Question

The mean of 30 consecutive natural numbers is x and the mean of the odd numbers in these 30 natural numbers is y. If the smallest number is an odd number, find the value of x - y.





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Answer 1

Given,

Mean of 30 consecutive natural numbers $\[ = x\]$

Mean of odd numbers up to 30 $\[ = y\]$

Solution,

Calculate the value of x.

$\[x = \frac{{1 + 2 + 3 + 4 + 5 + - - - - + 28 + 29 + 30}}{{30}}\]$

$\[ \Rightarrow x = \frac{{465}}{{30}}\]$

$\[ \Rightarrow x = 15.5\]$

Calculate the value of y.

$\[y = \frac{{1 + 3 + 5 + 7 + 9 + - - - - 27 + 29}}{{15}}\]$

$\[ \Rightarrow y = \frac{{225}}{{15}}\]$

$\[ \Rightarrow y = 15\]$

Calculate the value of x - y.

$\[\begin{array}{l} = x - y\\ = 15.5 - 15\\ = 0.5\end{array}\]$

Hence the value of x - y is $\[0.5\]$

Answer 2

Given : The mean of 30 consecutive natural numbers is x and the mean of the odd numbers in these 30 natural numbers is y.

The smallest number is an odd number,

To find :  the value of x - y.​

Solution:

The smallest number is an odd number,

Assume 30 natural numbers are

2k+1 , 2k + 2 , ...   , 2k + 29 , 2k + 30

15 odd natural numbers in these are

2k + 1 , 2k + 3 , .... , 2k + 29  

where k is non negative integer

Sₙ = (n/2) (first term + nth Term)  

Mean =  ( first Term + nth term)/2

Mean of 30 natural numbers

x= (2k +1  + 2k + 30)/2

x = 2k + 15.5

Mean of 15 odd natural numbers

y= (2k +1  + 2k + 29)/2

y = 2k + 15

x  - y  = 2k + 15.5 - (2k + 15)

=> x - y = 0.5

the value of x - y is 0.5

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