explained the Associative and Commutative Properties of Multiplication to a friend. You told them that they would get the same answer every time they multiplied the numbers in the problem.

Question

Imagine that your neighbor is scrolling through social media. Your cousin is considering making an in-app purchase. Your neighbor shows you advertisements for products that make bold claims:

“Get rich quick! Do you want to work fewer hours for the same amount of money per hour, and get even richer? Click here to see how you can relax and earn. You’ll work 10 fewer hours at the same wage, and you’ll still make more money!”
“Get hydrated now! Do you hate drinking water? Click here to see how drinking less water is actually the secret to having more water in your body. It’s the study the government doesn’t want you to see!”
Select one of the advertisements and advise your cousin:

Represent the advertisement’s claim with a sample mathematical statement.
Using the Associative and Commutative Properties of Multiplication, evaluate whether the claim is reasonable.
Explain your reasoning in your post.

Answer 1

Let's focus on the first advertisement: “Get rich quick! Do you want to work fewer hours for the same amount of money per hour, and get even richer? Click here to see how you can relax and earn. You’ll work 10 fewer hours at the same wage, and you’ll still make more money!”

Sample Mathematical Statement

Let's say your cousin currently works H hours per week at a wage of W dollars per hour. Therefore, their current weekly earnings can be represented as: \[ E_{current} = W \times H \]

According to the advertisement, they would work H - 10 hours at the same wage of W dollars per hour and somehow end up making more money. This can be represented as: \[ E_{new} = W \times (H - 10) \]

Evaluating the Claim Using the Properties of Multiplication

Now, let’s evaluate the claim about earning more money by rearranging and simplifying the mathematical statement using the Commutative and Associative Properties of Multiplication.

1. Commutative Property: This property states that changing the order of the factors does not change the product. For example: \[ W \times (H - 10) = (H - 10) \times W \]

2. Associative Property: This property states that how we group the numbers when multiplying does not change the product. However, in this case, the Associative Property may not be directly applicable to our case since we are dealing with a reduction of hours rather than regrouping factors.

Calculation:

Using the first equation:

• Current earnings: \[ E_{current} = W \times H \]

Using the advertisement's claim:

• New earnings: \[ E_{new} = W \times (H - 10) = W \times H - W \times 10 \]

So, we rewrite the new earnings: \[ E_{new} = W \times H - 10W \]

Conclusion

For the original claim to be true that your cousin would make "more money" after working 10 fewer hours, it needs to satisfy: \[ E_{new} > E_{current} \] This can be expressed as: \[ W \times (H - 10) > W \times H \] Which simplifies to: \[ -10W > 0 \]

This inequality does not hold true unless the wage W is less than zero, which is impossible in a typical employment situation. Therefore, it is unreasonable to claim that one can work fewer hours and still earn more money under those circumstances.

Advice

You should advise your cousin to be cautious about advertisements that make such bold claims, as they fundamentally contradict basic principles of arithmetic and economics. Working fewer hours at the same wage typically results in reduced earnings, not increased earnings.