Simple Thermodynamics question: Temperature of water in a cup after adding cooler water to it

[untilthecern]

Homework Statement

I want to be ensure that I solved the question true.

Relevant Equations

Basic thermodynamic equations

There is water in a cup (which thermally insulated) with 75 °C of temperature and the mass of water is 0,25 kg. Then we are adding 20 °C and 0,35 kg of water into the cup. The heat capacity of water is c = 4180 J/kgK. What is the temperature of cup in the end? I found it 40 °C but I feel like I made a mistake somewhere.

 

[Lnewqban]

Correction:
Then we are adding 0,35 kg of water at 20 °C into the cup.

Could you show us how did you arrive to that amount?

 

[untilthecern]

Correction:
Then we are adding 0,35 kg of water at 20 °C into the cup.

Could you show us how did you arrive to that amount?

I forgot the formula that gives the final temperature from the given quantities, and then I used the ratio to find the final value. Since the mass of the added water is greater, the final temperature should naturally be closer to the added one's temperature. I first took the ratio of the masses and, since both are water, I thought the temperature should be divided according to the ratio of the masses. I took the difference between the initial temperatures, which is 55 K, and using the ratio of the masses (7/5), I divided by 7 and multiplied by 5. The answer came out to be approximately 40 and it's not closer to added water's temp. than the temp. of water in cup. I think I really ruined it.

 

[Lnewqban]

I forgot the formula that gives the final temperature from the given quantities, and then I used the ratio to find the final value.

Since there is no energy loss to the environment in this case, the initial thermal energy of each mass of liquid becomes the final thermal energy of the mix.

Please, see:
https://www.engineeringtoolbox.com/heat-capacity-d_338.html

https://www.engineeringtoolbox.com/mixing-fluids-temperature-mass-d_1785.html