Question

If the mean of ten consecutive odd numbers is 120 then what is the mean of the first five odd numbers among them?

A) 113
B) 115
C) 114
D) 116

Answer 1

Hint: You need to assume any ten consecutive odd number in terms of x. Then, apply the formula: Mean of observations equals to the sum of all the observations divided the total number of observations.

Complete step by step solution: First, we need to assume any ten consecutive odd numbers in terms of x that is:

Let ten consecutive odd numbers be $2x + 1,2x + 3,..........................,2x + 19$
It is given that mean of ten consecutive odd numbers is 120

We will use the formula:

\[\boxed{Mean{\text{ }}of{\text{ observations = }}\dfrac{{{\text{Sum of all the observations}}}}{{{\text{Total no}}\;{\text{of\;observations}}}}}\]
Now, Mean of ten consecutive odd
\[{\text{Numbers = }}\dfrac{{{\text{Sum of all the odd consecutive numbers}}}}{{{\text{ Total number of odd consecutive number}}}}{\text{ }}\]
$\dfrac{{ = \left( {2x + 1} \right) + \left( {2x + 3} \right) + \left( {2x + 5} \right) + .................................. + \left( {2x + 17} \right) + \left( {2x + 19} \right)}}{{10}}$
$ = \dfrac{{20x + \left( {1 + 3 + 5 + 7 + .............. + 19} \right)}}{{10}}$
But mean of 10 consecutive odd number $ = \dfrac{{20x + 100}}{{10}}$
$ \Rightarrow \;\dfrac{{20x + 100}}{{10}} - 1200$
$ \Rightarrow \;20x + 100 - 200$
$ \Rightarrow 20x - 1200 = 100$
$ \Rightarrow \; 20x = 1100$
$ \Rightarrow $ Solving for x, we get
$ \Rightarrow \; x = \dfrac{{1100}}{{20}} = 55$
$\therefore \;\boxed{x = 55}$

∴ Our ten consecutive odd number are :
$2x + 1, 2x + 3, 2x + 5,................................., 2x + 19$
That is : $111,113,115,..........................129$

We are asked to find the mean of the first five odd consecutive numbers.

The first five odd consecutive numbers are: $111,113,115,117,119$ .
$\therefore $ Mean of first five odd \[{\text{consecutive number = }}\dfrac{{{\text{Sum of number}}}}{{\text{5}}}{\text{ }}\]
$ = \dfrac{{111 + 113 + 115 + 117 + 119}}{5}$
$ = \dfrac{{575}}{5}$
$ = 115$
Mean of first five consecutive odd numbers $ = 115$

∴ Correct option (B).

Note: We assumed our first odd number as $2x + 1$ because we know as $x$ can be any number, $2x$ is always even number & if we add $1\;{\text{to}}\;{\text{2x}}$ , it will become odd number; that is $2x + 1$ is always a odd number so, it is safe to assume $2x + 1$ as our first odd number. You must also be very careful in doing the calculations as you might end up making a silly mistake.