Transferring memory from 2011 iMac to 2017?

This is an embarrassing question, but I have a 2011 iMac that is sadly at the end of its life. I wondered if I can transfer the 16 GB of memory from this computer (1333 MHz) into a new 2017 iMac (specs on Apple say 1867 MHz). Not knowing what these numbers really mean, would there be a compatibility issue? Looking to save the $180 if possible.

First, the 2017 iMac has not yet been announced by Apple and likely won't be until sometime late April or May. So, you must be referring to the current iMac which is a late 2015 model. Anyway, the answer is no. The memory you have from the 2011 model is different than the memory used in the latest iMac.

Your 2011 iMac:

Maximum Memory 32 GB (Actual) 16 GB (Apple)
Memory Slots 4 - 204-pin PC3-10600 (1333 MHz) DDR3 SO-DIMM

The late 2015 iMac:

Maximum Memory 64 GB (Actual) 32 GB (Apple)
Memory Slots 4 - 204-pin PC3L-14900 (1867 MHz) DDR3 SO-DIMM

As you can easily see, they're not the same.

You might want to consider selling your older iMac if you want that to help go towards the cost of your new iMac.