He give someone money to buy some land for him, and that person invested it and made a profit; is he entitled to a share of the profit?

If someone takes money from someone else to buy something for him with it, the money becomes a trust (amaanah) that he is in charge of. If he does business with it without the permission of the owner, then he comes under the ruling on one who seizes wealth by force, and if he makes a profit, then the profit is to be shared between them, based on the standard percentage for profit sharing, according to the more correct scholarly view.

Some of the scholars are of the view that the entire profit belongs to the owner of the money, and some of them are of the view that it belongs to the one who seized it by force. The former view is more correct and is closer to fairness, because the profit stemmed from two things: the money and the work that was done with it, so it should be shared between them.

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: Whoever earns profit with this money, by investing it in trade and the like, it was said that the profit belongs to the owner of the capital; or that it belongs to the one who took it and traded with it, if he traded in his own name; or that they should both give it in charity, because it is profit that was earned in a wrong way; or that it should be divided between  him and the owner of the capital, like any profit-sharing investment,  as ‘Umar ibn Khattaab did with the wealth that Abu Moosa al-Ash‘ari lent to his two sons, to the exclusion of the other troops. This is the most appropriate and fair view."(Majmoo‘ al-Fataawa  30/29).

And he [Ibn Taymiyah] said: As for the capital that is taken by force, if the one who took it by force invested it and acquired profit, there are different scholarly views: Does the profit belong only to the owner of the capital? Or should they give it in charity? Or should it be shared between them as it would be if he did that by way of a profit-sharing business, musaaqaah (when trees are given to someone to tend and irrigate in return for a share of the harvest), or muzaara‘ah (where one person supplies land and another cultivates it), or when one person gives an animal to a worker who will look after it in return for some of its milk and offspring, or he gives the worker a set fee like that of one who does similar work, or whatever is in accordance with then established custom, like what ‘Umar ibn al-Khattaab did? That is what is fair, because this profit was earned by means of the capital of one of them and the work of the other. Therefore neither of them should keep the profit to himself, and they do not have to give it in charity, because it rightfully belongs to both of them and not to anyone else. Rather the profit should be shared between them, as if they were partners in a profit-sharing business."(Majmoo‘ al-Fataawa  30/322).

Shaykh Ibn Baaz (may Allah have mercy on him) was asked: Someone left some cash with me, so I made use of it and invested it. When the owner of the money came to me, I gave him all of his money, but I did not tell him about the profit that I had made from his money. Is what I did permissible or not?

He replied: If someone entrusted something to you, you have no right to dispose of it except with his permission, and you have to take care of it as people usually take care of such things. If you dispose of it without his permission, you have to seek his forgiveness; if he pardons you, all well and good, otherwise you should give him the profit that was made with his money, or reach an agreement with him to give him half, or less or more than that. Making a deal to bring about reconciliation between Muslims is permissible, except a deal that prohibits something that is permissible or permits something that is prohibited."(Fataawa ash-Shaykh Ibn Baaz 19/411).

You can see how profit sharing is done by looking at what people give to the owner of the capital and to the worker. If they give 60% to the owner of the capital, for example, and they give 40% to the worker, you can take 60% of the profit made by the one to whom you gave your money, and you can let him off some of it. You have the right to do that.

Then after that, you should look at the price for which you bought the land; if he bought it for the same amount that you paid, then you can divide what he earned by investing the money. But if he bought the land for more than what you gave him, then you may share the profit he made when he invested the money, then deduct from your share the extra amount that he paid for the land.

And Allah knows best.