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 $a > \dfrac{{m + f}}{2}$
B - If m > f , then $a < \dfrac{{m + f}}{2}$
C - If m < f , then $a < \dfrac{{m + f}}{2}$
D - $a = \dfrac{{m + f}}{2}$

Answer 1

Hint:
First let us consider that the number of person in the cricket league is $100$ Use the average formula that is = $\dfrac{{{\text{Sum of ages }}}}{{{\text{Total number of person}}}}$ hence Sum of male ages $ = m \times 65$ and Sum of female ages $ = f \times 35$ from here a = $\dfrac{{m \times 65 + f \times 35}}{{100}}$ now try insert $\dfrac{{m + f}}{2}$ 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,
= $\dfrac{{{\text{Sum of ages }}}}{{{\text{Total number of person}}}}$
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 = $\dfrac{{{\text{Sum of male ages }}}}{{{\text{Total number of male person}}}}$
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 = $\dfrac{{{\text{Sum of female ages }}}}{{{\text{Total number of female person}}}}$
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 = $\dfrac{{{\text{Sum of male ages + Sum of female ages }}}}{{{\text{Total number of person}}}}$
a = $\dfrac{{m \times 65 + f \times 35}}{{100}}$
Now divide in both numerator and denominator by $5$
$a = \dfrac{{13m + 7f}}{{20}}$
Now as in the option all are given in $\dfrac{{m + f}}{2}$ type so try to convert the above equation in this form

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

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.