A given set consists of precisely 7 ordered positive integers that satisfy each of the following conditions:
• The arithmetic mean of the set is equal to 4.
• The mode of the set is 1.
• The median of the set is 5, which occurs only once.
• The range of the set is 6.
Determine the 7 ordered positive integers in the set.
Solution to the Problem:
The median of the set is 5, which occurs only once.Therefore, the set must be:
* * * 5 * * *
The range of the set is 6 and the mode of the set is 1.
Thus, the maximum number is 1+6=7
Since the digit 5 occurs only once, we need to have at least two 1s for the mode to equal 1.
Hence, the set is:
1 1 * 5 * * 7
This implies that:
(5th position, 6th position) = (6, 6), or: (6, 7)
In the former case, the given set admits of precisely two 6s.
In the latter case, the given set admits of exactly two 7s
Therefore, for the mode to equal 1, we require one more 1 (in the 3rd place).
So, the set is:
Either: 1 1 1 5 6 6 7
Or : 1 1 1 5 6 7 7
The former case gives a total of 27, so that the mean is NOT equal to 4. This leads to a contradiction.
The latter case gives a total of 28, so that the mean is equal to 4.
Consequently, the required ordered set of 7 positive integers is given by:
{1, 1, 1, 5, 6, 7, 7}
Benny Varghese gets extra credit for coming up with an alternate solution. He writes:
"I can see 2 answers to this question. The perfect answer is : 1, 1, 1, 5, 6, 7, 7 where the mean is 4, the median is 5, and the range is 6. In this case, there is only one mode, which is 1.
But I feel this is also a valid answer: 1, 1, 2, 5, 6, 6, 7 where the mean is 4, the median is 5, and the range is 6. In this case, there are two modes: 1 and 6. This is a bimodal set."
Correctly solved by:
| 1. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 2. Davit Banana | Istanbul, Turkey |
| 3. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 4. Fulin Sun | Seattle, Washington |
| 5. Benny Varghese | Cochin, Kerala, India |
| 6. Mrs. Painter’s Honors Precalculus Class |
Broome High School, Spartanburg, South Carolina |
| 7. Ritwik Chaudhuri | Santiniketan, West Bengal, India |
| 8. Ivy Joseph | Pune, Maharashtra, India |
| 9. Corbin Carey |
Central High School, Grand Junction, Colorado |