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When each of the athletes at a sports event was given the same number of sports drinks from a crate containing 80 sports drinks, the number of sports drinks left in the crate was 4 less than the number of athletes at the sports event. Which of the following CANNOT be the number of athletes at the sports event?
Let's break down what's happening in plain English:
We start with a crate containing 80 sports drinks. These drinks are distributed equally among all the athletes - meaning each athlete gets the exact same number of drinks. After this equal distribution, some drinks are left over in the crate.
The key relationship we're told is: the number of leftover drinks = (number of athletes - 4)
So if there are 12 athletes, then there should be \(12 - 4 = 8\) drinks left over.
If there are 21 athletes, then there should be \(21 - 4 = 17\) drinks left over.
We need to find which of the given options cannot work as the number of athletes.
Process Skill: TRANSLATE - Converting the word problem into a clear mathematical relationship
Let's think about this step by step using everyday division:
When we divide 80 sports drinks equally among athletes, we can write this as:
\(80 = (\mathrm{number\ of\ athletes}) \times (\mathrm{drinks\ per\ athlete}) + (\mathrm{leftover\ drinks})\)
From the problem, we know:
\(\mathrm{leftover\ drinks} = \mathrm{number\ of\ athletes} - 4\)
Substituting this into our equation:
\(80 = (\mathrm{number\ of\ athletes}) \times (\mathrm{drinks\ per\ athlete}) + (\mathrm{number\ of\ athletes} - 4)\)
Let's call the number of athletes 'n' for simplicity:
\(80 = n \times (\mathrm{drinks\ per\ athlete}) + (n - 4)\)
\(80 = n \times (\mathrm{drinks\ per\ athlete}) + n - 4\)
\(84 = n \times (\mathrm{drinks\ per\ athlete}) + n\)
\(84 = n \times (\mathrm{drinks\ per\ athlete} + 1)\)
This means that n (the number of athletes) must be a divisor of 84.
Now we need to check if each option can actually divide 84 evenly. Let's find the factors of 84 first:
\(84 = 4 \times 21 = 2^2 \times 3 \times 7\)
The factors of 84 are: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Let's verify each answer choice:
Choice A: 12 athletes
• If n = 12, then each athlete gets \((84 \div 12) - 1 = 7 - 1 = 6\) drinks
• Total drinks used: \(12 \times 6 = 72\)
• Leftover: \(80 - 72 = 8\)
• Required leftover: \(12 - 4 = 8\) ✓
Choice B: 14 athletes
• If n = 14, then each athlete gets \((84 \div 14) - 1 = 6 - 1 = 5\) drinks
• Total drinks used: \(14 \times 5 = 70\)
• Leftover: \(80 - 70 = 10\)
• Required leftover: \(14 - 4 = 10\) ✓
Choice C: 21 athletes
• If n = 21, then each athlete gets \((84 \div 21) - 1 = 4 - 1 = 3\) drinks
• Total drinks used: \(21 \times 3 = 63\)
• Leftover: \(80 - 63 = 17\)
• Required leftover: \(21 - 4 = 17\) ✓
Choice D: 24 athletes
• 24 is NOT a factor of 84 \((84 \div 24 = 3.5)\)
• Since 24 doesn't divide 84 evenly, this scenario is impossible ✗
Choice E: 28 athletes
• If n = 28, then each athlete gets \((84 \div 28) - 1 = 3 - 1 = 2\) drinks
• Total drinks used: \(28 \times 2 = 56\)
• Leftover: \(80 - 56 = 24\)
• Required leftover: \(28 - 4 = 24\) ✓
From our systematic testing, we found that 24 athletes cannot work because 24 is not a factor of 84.
When we have a whole number of athletes and each gets the same whole number of drinks, the number of athletes must divide evenly into our derived value of 84.
Since \(84 \div 24 = 3.5\), this would mean each athlete gets 2.5 drinks, which contradicts the requirement that each athlete gets the same whole number of drinks.
Process Skill: APPLY CONSTRAINTS - Recognizing that the number of drinks per athlete must be a whole number
The answer is D. 24
24 cannot be the number of athletes because it would require distributing a fractional number of drinks to each athlete, which violates the constraint that each athlete receives the same whole number of sports drinks.
1. Misinterpreting the constraint about leftover drinks
Students often misread "the number of sports drinks left in the crate was 4 less than the number of athletes" as "4 drinks were left over" or "each athlete got 4 fewer drinks than expected." This fundamental misunderstanding of the relationship (leftover = athletes - 4) leads to setting up the wrong equation entirely.
2. Overlooking the "same number" constraint
Students may miss that each athlete gets the "same number" of sports drinks, meaning the drinks per athlete must be a whole number. This constraint is crucial because it means the total athletes must divide evenly into our derived equation, but students often skip this restriction.
3. Setting up division incorrectly
When translating "80 drinks distributed equally among athletes with some leftover," students frequently write \(80 \div \mathrm{athletes} = \mathrm{drinks\ per\ athlete}\), forgetting to account for the leftover drinks in their initial setup. They miss that the correct relationship is: \(80 = (\mathrm{athletes} \times \mathrm{drinks\ per\ athlete}) + \mathrm{leftover}\).
1. Arithmetic errors when finding factors of 84
Students often make calculation mistakes when determining if \(84 \div 24 = 3.5\), or when finding the complete list of factors of 84. They might incorrectly conclude that 24 is a factor of 84, or miss checking whether each answer choice actually divides 84 evenly.
2. Verification calculation errors
When checking each answer choice, students frequently make mistakes in the multi-step verification: calculating drinks per athlete, total drinks used, and comparing actual vs. required leftover. For example, with 12 athletes, they might incorrectly calculate \(12 \times 6 = 70\) instead of 72.
1. Selecting a choice that works instead of one that CANNOT work
Students often miss the word "CANNOT" in the question stem and select an answer choice that actually works (like 12, 14, 21, or 28) instead of the one that's impossible (24). This is a classic error where students solve correctly but answer the wrong question.