Question

The sum of the value of Alpha, Beta & Gamma is 111. The average value of Beta, Gamma, Delta & Epsilon is 48 and the average value of Beta, Gamma &
Epsilon is 52. If the value of Epsilon is 72 then what is the average value of Alpha, Delta & Epsilon.
OPTIONS
a.
48
b.
52
w
C. 50
d
45​

Answer 1

Given that :-

$\red{\rm :\longmapsto\: \alpha + \beta + \gamma = 111}$

$\red{\rm :\longmapsto\: Average \: of \: \beta, \gamma,\delta \: and \in \: is \:48 }$

$\red{\rm :\longmapsto\: Average \: of \: \beta, \gamma\: and \in \: is \:52 }$

$\red{\rm :\longmapsto\: \in \: = \:72 }$

To find :-

$\blue{\bf :\longmapsto\: Average \: of \: \alpha,\delta \: and \in }$

Formula Used :-

$\boxed{ \sf \:Average = \dfrac{Sum \: of \: observations}{Number \: of \: observations}}$

Solution :-

We have

$\red{\bf :\longmapsto\: \alpha + \beta + \gamma = 111} - - - (1)$

Also,

$\red{\rm :\longmapsto\: Average \: of \: \beta, \gamma,\delta \: and \in \: is \:48 }$

$\rm :\longmapsto\:\dfrac{ \beta + \gamma + \delta + \in}{4} = 48$

$\red{\rm :\longmapsto\: \beta + \gamma + \delta + \in = 192} - - - (2)$

Also,

$\red{\rm :\longmapsto\: Average \: of \: \beta, \gamma\: and \in \: is \:52 }$

$\rm :\longmapsto\:\dfrac{ \beta + \gamma + \in}{3} = 52$

$\red{\rm :\longmapsto\: \beta + \gamma + \in = 156} - - - (3)$

Now,

$\red{\rm :\longmapsto\: \in \: = \:72 }$

So,

Equation (3) becomes,

${\rm :\longmapsto\: \beta + \gamma + 72 = 156}$

${\rm :\longmapsto\: \beta + \gamma= 156 - 72}$

$\boxed{\bf\implies \: \sf \: \beta + \gamma = 84} - - - (4)$

On substituting equation (3) in equation (2), we get

${\rm :\longmapsto\: \delta + 156= 192}$

${\rm :\longmapsto\: \delta= 192 - 156}$

$\boxed{\bf\implies \: \sf \:\delta = 36} - - - (5)$

On substituting equation (4) in equation (1), we get

${\bf :\longmapsto\: \alpha + 84 = 111}$

$\boxed{\bf\implies \: \sf \: \alpha = 27} - - - (6)$

Now, we have

$\red{\rm :\longmapsto\: \alpha = 27} \\ \red{\rm :\longmapsto\:\delta = 36} \\ \red{\rm :\longmapsto\:\in = 72}$

Hence,

$\blue{\bf :\longmapsto\: Average \: of \: \alpha,\delta \: and \in }$

$\rm \: = \: \: \dfrac{ \alpha + \delta + \in}{3}$

$\rm \: = \: \: \dfrac{27 + 36 + 72}{3}$

$\rm \: = \: \: \dfrac{135}{3}$

$\blue{\rm \: = \: \: 45}$

Therefore,

$\blue{\bf :\longmapsto\: Average \: of \: \alpha,\delta \: and \in \: is \: 45 }$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf \: Option \: (d) \: is \: correct}}$