In your journal, you 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 analyze 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!"

Mathematical Representation:

Let's assume your cousin currently makes $W per hour and works H hours per week. This means their current weekly earnings can be represented as:

Current Earnings = W * H

According to the advertisement, if your cousin works 10 fewer hours, their new hours would be:

New Hours = H - 10

The advertisement claims they will still make more money even after working fewer hours, represented as:

New Earnings = W * (H - 10)

Now, the claim can be formulated as:

W * (H - 10) > W * H

If we simplify this inequality using the Distributive Property:

W * (H - 10) = W * H - W * 10

Thus, the inequality becomes:

W * H - W * 10 > W * H

If we subtract W * H from both sides:

-W * 10 > 0

Now, dividing by -10 (and reversing the inequality since we are dividing by a negative number):

W < 0

Evaluation of the Claim:

For this inequality (W < 0) to be true, the wage W must be negative, which is unreasonable in a typical work scenario. Therefore, it suggests that for a positive wage—meaning they are earning money—the equation does not hold.

Conclusion:

The claim in the advertisement is not reasonable. According to the Associative and Commutative Properties of Multiplication, or any standard arithmetic, if you reduce the number of hours worked (without increasing the hourly wage), the total earnings will decrease. So, if your cousin works fewer hours at the same rate, they will make less money, not more as the advertisement suggests. It's important to be cautious of claims that sound too good to be true, especially those that contradict basic mathematical principles.