In the emerging fifth generation (5G) technology, communication nodes are expected to support two crucial classes of information traffic, namely, the enhanced mobile broadband (eMBB) traffic with high data rate requirements, and ultra-reliable low-latency communications (URLLC) traffic with strict requirements on latency and reliability. The URLLC traffic, which is usually analyzed by a metric called the age of information (AoI), is assigned the first priority over the resources at a node. Motivated by this, we consider long-term average throughput maximization problems subject to average AoI and power constraints in a single user fading channel, when (i) perfect and (ii) no channel state information at the transmitter (CSIT) is available. We propose simple age-independent stationary randomized policies (AI-SRP), which allocate powers at the transmitter based only on the channel state and/or distribution information, without any knowledge of the AoI. We show that the optimal throughputs achieved by the AI-SRPs for scenarios (i) and (ii) are at least equal to the half of the respective optimal long-term average throughputs, independent of all the parameters of the problem, and that they are within additive gaps, expressed in terms of the optimal dual variable corresponding to their average AoI constraints, from the respective optimal long-term average throughputs.