Research has shown that, machine learning models might inherit and propagate undesired social biases encoded in the data. To address this problem, fair training algorithms are developed. However, most algorithms assume we know demographic/sensitive data features such as gender and race. This assumption falls short in scenarios where collecting demographic information is not feasible due to privacy concerns, and data protection policies. A recent line of work develops fair training methods that can function without any demographic feature on the data, that are collectively referred as Rawlsian methods. Yet, we show in experiments that, Rawlsian methods tend to exhibit relatively high bias. Given this, we look at the middle ground between the previous approaches, and consider a setting where we know the demographic attributes for only a small subset of our data. In such a setting, we design fair training algorithms which exhibit both good utility, and low bias. In particular, we show that our techniques can train models to significantly outperform Rawlsian approaches even when 0.1% of demographic attributes are available in the training data. Furthermore, our main algorithm can accommodate multiple training objectives easily. We expand our main algorithm to achieve robustness to label noise in addition to fairness in the limited demographics setting to highlight that property as well.