The average of several numbers is 7. The largest number was decreased by 24, and the smallest was increased by 24. Will the arithmetic mean change?

Solution:

Let the numbers be $$x_1, x_2, ..., x_n$$. The average is $$\frac{\sum_{i=1}^{n} x_i}{n} = 7$$.

Let $$x_{min}$$ be the smallest number and $$x_{max}$$ be the largest number.

The sum of the numbers is $$\sum_{i=1}^{n} x_i = 7n$$.

When the largest number is decreased by 24 and the smallest is increased by 24, the new sum becomes:

$$ \sum_{i=1}^{n} x_i - x_{max} + (x_{max} - 24) + x_{min} + (x_{min} + 24) - x_{min} - x_{max} = \sum_{i=1}^{n} x_i - 24 + 24 = \sum_{i=1}^{n} x_i$$

The new sum is $$(\sum_{i=1}^{n} x_i) - x_{max} + (x_{max} - 24) + x_{min} + (x_{min} + 24) - x_{min} - x_{max} = \sum_{i=1}^{n} x_i - x_{max} + (x_{max} - 24) + x_{min} + (x_{min} + 24) - x_{min} - x_{max} = \sum_{i=1}^{n} x_i$$.

The new sum is $$(\sum_{i=1}^{n} x_i) - x_{max} + (x_{max} - 24) + x_{min} + (x_{min} + 24) - x_{min} - x_{max}$$.

The change in the sum is $$(x_{max} - 24) + (x_{min} + 24) - x_{max} - x_{min} = x_{max} - 24 + x_{min} + 24 - x_{max} - x_{min} = 0$$.

Since the sum of the numbers does not change, the arithmetic mean will also not change.

However, this is only true if there are at least two numbers and the operation is performed on the absolute minimum and maximum values. If the number of elements is 1, then the minimum and maximum are the same, and the operation would be $$x_1 - 24$$ and $$x_1 + 24$$, which is not possible. If the numbers are identical, then $$x_{min}=x_{max}$$, and the change would be $$x_1 - 24 + x_1 + 24 = 2x_1$$.

Let's consider the change in the sum:

Original sum: $$S = \sum x_i$$

New sum: $$S' = S - x_{max} + (x_{max} - 24) - x_{min} + (x_{min} + 24) = S - 24 + 24 = S$$.

The sum remains the same. Therefore, the arithmetic mean remains the same.

• Question analyzed: Mathematical problem concerning the arithmetic mean.
• Category: Mathematics.
• Class: Likely middle school or early high school (based on concept of average).
• Concept: Arithmetic mean, properties of addition and subtraction.

Explanation:

• When a value is decreased by a certain amount and another value is increased by the same amount, the total sum of the numbers remains unchanged.
• Since the sum of the numbers is unchanged, and the number of elements ($$n$$) is also unchanged, the arithmetic mean (which is the sum divided by the number of elements) will also remain unchanged.

Final Answer:

• Yes, the arithmetic mean will not change.
• Reasoning: The net change to the sum of the numbers is zero (a decrease of 24 is exactly offset by an increase of 24). Since the sum and the count of numbers remain the same, the average will remain the same.

The original OCR text mentions '9 out of 11', suggesting this is part of a quiz or test. The options provided in the OCR (Да, Нет, Определить невозможно) are 'Yes', 'No', 'Impossible to determine'. Based on the mathematical logic, the answer is 'Yes'.