Question

$\sf{Prove \: That} \\& \sf{ x \% \: Of \: \: y \: \: = \: \: y\% \: Of \: x}$
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Answer 1

Answer:

Statement: Is X% of Y same as Y% of X ?( pause for a moment and think, then proceed)

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Let us verify above statement.

X% of Y = X/100 * Y

= X*Y/100 -- (1)

&

Y% of X = Y/100 * X

= Y*X/100 -- (2)

So, From equation 1 and 2 it is clear that X% of Y has exactly same value as Y% of X.

Answer 2

Multiplication is commutative, for most of the “everyday” abstract objects we call “numbers”. That is, for such objects, x × y = y × x .

The definition of x% of y is precisely x/ 100 × y .Since 1/100 is a regular, everyday number, and x/100 = x×1/100 = 1/ 100×x , we can shuffle the terms in the expression 1/100 × x × y around however we like, and we’ll still get the same result.

If you want to rearrange x/100 × y= y/100 × x and then re-interpret the right hand side as y% of x , it is no problem.

Commutativity of multiplication (for all “everyday” numbers, such as integers, rational numbers and so-called* “real” numbers) assures you of it.

* There’s nothing real about any kind of number, so don’t take the term “real” in “real number” as an adjective that ascribes to them some undue level of “concreteness” that certain other types of numbers somehow lack. All numbers are imaginary!