Confusion regarding lithium ion batteries.

Hi, I want to build a diy battery pack from lithium cells. I have read that discharging batteries at a higher rate will reduce their capacity much faster. So, suppose I have two identical batteries, apart from the fact that one battery is rated for 10c and the other for 30c. If I discharge both batteries at a rate of 10c, will the battery rated for 30c last longer or the same amount of time as the battery rated for 10c?

Sorry for the long explanation but I just wanted to make my point clear.