Is it worth getting multiple hard-wired smoke detectors connected to an alarm system?

We recently moved into a house that has a high quality alarm system already installed, so we’re planning to activate it. Currently there is only one smoke detector installed, and this one is hard-wired and connected to the alarm company. It’s in the middle of the house, which is ~1300 sq ft. We are planning to put smoke detectors in the bedrooms and possibly the living room, but we have to decide whether to get inexpensive but high-quality battery-powered ones ($25 each) or additional hard-wired ones ($150 each). These options should be more or less equal if a fire started while we were in the house.

The question is this: if a fire started in one end of the house while we were away, how long would it likely take for the alarm in the center of the house to detect it? Would there be enough of a delay that having additional hard-wired alarms would be likely to lead to protection of more of the house by alerting the alarm company sooner?

Thanks for any help!