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I was reading a text book and came across the following:

If a ratio $a/b$ is given such that $a \gt b$, and given $x$ is a positive integer, then $$\frac{a+x}{b+x} \lt\frac{a}{b}\quad\text{and}\quad \frac{a-x}{b-x}\gt \frac{a}{b}.$$

If a ratio $a/b$ is given such that $a \lt b$, $x$ a positive integer, then $$\frac{a+x}{b+x}\gt \frac{a}{b}\quad\text{and}\quad \frac{a-x}{b-x}\lt \frac{a}{b}.$$

I am looking for more of a logical deduction on why the above statements are true (than a mathematical "proof"). I also understand that I can always check the authenticity by assigning some values to a and b variables.

Can someone please provide a logical explanation for the above?

Thanks in advance!

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    I think you used the "logic" tag again in a previous question that was unrelated to logic so I'm posting a comment, the logic tag is used for questions that are related to mathematical logic (for more information see the tag description). From the way you are posing the question it looks to me that you could use the "intuition" tag (though I'm not sure).2011-06-18
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    Just as a note, assigning some values to $a$ and $b$ would *not* be a way to check the 'authenticity' (by which I assume you mean 'truth') of those statements; if you found $a$ and $b$ such that the statement failed to hold you would have shown the general statement false, but no amount of testing with constants can *prove* it to be true. Of course it can be a helpful way to build intuition about the question, though.2013-06-16

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