Question

The average of all odd numbers up to 100 is:
A. 49
B. 49.5
C. 50
D. 51

Answer 1

Hint: Here we want to find the average of all odd numbers up to 100. First, find the sum of all odd numbers up to 100. After that divide the sum by the number of odd numbers up to 100 by the mean formula $\dfrac{{\sum {{x_i}} }}{n}$. Try it and you will get the answer.

Complete step by step solution:
Odd numbers are the numbers that cannot be divided into two separate groups evenly. These numbers are not completely divided by 2, which means there is some remainder left after division.
Odd numbers are defined as any number which cannot be divided by two. In other words, several forms $2k + 1$, where $k \in Z$ (i.e. integers) are called odd numbers. It should be noted that numbers or sets of integers on a number line can either be odd or even. A few more key points:
An odd number is an integer which is not a multiple of 2.
If these numbers are divided by 2, the result or remainder should be a fraction or 1.
In the number line, 1 is the first positive odd number.
The average of odd numbers till a given odd number is a simple concept. You just need to find odd numbers till that number then take their sum and divide by the number.
If an average of odd numbers till $n$ is to be found. Then we will find odd numbers from 1 to $n$, then add and divide it by the number of an odd number.
The average can be calculated by the formula of mean,
$\bar x = \dfrac{{\sum {{x_i}} }}{n}$
The number of odd numbers up to 100 is 50.
The sum of odd number up to 100 is,
$ \Rightarrow \sum {{x_i}} = 1 + 3 + 5 + \ldots + 99$
Pair the first and last term, second and second last term, and so on,
$ \Rightarrow \sum {{x_i}} = \left( {1 + 99} \right) + \left( {3 + 97} \right) + \ldots $ up to 25 terms
So, multiply any pair with 25 to get the sum of the odd numbers,
$ \Rightarrow \sum {{x_i}} = 100 \times 25$
Multiply the terms,
$ \Rightarrow \sum {{x_i}} = 2500$
Substitute the values in the mean formula,
$ \Rightarrow \bar x = \dfrac{{2500}}{{50}}$
Divide numerator and denominator,
$\therefore \bar x = 50$
So, the average of all odd numbers up to 100 is 50.

So, the correct answer is “Option C”.

Note: Mean is the arithmetic average of the data given. Mean is also used to calculate the variance and standard deviation of the data in statistics. The mean is affected by extremely high or low values, called outliers.