12. 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?
To determine whether the student's claim is valid, we need to use the commutative property of multiplication and apply it to the scenario of a p% increase followed by a q% decrease.
Let's consider an example for better understanding:
Assume the original salary is $100.
According to the student's argument:
- A p% increase in salary is equivalent to multiplying the original salary by (1 + p/100). So, after the p% increase, the salary becomes:
$100 * (1 + p/100) = $100 + $100 * (p/100)
- A q% decrease in salary is equivalent to multiplying the adjusted salary by (1 - q/100). Therefore, after the q% decrease, the salary becomes:
($100 + $100 * (p/100)) * (1 - q/100) = (1 - q/100) * ($100 + $100 * (p/100))
Now, let's examine the scenario of a q% decrease followed by a p% increase in salary:
- A q% decrease in salary is equivalent to multiplying the original salary by (1 - q/100). So, after the q% decrease, the salary becomes:
$100 * (1 - q/100) = $100 - $100 * (q/100)
- A p% increase in salary is equivalent to multiplying the adjusted salary by (1 + p/100). Therefore, after the p% increase, the salary becomes:
($100 - $100 * (q/100)) * (1 + p/100) = (1 + p/100) * ($100 - $100 * (q/100))
Now, let's compare the two scenarios by simplifying them:
Scenario 1: (1 - q/100) * ($100 + $100 * (p/100))
Scenario 2: (1 + p/100) * ($100 - $100 * (q/100))
To determine whether the two scenarios are equivalent, we need to compare their results. By expanding and simplifying each expression, we can determine if they are equal or not.
If, after simplifying, Scenario 1 is equal to Scenario 2, then the student's argument would hold true. However, if they are not equal, then the student's claim is not valid, and the two scenarios are not equivalent.