Four brothers had inherited 4500 euros with a somewhat peculiar unequal cast.

The third of them said: "If the first one was given 200, the second one had 200 taken away, I doubled what I have and the room was cut in half, we would all have the same thing in the end."

**How much had each brother inherited?**

Extracted from the problemate.blogspot.com.es page

#### Solution

We will distinguish between before and after the changes proposed by the third of the brothers. It is clear that later, everyone will have the same. If the sum is maintained, it is clear that each would have the total divided by four, but the sum may not be maintained.

As we add 200 to the first and the second we remove 200, it is clear that the sum of both remains. However, we double what we have to the third and the second we take away half. It is clear that if in the end they have the same, the third must have less than the fourth. Specifically, it must have a quarter, because if not, in the end, they would not have the same.

Imagine a specific case, in which the third has 300 and the fourth has 1200. The sum of both is 1500. However, if we double the third, it has 600, and if we divide the fourth by two, it has 600, and the sum of both is now only 1200. Along the way, a quarter of what the room had was lost, that is, what the third had originally, or half of what it had at the end. Let's check again what happens. If one has 12 and the other 48, the sum is 60. If we double 12 and divide by 2 48, both have 24, but the sum has dropped by 12 and now only gives 48.

This is further complicated, because there are two others that ultimately have the same and the sum of both does remain constant. Imagine the previous case, in the end everyone has 24, that is, 12 are lost along the way, then the total at the beginning is 24 * 4 + 12 (the 12 that have been lost). That is to say, it is as if we had four and a half brothers, and by transforming money as the third brother asks, the capital of the medium would be lost.

In our example, 4500 divided by 4.5 gives 1000, that is, in the end everyone will have 1000 and 500 will have been lost. At the beginning, **the first brother would have 800, so that by adding 200 he gives 1000. The second brother will have 1200, so that by subtracting 200 he also gives 1000. The third one will have 500, so that when doubling it he will give 1000, and the fourth will have 2000, so that at divide it by 2 also give 1000**.

In total, we can verify that it adds 800 + 1200 + 500 + 2000 = 4500.