z
2375	1400	825	475	Nearly equal ratios (~1.7/1)
2500	1200	900	500	Nice round numbers
2635	984	981	527	Max where x > y > z

So there we have it, my best guess is the round numbers (though even ratios is nice):

2500 x $20 + 1200 x $50 + 900 x $100 = $200k

update: I found on https://www.coke.co.nz/cash/conditions-of-entry the answer (n = 300), which wasn't on the label:

 The instant win prizes available to be won (the Cash Prizes) are:

• (a) 1,500 x cash prizes valued at $20 each;
• (b) 2,800 x cash prizes valued at $50 each; and
• (c) 300 x cash prizes valued at $100 each.

Surprising that there is more $50s than $20s.