Question

Find the mean of the first six multiples of $5$.

Answer 1

First of all, we will find the first six multiples of $5$ . Then, we will apply the formula of arithmetic mean on the first six multiples of $5$ . Then, by simplifying further we can evaluate the mean of the first six multiples of $5$.

Formula used: The arithmetic mean of ungrouped data: If ${x_1},{x_2},{x_3},...,{x_n}$ are $n$ observations of a variable $X$ , then the arithmetic mean is denoted by $\bar X$ and is defined as $\bar X = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n}$.

Complete step by step solution:
Firstly, we know that a multiple of a natural number is obtained by multiplying that number by any whole number.
Now, we need to find the first six multiples of $5$, which are
Multiples of $5$ are $5 \times 0$ , $5 \times 1$ , $5 \times 2$ , $5 \times 3$ , $5 \times 4$ , $5 \times 5$ , $5 \times 6$ , …
i.e. $0$ , $5$ , $\;10$ , $\;15$ , $\;20$ , $\;25$ , $\;30$ , …
But, we will consider non-zero multiples only, so, we have
First six multiples of $\;5$ : $5$ , $\;10$ , $\;15$ , $\;20$ , $\;25$ , $\;30$
Now, as we need to find the mean of the first six multiples of $5$ .
So, we take it as
$X$ : $5$ , $\;10$ , $\;15$ , $\;20$ , $\;25$ , $\;30$
Now, let $\bar X$ be the arithmetic mean of the observations.
As we know that, $\bar X = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n}$
$\Rightarrow \bar X = \dfrac{{5 + 10 + 15 + 20 + 25 + 30}}{6}$
$\Rightarrow \bar X = \dfrac{{105}}{6}$
Simplifying on R.H.S., we get
$\Rightarrow \bar X = 17.5$

The mean of the first six multiples of $5$ is $17.5$.

Note: Arithmetic mean is also called simply mean. The arithmetic mean of a set of observations is defined as the sum of all observations divided by the total number of observations. The method of finding the arithmetic mean depends on the kind of data that is given whether the data is grouped or ungrouped.