If you're the kind of person who is naturally frugal, who tries to squeeze out every last penny from the tube of toothpaste and to identify which products are identical in the off-brand so you can save cost without sacrificing quality, you've probably already tried to figure out how to handle the end of the bar of soap. This is one case where it's easy to mislead yourself.
The most obvious thing is to stick the old bar to the new bar, but often this proves impractical. The skimpy remains of the old bar are breaking into fragments, and they're hard to make stick on the new bar; they splinter and dry up, or fall off.
So some frugalistas make up other approaches. The most popular: take an old container from liquid soap, cut or break up the broken bits at the end of the bar, throw them in, and add a bit of water. Keep adding new soap and a bit more water as you end each bar, and you keep having some liquid soap. It's uneven, and unless you put a lot of time into breaking up the bits very fine and dissolving them, clumpy; but it gets it all used.
Others will cut up the bits from one bar of soap and press them into a mold to make another bar of soap out of the leavings of every ten (or however many) bars. I've even seen products sold to help you do this.
But it turns out the first approach is actually the best one, and it's your frugal nature that's making it not work. The trick is that the more frugal-minded you are, and thus, the more likely to be using this approach, the more likely you are to try to use every last bit of the bar the "normal" way before you turn to finding another use for the last bits. That's why sticking it to the next bar is hard: by this time, the previous bar is just little fragments that dry out too fast.
If, instead, you add the new bar to the old bar somewhat earlier, when it's just starting to get soft enough to curve and bend in your hand without cracking, then it'll stick to the new bar perfectly. On the very first day that the old bar is flexible, use both it and the new bar; and when they're both wet, stick them together firmly, and set them aside. Don't touch them until the next day. By that point, they'll be fused nearly as perfectly as a single bar. No flaking bits, no fragments to lose, no pieces sliding off. So you get to use every particle of soap, without spending more than a few extra seconds on the process.