A biology lesson combined with a math lesson! How many times do you need to change the water when raising pearl oysters?

Hi everyone, I'm Pearl Professor. Recently, I asked Mr. Hui (@Mathemagic) a problem about applied mathematics. I asked him a very interesting question—"Contrast with water!" This isn't just a math puzzle; it's also a real step in pearl oyster farming. Today, I'd like to revisit this topic, firstly to share it with more friends, and secondly to invite Dick Sir or his team ( @dickhui_dhmp ) to add to the discussion, interact, or even solve it again using his "Mathemagic" method.

First, let's talk about what "water" is and why pearl oysters need it.

All living things (including humans) need time to adapt to a new environment when they move or change their habitat. Fish and shellfish are especially sensitive because even slight changes in water quality, temperature, pH, and dissolved oxygen can "shock" them to death.

When cultivating freshwater pearl mussels, we perform two stages of "acclimation":

1. Temperature adaptation : First, let the mussels slowly adapt to the temperature of the new tank (usually by using a float bag or slowly adding water).

2. Adapting to water quality : This is the key! Gradually change the old water and add new water, allowing the mussels to slowly get used to the chemical composition (hardness, minerals, etc.) of the new environment.

   
   

We were recently invited by a secondary school to try raising pearl oysters. To ensure the oysters' healthy survival, I completed the full acclimatization process before placing them in the school's aquarium. At that moment, I wondered: could this step be turned into a fun teaching activity? So, I designed a simple yet profound math problem for the students to practice their understanding of exponents and logarithms.

A little freshwater pearl oyster acclimated to the new environment.

The question is: how many times should I "add water"?

Assume the original water tank contains 1 liter of old water (100% old water, the pearl oysters are fully adapted). Maintain a 1 liter water level throughout the initial watering process, following these steps:

1. First, pour out 30% of the old water (i.e., pour out 0.3 liters and leave 0.7 liters of old water).

2. Add another 0.3 liters of fresh water to bring the volume back to 1 liter.

3. Wait 15 minutes for the clam to adjust, then repeat the above steps.

   
   

Q: How many operations are needed to reduce the proportion of old water to 10% or less (i.e., new water ≥ 90%)? Only then can the pearl oysters be considered to have basically adapted to the new environment.

Number of times n	The remaining proportion of old water (0.7^n)	New water ratio	Is ≥ 90% fresh water?
1	70.00%	30.00%	no
2	49.00%	51.00%	no
3	34.30%	65.70%	no
4	24.01%	75.99%	no
5	16.81%	83.19%	no
6	11.76%	88.24%	no
7	8.24%	91.76%	yes
8	5.76%	94.24%	yes

That's the answer for the intermediate level. So, how would Mr. Hui @Mathemagic calculate it?

Mr Hui is an expert and very passionate about math! Remember to follow his IG !

IG: Pearl Professor