The Mishna in Shekalim writes that although a katan is exempt from the mitzvah of giving machtzis hashekel, if he voluntarily gives, his money is accepted. The Sha’ar haMelech asks: M’doraysa a katan has no power to be makneh anything to others. Therefore, the machtzis hashekel money which he gives is owned by hekdesh only m’derabbanan. Since korbanos tzibur must be bought from the public funds, funds that belong to the tzibur, how is the tzibur yotzei if they use this machtzis hashekel that (m’doraysa) still belongs to the katan?
The Sha’ar haMelech’s suggests that the katan’s machtzis hashekel is but a drop in the bucket compared with thousands of other shekalim donated and is therefore bateil b’rov.
Many Achronim since have debated how to make sense of this chiddush. The gemara in Beitzah (38) writes that if I contribute some ingredients to a Yom Tov dish you are cooking, you can’t move the dish outside the limits of my techum; my portion of the ingredients are not bateil. Just because most of the dish belongs to you doesn’t mean the entire dish belongs to you. So how is it that just because the majority of the shekalim funds used to buy korbanos belong to the tzibur, the percentage that may belong to a katan can be ignored? Aside from the gemara, it’s hard to understand how logically the sevara works. If I mix $10 of yours with a bunch of other bills in my wallet, does your $10 become mine because it is bateil b’rov? Of course not. Ownership cannot be bateil. What does the Sha’ar haMelech mean?
(Aside from the dinei mamonos issue, there is an issue that impacts issur v’heter here as well. Oneg Yom Tov discusses whether bitul b’rov helps in a case where strings for tzitzis not made lishma become mixed into a rov of those that were made lishma – does bitul mean that the miyut takes on the properties of the rov, or does bitul simply render a miyut as if it did not exist? It seems that the Sha’as haMelech comes down squarely on the side of saying that the miyut takes on the property of the rov, meaning the money of the katan is treated a tzibur money because that is the quality of the majority of funds.)
I think the easiest answer in a nutshell is that the difference between the gemara in Beitzah and this din in Shekalim is that the gemara is speaking about bitul b’rov – when a whole mixture is present, is a miyut counted apart from the whole? The Sha’ar haMelech’s question is one of kol d’parish – when the gizbar pulls a bunch of coins from the lishka to buy new korbanos (terumas halishka), is there a potential “bad” shekel that belongs to a katan in the mix? These are two different issues entirely. To return to the analogy of a wallet, it’s true that if you take your friend’s $10 and mix it into your wallet, that $10 doesn’t become yours, but that doesn’t change the fact that if you pull put a $10 bill at random, odds are that it is one that truly is yours and not the bill you took from your friend. But again, lots of bigger minds than mine have what to say on these issues (and yes, the Mishna at the end of Me'ila will pose a problem that I don't have time to work out right now.)