To find a set of five positive integers that satisfy the given conditions, let's break down and analyze each requirement step by step:
1. Mean of 4:
For a set of five numbers, the mean (average) is the sum of the numbers divided by 5. Therefore, the sum of the five numbers must be:
[tex]\[
\text{Sum} = \text{Mean} \times 5 = 4 \times 5 = 20
\][/tex]
2. Median of 6:
The median of a set of five numbers is the third number when the numbers are arranged in ascending order. Therefore, the third number in our set must be 6.
3. Mode of 6:
The mode is the number that appears most frequently in the set. Since the mode is 6, at least two out of the five numbers must be 6.
4. Range of 5:
The range is the difference between the largest and smallest numbers in the set. Hence, the largest number minus the smallest number must be 5.
Now, combining all these conditions:
- We need the sum to be 20.
- The third number (median) must be 6.
- At least two instances of 6 (for the mode).
- The largest minus the smallest number must be 5.
Let's consider a possible set:
We will denote the five numbers as [tex]\(a, b, c, d, e\)[/tex] such that [tex]\(a \leq b \leq c \leq d \leq e\)[/tex] and
we know:
[tex]\(c = 6\)[/tex] (since the median is 6),
there are at least two 6s for the mode condition (let's assume three 6s).
To satisfy the conditions:
Let's assume the numbers in the set are:
[tex]\[
[1, 1, 6, 6, 6]
\][/tex]
Check calculations:
- Sum: [tex]\( 1 + 1 + 6 + 6 + 6 = 20 \)[/tex]
- Mean: [tex]\(\frac{20}{5} = 4\)[/tex]
- Median: The third number is 6.
- Mode: 6 appears most frequently.
- Range: [tex]\(6 - 1 = 5\)[/tex]
Thus, the set of numbers [tex]\( [1, 1, 6, 6, 6] \)[/tex] meets all the given conditions. Let's list the results:
- Mean: 4.0
- Median: 6
- Mode: 6
- Range: 5
The set of five positive integers that meet all the criteria is [tex]\(1, 1, 6, 6, 6\)[/tex].
