Is this silly?A mechanism of action

[AlexMason]

Why exactly this to this and not vice versa. For example problem : T-shirt costs 10 dollars. How many of these T-shirts can John buy if he has 30 dollars.Usually in the problems the solution is described: you need to divide 30 by 10. And why exactly so and not vice versa(10 / 30), how should you make a start in solving such problems, what is the formula or mechanism?

[Guest]

Oh, I see your problem! You ask about a "formula". Entirely too many people think you solve math problems by plugging numbers into formulas. You don't! You solve math problems by thinking!

"T-shirt costs 10 dollars. How many of these T-shirts can John buy if he has 30 dollars"

There are a number of different ways to think about this. One is to say that John could put his 30 dollars in three stacks, each with 10 dollars, so could by 3 shirts.

Another is to interpret "T-shirt costs 10 dollar" as meaning that the cost is [tex]10\frac{dollar}{shirt}[/tex]. We want to go from a number of "dollars" to a number of "shirts" so first we need to invert that fraction to [/tex]\frac{shirt}{dollar} to get "shirts" on top, then multiiply by "dollars" to cancel the "dollars in the denominator. Of course, "inverting" a fraction and the multiplying by it is the same as dividing by the fraction:
[tex]\frac{30 dollars}{10 \frac{dollar}{shirt}}= (30 dollar)\left(\frac{1}{10}\frac{shirt}{dollar}\right)= 10 shirts[/tex].