# A is x% more than B

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!

## 2 Answers

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 ...

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 $$B$$.

• I would rather see $A = B + (x/100)B$ so that the "per cent" modifies $x$. Even better is $A = (1 + x/100)B$. – Ethan Bolker Feb 11 at 2:17
• Well, it's not hard to get those after some rather very basic algebraic manipulations. Thanks. – Michael Rybkin Feb 11 at 2:20
• They are algebraically equivalent but one reflects the semantics better. – Ethan Bolker Feb 11 at 2:31
• 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. – Michael Rybkin Feb 11 at 2:58