Question

In a cricket league match , there are both male and female members . If m is the average age of male members and f is the average age of female members , a is the total average age and $65$ percent of the leagues members are male , then which one of the following statements relate about m , f and a ?
A - If m < f , then $a>{\displaystyle \frac{m+f}{2}}$
B - If m > f , then $a<{\displaystyle \frac{m+f}{2}}$
C - If m < f , then $a<{\displaystyle \frac{m+f}{2}}$
D - $a={\displaystyle \frac{m+f}{2}}$

Answer 1

Hint:
First let us consider that the number of person in the cricket league is $100$ Use the average formula that is = ${\displaystyle \frac{\text{Sum of ages }}{\text{Total number of person}}}$ hence Sum of male ages $=m\times 65$ and Sum of female ages $=f\times 35$ from here a = ${\displaystyle \frac{m\times 65+f\times 35}{100}}$ now try insert ${\displaystyle \frac{m+f}{2}}$ in this equation because all options are given in this form after that compare it with options.

Complete step by step solution:
As we know that the average of any given data is,
= ${\displaystyle \frac{\text{Sum of ages }}{\text{Total number of person}}}$
So let us consider that the number of person in the cricket league is $100$ and it is given that the $65$ percent of the leagues members are male
Hence $65$ person are male and $35$ person are female ,
As it is given that m is the average of the male person , and total number of male is $65$
therefore m = ${\displaystyle \frac{\text{Sum of male ages }}{\text{Total number of male person}}}$
Sum of male ages $=m\times 65$
As it is given that f is the average of the female person , and total number of male is $35$
therefore f = ${\displaystyle \frac{\text{Sum of female ages }}{\text{Total number of female person}}}$
Sum of female ages $=f\times 35$

As it is given that a is the average of the total person and total number of person is $100$
a = ${\displaystyle \frac{\text{Sum of male ages + Sum of female ages }}{\text{Total number of person}}}$
a = ${\displaystyle \frac{m\times 65+f\times 35}{100}}$
Now divide in both numerator and denominator by $5$
$a={\displaystyle \frac{13m+7f}{20}}$
Now as in the option all are given in ${\displaystyle \frac{m+f}{2}}$ type so try to convert the above equation in this form

 $a={\displaystyle \frac{10m+10f+3m-3f}{20}}$
$a={\displaystyle \frac{10m+10f}{20}}+{\displaystyle \frac{3m-3f}{20}}$
On solving further we get it as
$a={\displaystyle \frac{m+f}{2}}+{\displaystyle \frac{3}{20}}(m-f)$
So if the $m>f$ , $a>{\displaystyle \frac{m+f}{2}}$
else if , $m<f$ , $a<{\displaystyle \frac{m+f}{2}}$

Hence from the above option C is the correct answer.

Note:
In an average type of question we will take any number for solving this as in this question we take $100$ person in the cricket league we also take $10,20$ etc any number but on choosing $100$ the calculation becomes easier.
In this type of question first see the option as in the option all the data is given in the relation of a , m and f. So only the total average age is given this relation so go through that and try to find out the relation between them.