- Translate the problem requirements: We have 8 integers with smallest = 5, range = 14 (so largest = 19). We need to find the difference between the maximum possible average and minimum possible average, then express this as a fraction of the range.
- Identify the constraints and boundaries: All 8 integers must be between 5 and 19 inclusive, with at least one integer equal to 5 and at least one equal to 19.
- Maximize the average strategically: To get the highest average, make as many numbers as possible equal to 19 (the maximum), while satisfying the constraint that the smallest must be 5.
- Minimize the average strategically: To get the lowest average, make as many numbers as possible equal to 5 (the minimum), while satisfying the constraint that the largest must be 19.
- Calculate the difference and express as fraction of range: Find the difference between max and min averages, then divide by the range (14) to get the final answer.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in simple terms:
- We have 8 integers (whole numbers)
- The smallest number is 5
- The range is 14, which means: \(\mathrm{largest\,number - smallest\,number = 14}\)
- So the \(\mathrm{largest\,number = 5 + 14 = 19}\)
- We need to find two things: the highest possible average and the lowest possible average
- Then we need to see what fraction of the range (14) their difference represents
Process Skill: TRANSLATE - Converting the problem's language into clear mathematical understanding
2. Identify the constraints and boundaries
Now let's understand our limitations:
- All 8 numbers must be between 5 and 19 (including 5 and 19)
- At least one number must equal 5 (the minimum)
- At least one number must equal 19 (the maximum)
- The remaining 6 numbers can be anywhere between 5 and 19
Think of it like this: we have 8 slots to fill, and we know the boundaries but have flexibility with 6 of the numbers.
Process Skill: APPLY CONSTRAINTS - Understanding the boundaries that limit our choices
3. Maximize the average strategically
To get the highest possible average, we want to make our numbers as large as possible:
- We must have one number equal to 5 (constraint)
- We can make the other 7 numbers all equal to 19
- So our 8 numbers would be: 5, 19, 19, 19, 19, 19, 19, 19
Let's calculate this maximum average:
\(\mathrm{Sum = 5 + (7 \times 19) = 5 + 133 = 138}\)
\(\mathrm{Maximum\,average = 138 \div 8 = 17.25}\)
4. Minimize the average strategically
To get the lowest possible average, we want to make our numbers as small as possible:
- We must have one number equal to 19 (constraint)
- We can make the other 7 numbers all equal to 5
- So our 8 numbers would be: 5, 5, 5, 5, 5, 5, 5, 19
Let's calculate this minimum average:
\(\mathrm{Sum = (7 \times 5) + 19 = 35 + 19 = 54}\)
\(\mathrm{Minimum\,average = 54 \div 8 = 6.75}\)
5. Calculate the difference and express as fraction of range
Now we find the difference between our maximum and minimum averages:
\(\mathrm{Difference = Maximum\,average - Minimum\,average}\)
\(\mathrm{Difference = 17.25 - 6.75 = 10.5}\)
The problem asks for this difference as a fraction of the range:
\(\mathrm{Fraction = Difference \div Range = 10.5 \div 14}\)
Let's simplify this fraction:
\(\mathrm{10.5 \div 14 = \frac{21}{2} \div 14 = \frac{21}{2} \times \frac{1}{14} = \frac{21}{28} = \frac{3}{4}}\)
Final Answer
The difference between the greatest possible average and the smallest possible average is \(\frac{3}{4}\) of the range.
This matches answer choice E. \(\frac{3}{4}\)
Verification: Our maximum average was 17.25, minimum was 6.75, difference is 10.5. The range is 14. So \(\frac{10.5}{14} = 0.75 = \frac{3}{4}\) ✓
Common Faltering Points
Errors while devising the approach
- Misunderstanding the constraint requirements: Students often fail to recognize that BOTH the minimum value (5) AND maximum value (19) must be present in the set. They might think they can use all 5s for minimum average or all 19s for maximum average, forgetting that the range constraint requires both endpoints to actually appear in the data.
- Confusing range definition: Some students may incorrectly interpret range as the difference between any two numbers in the set, rather than specifically (largest number - smallest number). This leads to incorrect identification of the maximum possible value as something other than 19.
- Misinterpreting the optimization goal: Students might not realize they need to find two separate scenarios - one that maximizes the average and another that minimizes it. They may try to find a single set of numbers that somehow satisfies both conditions simultaneously.
Errors while executing the approach
- Arithmetic errors in average calculation: When calculating averages like \(138\div8\) or \(54\div8\), students frequently make division errors, especially when dealing with decimal results like 17.25 and 6.75. These computational mistakes carry through to the final answer.
- Incorrect fraction simplification: Converting \(10.5\div14\) to \(\frac{3}{4}\) involves multiple steps (\(10.5 = \frac{21}{2}\), then \(\frac{21}{28}\), then \(\frac{3}{4}\)). Students often make errors when working with mixed decimals and fractions, or fail to properly reduce the final fraction.
- Wrong sum calculations: Students may incorrectly calculate sums like \(5 + (7\times19) = 138\) or \((7\times5) + 19 = 54\), particularly when one number appears multiple times. Counting errors or multiplication mistakes are common here.
Errors while selecting the answer
- Reporting the difference instead of the fraction: After correctly calculating that the difference between averages is 10.5, students might look for 10.5 among the answer choices instead of computing \(10.5\div14 = \frac{3}{4}\). They solve most of the problem correctly but miss the final step of expressing the result as a fraction of the range.
- Using the wrong denominator: Students might divide their difference by something other than the range (14). For example, they might divide by 8 (number of integers) or by one of the boundary values, leading to incorrect fractions that might coincidentally match wrong answer choices.
