To determine a possible set of 5 numbers on Marley's cards satisfying the given conditions (median is 9, mode is 12, and range is 7), let's go through it step-by-step:
1. Median Calculation:
- The median of a set with 5 numbers is the third number when they are arranged in ascending order.
- Therefore, the third number must be 9. This means, when sorted, the set looks like [tex]\([a, b, 9, c, d]\)[/tex].
2. Mode Calculation:
- The mode is the number that appears most frequently in the set.
- Given that the mode is 12, the number 12 must appear at least twice in the set.
3. Range Calculation:
- The range is calculated as the difference between the maximum and minimum numbers in the set.
- The range given is 7. Therefore, if the smallest number is [tex]\(a\)[/tex], then the largest number is [tex]\(a + 7\)[/tex].
4. Constructing the Set:
- With the above points in mind, let's construct a possible set of numbers:
- Since the sorted set is [tex]\([a, b, 9, c, d]\)[/tex], and 12 being the mode, we need to place 12 in such a way that it appears at least twice.
- This possible placement can be [tex]\([a, 12, 9, 12, d]\)[/tex].
- To satisfy the range of 7, if [tex]\(a\)[/tex] is the smallest number, [tex]\(d\)[/tex] will be [tex]\(a + 7\)[/tex].
5. Choosing Numbers:
- Let's choose [tex]\(a = 5\)[/tex] (as it needs to be lower than 9 and also the smallest to make calculations straightforward).
- Given [tex]\(a = 5\)[/tex], the largest number [tex]\(d\)[/tex] would be [tex]\(5 + 7 = 12\)[/tex].
6. Final Set:
- So now, the set looks like: [tex]\([5, 12, 9, 12, 12]\)[/tex].
- This set satisfies:
- Median: The third number is 9.
- Mode: 12 appears the most frequently.
- Range: The difference between the highest (12) and lowest (5) numbers is 7.
Therefore, a possible set of numbers on Marley's cards is:
[tex]\[ \{5, 12, 9, 12, 12\} \][/tex]
