Three Positive Integers Have A Mean Of 4 And A Range Of 7. Find The Three Numbers.
Understanding how to find three positive integers based on their mean and range involves applying fundamental concepts of algebra, number properties, and problem-solving strategies. This type of problem appears frequently in math competitions, standardized tests, and classroom exercises, making it a valuable example for students learning how to analyze and solve word problems involving integers.
In this comprehensive article, we will explore the step-by-step process of determining the three positive integers given that their mean is 4 and their range is 7. We'll delve into key mathematical concepts, provide detailed explanations, and include illustrative examples to clarify the solution process. Whether you're a student preparing for exams or someone interested in sharpening your problem-solving skills, this guide will help you understand the underlying principles and improve your ability to tackle similar problems.
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Understanding the Problem
Before diving into calculations, it's essential to understand what the problem is asking and what information is provided.
Given:
- The three numbers are positive integers.
- Their mean (average) is 4.
- Their range is 7.
Goals:
- Find the three integers.
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Key Concepts Involved
To solve this problem, several mathematical concepts are involved:
1. Mean (Average)
The mean of three numbers \(a\), \(b\), and \(c\) is calculated as: \[ \text{Mean} = \frac{a + b + c}{3} \] Given the mean is 4: \[ \frac{a + b + c}{3} = 4 \] which implies: \[ a + b + c = 12 \]2. Range
The range of a set of numbers is the difference between the largest and smallest numbers: \[ \text{Range} = \text{Maximum} - \text{Minimum} \] Given the range is 7: \[ \text{Maximum} - \text{Minimum} = 7 \]3. Positivity of Integers
All numbers \(a\), \(b\), and \(c\) are positive integers, i.e., greater than 0.---
Step-by-Step Solution Approach
To find the three integers, we need to systematically analyze the constraints and possible scenarios.
Step 1: Assign Variables to the Numbers
Let:- \(a\) = the smallest positive integer,
- \(c\) = the largest positive integer,
- \(b\) = the middle number (which could be either the smallest, middle, or largest depending on the values).
Given that the sum of all three is 12:
\[
a + b + c = 12
\]
And the range:
\[
c - a = 7
\]
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Step 2: Express the Variables in Terms of One Another
From the range: \[ c = a + 7 \]Substitute into the sum:
\[
a + b + (a + 7) = 12
\]
which simplifies to:
\[
2a + b + 7 = 12
\]
\[
b = 12 - 7 - 2a = 5 - 2a
\]
Now, since \(a\), \(b\), and \(c\) are positive integers, and \(a \leq b \leq c\), the following conditions must be satisfied:
- \(a \geq 1\) (positive integers)
- \(b \geq a\)
- \(b \leq c = a + 7\)
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Step 3: Determine Valid Values for \(a\)
Because \(b = 5 - 2a\), and \(b\) must be positive:
\[
b \geq 1
\]
\[
5 - 2a \geq 1
\]
\[
-2a \geq -4
\]
\[
2a \leq 4
\]
\[
a \leq 2
\]
Also, \(a \geq 1\), so possible values for \(a\) are:
- \(a = 1\)
- \(a = 2\)
Let's analyze each case separately.
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Case 1: \(a = 1\)
Compute \(b\):
\[
b = 5 - 2(1) = 5 - 2 = 3
\]
Compute \(c\):
\[
c = a + 7 = 1 + 7 = 8
\]
Check if the numbers satisfy the ordering:
\[
a = 1, \quad b = 3, \quad c = 8
\]
Order: \(1 \leq 3 \leq 8\) — valid.
Verify the sum:
\[
1 + 3 + 8 = 12
\]
And the range:
\[
8 - 1 = 7
\]
All conditions are met.
---
Case 2: \(a = 2\)
Compute \(b\):
\[
b = 5 - 2(2) = 5 - 4 = 1
\]
Compute \(c\):
\[
c = 2 + 7 = 9
\]
Check order:
\[
a = 2, \quad b = 1, \quad c = 9
\]
Order: \(1 \leq 2 \leq 9\)? No, because \(b = 1\) is less than \(a = 2\).
But since in the original assignment, \(a\) was defined as the smallest number, the order should be \(a \leq b \leq c\). Here, \(b = 1\) is less than \(a=2\), which violates the ordering.
Therefore, this set is invalid unless we reassign the labels, but since the problem specifies the numbers are positive integers with a specific range and mean, and the smallest is \(a\), the set with \(a=2\), \(b=1\), \(c=9\) does not satisfy the ordering assumption.
Conclusion: For the purpose of the problem, the set \((1, 3, 8)\) is valid, and the set with \(a=2\) is invalid because it violates the assumption that \(a\) is the smallest.
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Final Solution and Valid Set of Numbers
Based on the above analysis, the only valid set of three positive integers satisfying the conditions is:
\(\boxed{1, 3, 8}\)
- Sum check: \(1 + 3 + 8 = 12\), which yields a mean of \(\frac{12}{3} = 4\).
- Range check: \(8 - 1 = 7\).
Thus, the three numbers are 1, 3, and 8.
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Additional Insights and Variations
While the specific problem yields a unique set of numbers under the assumptions, similar problems can lead to multiple solutions or require different approaches.
1. Considering Different Orderings
If the problem does not specify the order of the numbers, permutations may be valid as long as they satisfy the sum and range constraints.2. Extension to Other Averages and Ranges
Changing the mean or range values introduces different equations and solutions. For example:- If the mean were 5, the sum would be \(3 \times 5 = 15\).
- If the range were 5 or 10, similar algebraic steps could be used.
3. General Approach for Similar Problems
- Assign variables to the minimum and maximum.
- Express the middle number in terms of the other variables.
- Use the sum and range constraints to find possible values.
- Verify the ordering and positivity.
Conclusion
In solving the problem of finding three positive integers with a mean of 4 and a range of 7, a systematic approach involving algebraic expressions and logical reasoning is essential. By defining variables, establishing relationships, and applying constraints, the solution reveals that the three numbers are 1, 3, and 8. This problem exemplifies how combining basic properties of numbers with algebraic techniques can lead to clear, logical solutions.
Understanding these methods enhances problem-solving skills and prepares students to tackle a wide range of similar mathematical challenges. Whether for academic purposes or personal interest, mastering such problems deepens comprehension of number properties and algebraic reasoning.
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Keywords: positive integers, mean, range, algebra, problem-solving, mathematical reasoning, number properties, equations, sets of numbers
