A student argues that a p% increase in salary followed by a q% decrease is equivalent to a q% decrease followed by a p% increase because of the commutative property of multiplication. How do you respond?
In the first case, we have
(1+p)(1-q)
In the second case, we have
(1-q)(1+p)
Looks the same to me, and for the stated reason.
To respond to the student's argument, we can examine the scenario mathematically. Let's say the student's initial salary is denoted by S.
According to the student, a p% increase in salary would result in a new salary of S + (p/100) * S = S(1 + p/100).
Then, a q% decrease would be equivalent to multiplying the new salary by (1 - q/100), resulting in a final salary of S(1 + p/100)(1 - q/100).
Now, let's consider the scenario where a q% decrease is followed by a p% increase.
A q% decrease in salary would result in a new salary of S - (q/100) * S = S(1 - q/100).
Then, a p% increase is equivalent to multiplying the new salary by (1 + p/100), resulting in a final salary of S(1 - q/100)(1 + p/100).
If we apply the commutative property of multiplication, which states that the order of multiplication can be rearranged without changing the result, then we would expect S(1 + p/100)(1 - q/100) to be equal to S(1 - q/100)(1 + p/100).
However, when we expand and simplify both expressions, we find that they are not equal:
S(1 + p/100)(1 - q/100) = S(1 - q/100 + p/100 - (pq/10000))
S(1 - q/100)(1 + p/100) = S(1 + p/100 - q/100 - (pq/10000))
As you can see, these two expressions are not identical due to the additional terms (pq/10000) in each expression. Therefore, the student's argument is not supported by the commutative property of multiplication.
In conclusion, a p% increase in salary followed by a q% decrease does not give the same result as a q% decrease followed by a p% increase.