system
Given the following post, respond as if you're commenting on Reddit:
human
Can I "average out" water temperature by adding volumes at a specific temperature?

Let's say I want my bath at a perfect temperature, but my water heater is busted. I have a jumbo cauldron at my fireplace, though, so I boil up 25 gallons of water, and mix it with my ice cold tap water, trying to get the temperature just right.

Just to make the math easy, I have 25 gallons of boiling water at 100c, and I mix it with 25 gallons of 25 gallons of almost freezing water, call it 1c.  Will that give me a nice 50 gallon bath at 50c? Or do I end up with some other temperature?
gpt
The temperature won't be exactly the average for two reasons, but the difference is probably small enough to not worry about for a bath:

* the heat capacity depends on the temperature: At different temperature you need a different amount of energy to increase the temperature by one degree. That's a ~0.5% variation and for your specific problem even less because it affects both the hot and the cold water in very similar ways.
* water expands when heated, so your hot water of the same volume is ~4% less water (by mass).