"reduce sth to sth" vs "reduce sth by sth"

[Nessie]

Hi,

What's the difference between "reduce something (A) to something (B)" and "reduce something (A) by something (B)"?

I guess the former structure has the meaning that we reduce (A) so that the remain is (B), whereas the latter means the whole is (A), and we reduce part of it which is equal to (B). What do you think?

By the way, if my explanation is correct, then the two usages must be both applicable in this sentence, mustn't it:

In the last 10 years, Mexican government has reduced the number of its state-owned companies to/by about half. (of course, each usage has a different meaning)

Many thanks in advance.
Nessie.
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Sorry seems to be the hardest word...

[Haihao]

"reduce something (A) to something (B)": reduce art to show;
"reduce something (A) by something (B)": reduce the water by a pump

[Inga]

Hi, can you explain?

[Crochooked]

Try to see this better by using an example: You have 10 shirts and you want to reduce them TO 2. You will get rid of 8 of them, leaving 2.
If you have 10 shirts and want to reduce them BY 2, you will get rid of the 2, leaving 8.
In the example you give, either to or by will be correct as the example uses 1/2, and if you reduce by 1/2 you are left with 1/2. If you reduce to 1/2, you are still left with 1/2. If the example had used a different fraction, that would make a difference.
Another example: Let's say that you have 8 shirts and want to reduce them BY 1/4. 1/4 of 8 is 2, so you get rid of 2, leaving 6.
If you have 8 shirts and want to reduce them TO 1/4, that means you want to reduce them to 2, so you have to get rid of 6 to leave 2.

[Inga]

Oh, Math is your favourite subject?
Though I always liked it at school, I didn't guess to give such good examples.
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I am an incurable optimist.

[Nessie]

Thanks a lot, everybody
(good to see that I got the point right )
_________________
... something we never have again, I know... I guess I really really know..

Sorry seems to be the hardest word...