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In some observational studies of treatment effects, matched samples are created so treated and control groups are similar in terms of observable covariates. Traditionally, such matched samples consist of matched pairs. However, alternative forms of matching may have desirable features. One strategy that may improve efficiency is to match a variable number of control units to each treated unit. Another strategy to improve balance is to adopt a fine balance constraint. Under a fine balance constraint, a nominal covariate is exactly balanced, but it does not require individually matched treated and control subjects for this variable. Here, we propose a method to allow for fine balance constraints when each treated unit is matched to a variable number of control units, which is not currently possible using existing matching network flow algorithms. Our approach uses the entire number to first determine the optimal number of controls for each treated unit. For each stratum of matched treated units, we can then apply a fine balance constraint. We then demonstrate that a matched sample for the evaluation of the Peer Health Exchange, an intervention in schools designed to decrease risky health behaviors among youths, using a variable number of controls and fine balance constraint is superior to simply using a variable-ratio match.