Question

Find the mean of the first five odd multiples of \[5\].

Answer 1

Hint: First of all, note down all the first ten multiples of \[5\], then note the first five odd multiples of \[5\]. After this, to find their mean we can simply find the summation of the first five multiples of \[5\] and then divide them by \[5\] as the summation done is of five values.

Complete step-by-step answer:
We will first write down the first \[10\] multiples of \[5\].
They are \[5,10,15,20,25,30,35,40,45,50\].
Now out of these the first five odd multiples are \[5,15,25,35,45\].
As we have identified the first five odd multiples of \[5\] we can find its mean. To find its mean we will first have to find the summation of the multiples. Their summation is \[5 + 15 + 25 + 35 + 45 = 125\].
Now, to find the mean of five numbers we divide their summation by five.
This will give us \[\dfrac{{125}}{5} = 25\].
Thus, the mean of the first five multiples of \[5\] is \[25\].

Note: Here, it must be noted that the reason why we divide the summation of the first five odd multiples of \[5\] by \[5\] is because there are five values considered, not because they are multiples of \[5\].
Another simple method could be by taking \[5\] common from the summation and then dividing.
First five odd multiples will be given by \[5 \times 1,5 \times 3,5 \times 5,5 \times 7\] and \[5 \times 9\]. Now when we add these multiples we can simply take out \[5\] as common and then divide.
The summation will be written as \[5 \times 1 + 5 \times 3 + 5 \times 5 + 5 \times 7 + 5 \times 9 = 5(1 + 3 + 5 + 7 + 9)\]
This when solved will give \[5(25)\], now when we divide the summation by \[5\], both the \[5\] get cancelled from the numerator and denominator.
On solving it gives,
\[\dfrac{{5(25)}}{5} = 25\]
Thus, the mean will be \[25\].
This method is not completely different from the previous one, but it still decreases the time to solve up to some extent as the numbers you have to add here are much smaller.