solution

A private foundation has offered $3 million to allocate to cities to help fund programs that aid the homeless. Grant proposals were received from cities A, B, and C seeking assistance of $750,000, $1.2 million, and $2.5 million, respectively. In the grant proposals, cities were requested to quantify the number of assistance units that would be provided using the funds (an assistance unit is a night on a bed in a shelter or a free meal). Cities A, B, and C reported they could provide 485,000, 850,000, and 1.5 million assistance units, respectively, with the funds requested during the coming year. Assume that the assistance units provided vary linearly with the amount of funds allocated to the grant. That is, if a grant is funded with half of its requested amount, it can provide half of the stated number of assistance units. The directors of the foundation want to maximize the number of assistance units obtained with the $3 million. However, they also want to help each of the cities by funding as much of their individual requests as possible (this might be done by maximizing the minimum percentage of funding received by any city).