Is it permissible for a brother to exempt his brother from paying the remaining price of the taxi jointly owned by them after he saw that it did not yield a significant profit, and is this partnership considered an injustice to the wife and children?

Participating with your brother in buying a car, where each of you owns half, and your brother lends you the amount needed to complete your half of the capital, and you agree to divide its income, is permissible if several conditions are met. These include: neither of you guaranteeing the other's capital, and clearly defining in advance the percentage each of you is entitled to from the income.

If you are working on the car, there is no harm in your profit percentage being higher than your brother's, or in having an equal profit percentage.

As for the amount your brother exempted you from repaying, it can be interpreted in two ways: 1. It could be the money he lent you for your share and his entitlement from the car's income. In this case, the partnership is valid, and the exemption is permissible. This is not considered an injustice to his wife and children as long as your brother does not need it for their sustenance; rather, it is an act of maintaining kinship ties. 2. It could be a fixed income that he is entitled to from you, which neither increases nor decreases. In this case, the partnership is invalid. Your brother cannot exempt you from anything except half of the actual income corresponding to his ownership of half the car, provided he does not need it for the sustenance of those whose maintenance is obligatory upon him.