Question

Explain by taking a suitable example, how the arithmetic mean alters by multiplying each term by a constant $k$.

Answer 1

Hint:
Here we have to check what will happen to the arithmetic mean when we multiply each term by a constant. For that we will take an example in which we will take five numbers and we will find the arithmetic mean of these five numbers by first finding its sum and then dividing the sum by total number of terms. We will multiply each of the terms by a constant and then we will find the sum of these terms and then we will divide it by the number of terms to calculate its arithmetic mean again. We will find that the resulting arithmetic mean will be a product of constant and the original arithmetic mean.

Complete step by step solution:
Let’s first take the example to explain it.
Let the numbers be$10,5,20,20\And 15$
Here, the number of terms $=5$
Now, we will find the arithmetic mean of these numbers.
Formula for arithmetic mean $=\dfrac{\text{sum of numbers}}{\text{number of terms}}$
Now, we will find the sum of numbers.
Sum of numbers $=10+5+20+20+15$
$=70$
Putting value of sum of numbers and number of terms in the formula of arithmetic mean, we get
Arithmetic mean $=\dfrac{70}{5}$
We will divide 70 by 5 now.
Arithmetic mean $=14$
So the arithmetic mean is $14$
Now we will multiply each term by a constant. So let’s take $4$ as a constant.
We will multiply each term by $4$ now.
So the terms after multiplication will be
$10\times 4,5\times 4,20\times 4,20\times 4\And 15\times 4=40,20,80,80\And 60$
Now, the terms are $40,20,80,80\And 60$
Here the number of terms is the same i.e. 5.
Now, we will find the sum of numbers.
Sum of numbers $=40+20+80+80+60$
$=280$
Putting value of sum of numbers and number of terms in the formula of arithmetic mean, we get
Arithmetic mean $=\dfrac{280}{5}$
We will divide 280 by 5 now.
Arithmetic mean $=56$
So the arithmetic mean is $56$
56 can also be written as $4\times 14=4\times \text{original arithmetic mean}$

We can say that the arithmetic mean after multiplying each term by a constant will be equal to constant times the original arithmetic mean.

Note:
Since we have obtained arithmetic mean here, so we need to know its definition.
An arithmetic mean is also called as average or simply mean and it is defined as the ratio of the sum of terms to the number of terms. An arithmetic mean is easy to understand and it is easy to use.