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I didn’t think this was such a difficult concept but I guess it is?

First off let’s agree on what is better.

If I do a 20% water change, the resulting water is:

(1-.2) = .8 or 80 % of the original water is left.

If I do a 50% water change, the resulting water is:

(1-.5) = .5 or 50% of the original water is left

I am assuming we all agree that 50% changes more water than 20%?

Please note: The lower the resulting number in this calculation (which is the amount of the original water left) the more water is changed.

Now back to the question:

4 x 20% changes

(1-.2)*(1 -.2)*(1 -.2)*(1-.2) = .4096

(Only 41% of the original water is left)

1 x 50% changes

(1-.5)^1 = .5

50% of the original water is left.

So 4 x 20% changes more water than 1 x 50%

In general to find out how much of the original water is left, use the following formula:

(1 – A)*(1 - B)* (1 – C) * …

Where A, B, C, … is the % water changed divided by 100.

If you always change the same amount of water then the expression simplifies to:

(1 – A)^N

Where A is the % water changed divided by 100.

And N is the number of changes.

Which plan is more work is a relative thing.

If you have a 10 gal tank changing 50% means only changing 5 gal of water. Making, conditioning, adjusting and handling 5 gal is manageable for most people.

If you have a 100 gal tank, changing 50% means changing 50 gal of water. Making, conditioning, adjusting and handling 50 gal is going to require a lot more concentrated effort and even specialized equipment. You probably would want to do more frequent smaller changes.

Changing water is the best thing you can do for your fish/plants. It is also the riskiest and the more water you change at one time, the greater is the chance of making a fatal mistake.

For example:

Say your make up water is 22°C and your tank water is 32°C. If you do a 50% water change the new tank temp is:

.5 * 22°C + .5 * 32°C = 27°C (A change of 5°C)

A 10% change results in:

.1 * 22°C + .9 * 32°C = 31°C (A change of only 1°C)

So be very careful changing water!