Hi there!
In retrospect, I used to be surrounded by many crabby or “mean” bosses
or seniors while working at a trading company. It was a kind of nightmare
in my early twenties; however, I am flabbergasted to have faced the music to
settle the “mean” problem in mathematics a few decades later. Isn’t it funny?
Here comes the word problem today.
The following data set has a mean of 14 and no mode. All the six values
are positive integers, and the data are listed in ascending order. What is
the largest possible value for A?【Data Set】[A, B, C, 15, 18, 22]
Bear in mind:
1. mean = an average of n numbers computed by adding some function
of the numbers and dividing by some function of n
2. mode = the number which appears most often in a set of numbers.
Example) in {5, 2, 8, 5, 3, 6, 5, 7} the Mode is 5 (it occurs most often)
Here is my solution to this problem.
Let me use the formula for the mean \(\overline{x}\sum\frac{x}{n}\).
I substitute the values of this data set and get
\(14 = \frac{A + B + C + 15 + 18 + 22}{6}\),
which becomes \(84 = A + B + C + 55\), and finally 29 = A + B + C.
Now let’s look at the three values, A, B, and C.
First, since there is no mode and the values are in increasing order, the
value of C is no larger than 14, which would make the value of B no larger
than 13.
If C were 14 and B were 13, then A would be 2 since the sum of A, B, and C
must be 29. But there are other values that these three integers can take on:
C could be 13, B could be 12, and A would be 4.
Another option is that C is 12, B is 9, and A is 8, and these values result in a
maximum value for A. For the data set to have a mean of 14 and no mode,
the largest value that A can take on is 8.
I’ve reached the answer naturally because the positive integers mean the
natural numbers.
Wow, this silly joke has taken my coolness away.
Stay tuned, and expect to see my next post.
Keep well.
Frank Yoshida
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【グローバルビジネスで役立つ数学】でもっと学習する b＾＾)
【参考図書】『もう一度高校数学』（著者：高橋一雄氏）株式会社日本実業出版社
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