The idiom "don’t throw the baby out with the bathwater" means to be careful not to throw away things you still want when you are throwing away things you don't want.
In the middle ages, people lived without the luxury of running water. Back in those days, everyone shared the same bathwater. The father of the house bathed first, and the mother bathed next. The children would bathe last. By this time, the water was so dirty that you couldn’t see through it and might accidentally throw the baby out with the bathwater.
The earliest known use of the German phrase was in 1512 in Thomas Murner's Narrenbeschwörung (Appeal to Fools). The phrase was first recorded in English in 1853 by Thomas Carlyle.
- https://deepenglish.com/lessons/dont-throw-the-baby-out-with-the-bathwater/# with editing
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When I read the history of the idiom, I was shocked!!! How could people in the middle ages bath their babies last? 
No worder the high baby mortality rate then and the name of " medieval ". 哼！
I'm so glad that I'm livng in the modern era where young ones are given priority, and then "lady first". 
Back to the topic, be careful when you throw your trash out next time. Even you don't have baby at home, there could be treasures hidden in it!!
