Frank
Hi, this is Frank.
Thinking back, when I was young and working at a trading company, I was always surrounded by grumpy and “mean” supervisors and senior colleagues. For me in my early twenties, that situation was nothing short of a nightmare.
But surprisingly, decades later, I would face a completely different kind of “mean” in the world of mathematics—the average! Life truly has a humorous way of surprising us.
Now, let’s introduce today’s problem.
【Problem】
The mean of the following dataset is 14, and there is no mode. All six numbers are positive integers listed in ascending order. What is the maximum value A can take? The dataset is [A, B, C, 15, 18, 22]
First, let’s review the basics.
1. mean = sum of numbers divided by the count
2. mode = the number that appears most frequently
Example: {5, 2, 8, 5, 3, 6, 5, 7} → Mode is 5
Now, let’s solve the problem.
【Solution】
Using the formula for the mean \(\overline{x}=\sum x/n\) and substituting the data:
\(14 = \frac{A + B + C + 15 + 18 + 22}{6}\)
Simplifying gives:
\(84 = A + B + C + 55\)
So A + B + C = 29.
Next, consider the relationships among A, B, and C.
Since the dataset is in ascending order and has no mode, no numbers can be repeated. Therefore:
* C must be ≤ 14
* B must be ≤ 13
If C = 14 and B = 13, then A = 2 to satisfy the sum.
This works.
Other combinations are possible too, for example:
* C = 13, B = 12, A = 4
Furthermore,
* C = 12, B = 9, A = 8
Here, A reaches its maximum value of 8.
Given that A, B, and C are positive integers, this is the natural solution.
Ah, a little “mean” joke in mathematics seems to have stolen some of my coolness today.
Stay tuned for the next article, and take care!
Learn more at Global Business Math b＾＾)
Reference book: ‘High School Math Revisited‘ by Kazuo Takahashi, Japan Jitsugyo Publishing
For online English lessons, see here.
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