Suppose I have a till. I have a special that I give a 10 percent discount on all dinner meals. I want to know that if I calculate the discount on each dinner item and round up the discount and apply it to each meal, I will not lose money than if I first added up the dinner items and then applied the 10 percent discount and then rounded up the answer.
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The consumer benefits from roundup of individual meals because $$\lceil .1(x_1+x_2+\cdots +x_n)\rceil \le \lceil .1 x_1\rceil + \lceil .1 x_2\rceil + \cdots + \lceil .1 x_n\rceil$$ where $\lceil\quad\rceil $ is the ceiling function. So if you're the restauranteur you'd prefer to round up after totaling the meals.
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Thanks for that. Also $$x_1 \le \lceil x_1 \rceil$$. Sum up both sides for $x_2, x_3$ and so on and this will largely explain where that formula comes from. – Excalibur2000 Nov 19 '13 at 20:47