Understanding the Impact of Multiplying Original Numbers on Their Average

The average (or arithmetic mean) of a set of numbers is a fundamental concept in mathematics and statistics. Let's explore a specific scenario involving the multiplication of original numbers and the resulting impact on their average.

Introduction

Given a set of ten original numbers, if each of these numbers is multiplied by 12, how does this affect their average? We will walk through the steps to calculate this change and see why the new average is simply the original average multiplied by 12.

Step-by-Step Calculation

Suppose the original ten numbers have an average of 7. This means the sum of these ten numbers is:

text{Sum of 10 original numbers} 10 times 7 70 end{latex}

Now, if each of these ten original numbers is multiplied by 12, the sum of the new numbers will be:

text{New sum} 12 times 70 840 end{latex}

The new average of these 10 numbers is then calculated by dividing this new sum by 10:

text{New average} frac{840}{10} 84 end{latex}

This shows that the average of the new numbers is indeed 84, which is the same as the original average (7) multiplied by 12.

Mathematical Insight

Mathematically, let's denote the original ten numbers as (a, b, c, d, e, f, g, h, i, j). The average of these numbers is:

text{Average} frac{a b c d e f g h i j}{10} 7 end{latex}

If each number is multiplied by 12, the new numbers will be (12a, 12b, 12c, 12d, 12e, 12f, 12g, 12h, 12i, 12j). The new average will be:

text{New average} frac{12a 12b 12c 12d 12e 12f 12g 12h 12i 12j}{10} end{latex}

This can be simplified as:

text{New average} frac{12(a b c d e f g h i j)}{10} end{latex}

Substituting (a b c d e f g h i j 70) (from the original average calculation), we get:

text{New average} frac{12 times 70}{10} 84 end{latex}

This confirms that the new average is 84, which is the original average (7) multiplied by 12.

Conclusion

The key insight here is that when each number in a set is multiplied by a constant, the new average is the original average multiplied by that same constant. In our example, multiplying each of the original ten numbers by 12 results in a new average of 84. This property holds true and can be very useful in many mathematical and statistical applications.

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