Many of the items I like to build have the BF Sword as a component. This is one of the most expensive single components in the store, meaning I have to farm for quite a while before I can buy it. Meanwhile, everyone else in the game has bought two or three components for the same price, and so they are in general doing a little better than me. (Because they have more items and I just have a bunch of gold.)

I'm thinking about "trading up" to the BF Sword. What I mean by this is, I buy some cheaper +dam item first (or really, any cheaper item, I say +dam just because BF Sword is +dam) then sell it when my gold plus the sell amount would equal the amount for the BF sword.

This will cost me a little extra gold--the gold lost in the selloff of the "trade in" item--but on the other hand, I won't be handicapped the whole time I'm trying to save up for the BF sword.

Does this seem like a viable idea to you guys?

The idea isn't too great, but your logic is good, you just missed a bit in the last part XD

But congratulations - you're among the few new players to notice on their own that saving gold and getting BF sword first will generally be negative for you, as while you're saving you will die/not kill because your oponents would have traded gold for items.

The solution to this problem is to not start with infinity edge/bloodthirster/black cleaver. (those are the only items that need BF sword btw). You need to boost your damage with smaller items first, and choose them good, because in the perfect scenario you won't need to sell them. You can always invest in a long sword. It gives you 10 damage, and can be build into :
-phage (itself build to frozen mallet or trinity force)
-brutalizer (itself build into ghostblade)
-last whisper (good item in general)
-Mardred's Razors (build into bloodrazors - one of the top dmg items)

As you can see, you have options. Get a brutalizer. +25 dmg and +20 armor pen is incredible boost for just 1500, and is build from two long swords. Then you can get an avarice, which will boost your crit, and gold income. Keep it in your inventory for the gold, and when you need the slot merge them to ghostrblade.

There are VERY few champions that can start with infinity edge, and usually this involves geting a bunch of avarice blades first to help you farm the gold. In general, the items that involve BF sword are left for later.

P.S. I said your idea is bad because generally it's not recommended to sell items. You can always plan your build and use itemization routs that will allow you to start small and upgrade later, without having to sell anything.

I end up getting IE much later in the game with Twitch, and by that time gold isn't really an issue. I think if gold is an issue, it might be too early in the game. Plus you could just be losing, which would mean I'd never get to the IE in my build.

It's all situational.

It all depends on timing. If you are having a rough game then ya, it can be hard to get a BF sword really quickly.

However if you are good at last hitting minions, its actually not to hard to try and rush a bf sword if you want to and have it and rank 1 boots as your first 2 items. You may be a tad weaker for a couple minutes depending on your timing back to base, but it is very feasible.

That is an important point though. It depends on how your game is going and the timing of when you back to base. You will have to use your own judgement.

I would never buy damage items and then sell them off later, in my opinion it really would be wasting to much gold to do so. However I would give myself the option to buy other cheaper items - as someone else mentioned something like a brutalizer - as a perfect item that has cheap components and is cheap to build which can bridge the game for those long drawn out periods.

I would not recommend buying items that you intend to sell later... instead, I would invest 300 gold into an elixir (if you are on route to a BF sword, you'd prolly want the red one). For a low price, elixirs are able to give you a substantial boost that may be strong enough to net you a gank or 2 - and a single kill will earn you back the money you spent on it in the first place.

Or, if you want to spend your money on something more permanent, you can start by buying a longsword, vampiric scepter or a pickaxe before the pricey BF sword. These items will be required when you decide to upgrade your BF sword into a Black Cleaver (Longsword), Bloodthirster (Vampiric Scepter) or Infniities Edge (Pick Axe).

While I do see the advantage to having an increase of 50 AD early on, 1850 is a LOT of gold at that point, and I would usually recommend purchasing a smaller item or two before moving on to the BF.

I'd like to point something else out that the others seem not to have mentioned. If you sell an item you get less gold than you originally spent on it.

So if you were buying a few long swords first and then sold them to make money for your B.F. Sword it would actually cost more than if you had just saved up money for the B.F. sword.

But like an above post already mentioned its generally not a good idea to rush the purchase of a B.F. Sword before other items early in the game.

The only items you should really ever find yourself selling are possibly the Doran's items. Since they give good early bonuses but don't build into anything else you might find yourself needing to sell them later in order to make room for other items.

So, what should I start my AD kat with? BF sword cost is always my biggest enemy, when I practicaly have nothing but boots and doran at lvl 6

Well, Kat is kind of a special case. As most of her DPS comes from spells that all deal magic damage, all of the items Somnium suggested, while being good in themselves, don't do too much for her.
So for Kat it's a better idea to get some early Magic Pen meaning Sorc Boots, Haunting Guise or both. They're both not expensive and increase your damage nicely through all game.
If you think your game is going well you might also get a quick Sword of the Occult, but getting snowball items is always risky, especially on a champ like Kat

Kat is EXCEPTIONAL at last hitting minions early in the game. Before I first go back to base I almost always have 2000-3000 gold sitting around on her. Most of the time I end up buying a BF sword + normal boots.

If you can take advantage of that farming potential and getting good at last hitting, you can easily farm up a BF sword early on her without having to worry about intermediary items.

If for some reason you are struggling though you can start by finishing sorc boots and then trying hard to farm and save up.

Occult is another option for a cheaper item, but it doesnt help your strength early if thats a problem. It needs to snowball and get strong.

As people have mentioned, Kat is a bit of an exception here. however, in the cases when you haven't farmed up your BF sword, you could use phage. It gives you damage, some health, a nice bonus, and can be build into frozen mallet later on. When you have tons of damage, investing in some sort of def pays well off. If i don't have for BF, i like to get phage, which i upbrage after bloodthirster.