- Translate the problem requirements: We need to find how much money to spend on advertising to generate $100,000 in revenue. We're given that $100 revenue comes from 1,000 views, and views follow the equation \(\mathrm{y = 100x - 100,000}\) where x is dollars spent.
- Convert revenue target to required views: Use the given rate to determine how many advertisement views are needed to generate $100,000 in revenue.
- Set up the views equation: Use the formula \(\mathrm{y = 100x - 100,000}\) where y is the required number of views we found, and solve for x (dollars spent).
- Solve and approximate: Calculate the value of x and compare with answer choices to find the nearest match.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in plain English:
- We want to generate $100,000 in revenue from our advertisement
- We know that every 1,000 views of the ad brings in $100 in revenue
- We also know there's a relationship between how much we spend on the ad (x dollars) and how many times people will see it (y views): \(\mathrm{y = 100x - 100,000}\)
- Our goal is to find out how much money (x) we need to spend to hit our revenue target
Think of this like planning a party: we know how much each guest contributes to our goal, we know how spending on invitations affects attendance, and we want to figure out our invitation budget to reach our target.
Process Skill: TRANSLATE - Converting the business scenario into clear mathematical relationships
2. Convert revenue target to required views
Now let's figure out how many people need to see our ad to generate $100,000 in revenue.
We know that:
- 1,000 views = $100 in revenue
- We want $100,000 in revenue
Using simple proportion thinking: if 1,000 views give us $100, then to get $100,000 (which is 1,000 times more money), we need 1,000 times more views.
So we need: \(\mathrm{1,000 \times 1,000 = 1,000,000}\) views
We can verify this: \(\mathrm{1,000,000 \div 1,000 = 1,000}\) groups of 1,000 views
\(\mathrm{1,000 \times \$100 = \$100,000}\) ✓
3. Set up the views equation
Now we know we need 1,000,000 views. We can use the given formula to find how much we need to spend.
The formula tells us: \(\mathrm{y = 100x - 100,000}\)
Where:
- y = number of views (we need 1,000,000)
- x = dollars spent (this is what we're looking for)
Substituting what we know:
\(\mathrm{1,000,000 = 100x - 100,000}\)
4. Solve and approximate
Let's solve for x step by step:
Starting with: \(\mathrm{1,000,000 = 100x - 100,000}\)
First, add 100,000 to both sides:
\(\mathrm{1,000,000 + 100,000 = 100x}\)
\(\mathrm{1,100,000 = 100x}\)
Now divide both sides by 100:
\(\mathrm{x = 1,100,000 \div 100}\)
\(\mathrm{x = 11,000}\)
So we need to spend $11,000 on the advertisement.
Let's verify: If x = 11,000, then \(\mathrm{y = 100(11,000) - 100,000 = 1,100,000 - 100,000 = 1,000,000}\) views ✓
And 1,000,000 views generates $100,000 in revenue ✓
Final Answer
We need to spend $11,000 on the advertisement to generate $100,000 in revenue.
Looking at our answer choices:
- $900
- $1,100
- $2,000
- $9,000
- $11,000
Our calculated answer of $11,000 exactly matches choice E.
The answer is E. $11,000
Common Faltering Points
Errors while devising the approach
- Misinterpreting the revenue-to-views relationship: Students might incorrectly think that $100 generates 1,000 views instead of understanding that 1,000 views generate $100 in revenue. This reversal of cause and effect leads to setting up the wrong proportion.
- Confusing the variables in the equation: Students may mix up what x and y represent in the formula \(\mathrm{y = 100x - 100,000}\), thinking that x represents views and y represents spending, rather than x being spending and y being views.
- Missing the two-step process: Students might try to directly relate spending to revenue without recognizing they need to first convert revenue to required views, then use the views equation to find spending.
Errors while executing the approach
- Arithmetic errors in proportion calculation: When calculating how many views are needed for $100,000 revenue, students might incorrectly compute \(\mathrm{100,000 \div 100 = 1,000}\) views instead of recognizing they need \(\mathrm{(100,000 \div 100) \times 1,000 = 1,000,000}\) views.
- Sign errors when solving the linear equation: Students may forget to add 100,000 to both sides or make errors with the negative term, getting \(\mathrm{1,000,000 = 100x + 100,000}\) instead of \(\mathrm{1,000,000 = 100x - 100,000}\).
- Incorrect algebraic manipulation: When isolating x, students might divide incorrectly or forget to perform the same operation on both sides of the equation, leading to wrong intermediate values.
Errors while selecting the answer
- Selecting an order-of-magnitude error: Students who make calculation mistakes might arrive at $1,100 (choice B) instead of $11,000, essentially being off by a factor of 10 but still seeing their number in the answer choices.
- Choosing a partial calculation result: Students might select $1,100 if they calculated the intermediate step \(\mathrm{(1,100,000 \div 100)}\) but forgot to complete the division, or if they confused this with their final answer.
