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. Then read your peers’ advice. Respond to two of your peers: For each post, evaluate your peer’s sample mathematical statement and determine whether it is accurate. Tell your peers what you agree with and disagree with in their advice.
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 Statement: Let's assume the cousin currently works x hours and earns $y per hour. The advertisement claims that if the cousin works 10 fewer hours and still earns the same wage per hour, they will make more money.
Based on the description, the mathematical statement can be written as:
(x - 10 hours) * y < x * y
Evaluation using the Associative Property of Multiplication:
(x - 10 hours) * y = x * y - 10y
So, the statement becomes:
x * y - 10y < x * y
Evaluation using the Commutative Property of Multiplication:
Since multiplication is commutative, the order in which we multiply doesn't affect the result. Therefore, the claim can be evaluated as:
-10y < 0
To evaluate whether the claim is reasonable, we need to compare the implications of working 10 fewer hours with the same wage to the current situation. If the cousin works fewer hours, their total earnings would decrease. Therefore, it is not reasonable to expect that by working 10 fewer hours at the same wage, they would make more money.
Peer Response:
Response 1:
The statement is accurate, and I agree that by working fewer hours at the same wage, it is unlikely for someone to make more money. The cousin's total earnings would decrease due to the reduced working hours.
Response 2:
I also agree with the evaluation that the claim is not reasonable. Working fewer hours at the same wage would result in a smaller total income, not more money. The mathematical statement accurately represents the situation and proves the claim to be false.
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 representing the claim: Let's say the cousin currently works 40 hours a week and earns $20 per hour. The claim states that by working 10 fewer hours, the cousin will still make more money. So, we can represent this with the following equation:
(Current weekly wage) - (10 hours * Hourly wage) > (New weekly wage)
Using the Associative and Commutative Properties of Multiplication, we can evaluate whether the claim is reasonable.
Evaluation: By applying the Commutative Property of Multiplication, the equation can be rearranged as:
(Current weekly wage - 10 hours) * Hourly wage > (New weekly wage)
Next, using the Associative Property of Multiplication, we can further simplify:
(Current weekly wage * Hourly wage) - (10 hours * Hourly wage) > (New weekly wage)
Now, let's substitute values to evaluate the claim. If the current weekly wage is $800 (40 hours * $20), then the equation becomes:
($800) - (10 hours * $20) > (New weekly wage)
Simplifying further, we have:
$800 - $200 > (New weekly wage)
$600 > (New weekly wage)
Based on this evaluation, as long as the new weekly wage is less than $600, the claim seems reasonable. However, it's essential to consider if the workload reduction of 10 hours is feasible and if any other factors may affect the outcome.
Responses to peers:
Peer 1: I agree with your mathematical statement and evaluation. Your reasoning is sound, and you've accurately represented the claim using mathematics. Good job!
Peer 2: While your mathematical statement is accurate, I disagree with your evaluation. You only focused on the direct calculation of the wages without considering other factors. It's essential to take into account any potential changes in productivity or additional expenses that may affect the outcome.
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 assume the cousin currently works x hours a week and earns $y per hour. The advertisement claims that by working 10 fewer hours a week, the cousin can still make more money.
To evaluate the claim, we can use the Associative Property of Multiplication:
Original scenario:
Earnings = (x hours) * ($y per hour) = $xy
Claimed scenario:
Earnings = (x - 10 hours) * ($y per hour) = $(xy - 10y)
Now let's compare the two scenarios:
If $(xy - 10y) is greater than $xy, then the claim is reasonable. However, if it is less than $xy, the claim is not reasonable.
By applying the Commutative Property of Multiplication, we can simplify the comparison:
$(xy - 10y) = $yx - $10y
If $yx - $10y > $xy, then the claim is reasonable.
However, since both terms involve the same variables, it is not possible for $yx to be greater than $xy while having a lower hourly wage ($y per hour) in the claimed scenario. Therefore, the claim in the advertisement is not reasonable.
I will now read my peers' advice and provide feedback to two of them.