- Translate the problem requirements: We need to find Shannon's contribution per minute of listening time, which means dividing her total annual contribution (\(\$35\)) by her total annual listening time in minutes.
- Convert listening time to consistent units: Transform weekly listening hours into total annual minutes to match the desired output unit.
- Calculate the rate using division: Divide total contribution by total listening minutes to find dollars per minute.
- Compare with answer choices for closest match: Since answer choices are spread apart in magnitude, identify which option our calculated value is nearest to.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we're looking for in everyday terms. Shannon made a contribution to a radio station, and we want to know how much she paid for each minute she listened.
Think of it like calculating the cost per minute of a phone call - if you paid \(\$35\) for all your phone calls in a year, and you talked for a certain number of minutes total, you'd divide the total cost by the total minutes.
Here, Shannon:
• Contributed \(\$35\) total for the year
• Listened 10 hours per week
• We need: dollars per minute of listening
So we need to find: Total contribution ÷ Total listening time in minutes
Process Skill: TRANSLATE - Converting the word problem into a clear mathematical relationship
2. Convert listening time to consistent units
Right now we have Shannon's listening time in "hours per week" but we need her total listening time in "minutes for the whole year."
Let's work through this step by step:
First, let's find her total hours for the year:
• 10 hours per week
• 52 weeks in a year
• Total hours = \(10 \times 52 = 520\) hours per year
Now let's convert hours to minutes:
• 1 hour = 60 minutes
• 520 hours = \(520 \times 60 = 31,200\) minutes per year
So Shannon listened for 31,200 minutes total last year.
3. Calculate the rate using division
Now we can find her contribution per minute:
Contribution per minute = Total contribution ÷ Total minutes
Contribution per minute = \(\$35 ÷ 31,200\) minutes
Let's calculate this:
\(\$35 ÷ 31,200 = \frac{35}{31,200}\)
To make this easier, let's simplify:
\(\frac{35}{31,200} = \frac{35}{31.2 \times 1,000} = \frac{35}{31.2} ÷ 1,000\)
\(35 ÷ 31.2 \approx 1.12\)
So: \(1.12 ÷ 1,000 = 0.00112\)
Therefore, Shannon's contribution per minute is approximately \(\$0.00112\)
4. Compare with answer choices for closest match
Our calculated value is \(\$0.00112\), and we need to find which answer choice is closest:
- \(\$0.001 = \$0.00100\)
- \(\$0.010 = \$0.01000\)
- \(\$0.025 = \$0.02500\)
- \(\$0.058 = \$0.05800\)
- \(\$0.067 = \$0.06700\)
Comparing \(\$0.00112\) to each choice:
• Distance from A (\(\$0.001\)): \(|0.00112 - 0.00100| = 0.00012\)
• Distance from B (\(\$0.010\)): \(|0.00112 - 0.01000| = 0.00888\)
• All other choices are much further away
Choice A (\(\$0.001\)) is clearly the closest to our calculated value of \(\$0.00112\).
Final Answer
Shannon's contribution per minute of listening time was approximately \(\$0.001\).
The answer is A.
Common Faltering Points
Errors while devising the approach
- Misinterpreting what "per minute" means: Students might think they need to find how much Shannon contributed each minute (as in a regular payment), rather than understanding this is asking for a rate calculation - total contribution divided by total listening time.
- Getting confused about time period consistency: The problem gives listening time in "hours per week" but asks for contribution "per minute." Students may struggle to recognize they need to convert the weekly listening time to total annual minutes before calculating the rate.
Errors while executing the approach
- Unit conversion errors: Students frequently make mistakes when converting 10 hours/week to total minutes per year. Common errors include forgetting to multiply by 52 weeks, or incorrectly converting hours to minutes (using 100 instead of 60 minutes per hour).
- Division calculation mistakes: When calculating \(35 ÷ 31,200\), students often struggle with the decimal placement or make arithmetic errors, especially when dealing with small decimal results like 0.00112.
- Incorrect order of operations: Some students might divide by hours first, then try to convert minutes, or perform the conversions in the wrong sequence, leading to incorrect intermediate results.
Errors while selecting the answer
- Misreading decimal places in answer choices: Students may confuse 0.001 with 0.01 or 0.1, especially under time pressure, not paying careful attention to the number of decimal places.
- Choosing an answer that "looks reasonable" without comparison: Instead of comparing their calculated value (0.00112) to each answer choice, students might select 0.01 because it "seems like a reasonable contribution per minute" without doing the proximity analysis.
