tax and discount problem Problem : In the total purchase amount $z, x\%$ is tax and $y\%$ is discount. Even if the tax is applied first and then discount or if discount is applied and then tax, the final amount is always same. How to prove this mathematically ?

Adding (or subtracting) tax or discount is in fact a multipication. To add $x\,\%$, you multiply with $1+\frac x{100}$; to subtract $y\,\%$, you multiply with $1-\frac{y}{100}$. So waht you observe is just that the order of multiplication does not matter ("commutativity of multiplication"). (Well, yes, associativity is also used).

Strictly speaking, due to rounding effects of intermediate results, minor differences in cents may occur.