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I am taking the AMC10 test, and I don’t want to lose points on silly misunderstandings. When a question says “A is x% greater/less than B”, or things like that sometimes with money, which respect should we take if it doesn’t tell me which? And please write a formula to make it clear. Thanks!

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Whenever you are given that a quantity $A$ is $x\%$ greater or less than $B$ the first thing is that the comparision is done with respect to the quantity $B$.

Look it in another way. So how do you calculate the $\%$ increase or decrease in a quantity with respect to another quantity.

If it is said that the quantity $A$ is $x\%$ greater than quantity $B$ what you mean is : $$\frac{A-B}{B}=\frac{x}{100}$$ And if the quantity is $x\%$ less than $B$ then : $$\frac{B-A}{B}=\frac{x}{100}$$ Hope this helps ...

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Not exactly sure what you're asking, but, I guess, a statement like "$A$ is $x\%$ greater than B" formulaically should be interpreted as follows:

$$ A=B+\frac{B}{100}\cdot x=B\left(1 + \frac{x}{100}\right) $$

Likewise, "$A$ is $x\%$ less than B" should be understood like this:

$$ A=B-\frac{B}{100}\cdot x=B\left(1 - \frac{x}{100}\right) $$

So, I guess you could say you're calculating percents with respect to $B$.

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  • $\begingroup$ I would rather see $A = B + (x/100)B$ so that the "per cent" modifies $x$. Even better is $A = (1 + x/100)B$. $\endgroup$ – Ethan Bolker Feb 11 at 2:17
  • $\begingroup$ Well, it's not hard to get those after some rather very basic algebraic manipulations. Thanks. $\endgroup$ – Michael Rybkin Feb 11 at 2:20
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    $\begingroup$ They are algebraically equivalent but one reflects the semantics better. $\endgroup$ – Ethan Bolker Feb 11 at 2:31
  • $\begingroup$ When I think of something like $10\%$ of $X$, I immediately divide $X$ by $100$ and then multiply the result by $10$. Because $10\%$ of $X$ is nothing more than $10$ pieces out of the $100$ pieces that you get when $X$ is divided by $100$. But that's just me. $\endgroup$ – Michael Rybkin Feb 11 at 2:58

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