In the cake cutting problem, a cake needs to be divided among players with possibly different valuations for different pieces of the cake. We consider a variation where the cake represents chores that need to be allocated. We assume the players to be work-averse, i.e., intending to get as little of the cake as possible. We further assume that the chores are indivisible and that they are ordered along a line. We seek a fair allocation of the chores. We study the loss in social welfare due to fairness using the price of fairness concept. Previous work has considered fair division of cakes and chores with non-connected and connected pieces as well as fair division of indivisible contiguous blocks of indivisible items (with positive utility) on a line. We complete the picture by providing price of fairness results for the allocation of contiguous blocks of indivisible chores.