It's not an easy thing to argue for purchasing something when your office has free software to do the job.
When I've had to do this, I've broken it down into percentage productivity increase (say even a conservative small number like 2% - which is about 10 minutes per workday), and apply that to your wage, showing how long it will take for the purchase to be worth it.
For example, say you earn $5000/month - 2% productivity gain works out to $100, meaning that if the purchase is under $100, it's already paid for itself by the end of the first month. This benefit can increase a lot if multiple people will take advantage of the purchase for similar productivity gains.
As you can see, it's usually a very short timeframe for things like this. Breaking down a purchase into cost/benefit and showing the raw numbers usually makes things like this incredibly clear.