The Gateway to Computer Science Excellence
+3 votes
348 views

The product of three integers $X$, $Y$ and $Z$ is $192$. $Z$ is equal to $4$ and $P$ is equal to the average of $X$ and $Y$. What is the minimum possible value of $P$?

  1. $6$
  2. $7$
  3. $8$
  4. $9.5$
in Numerical Ability by
edited by | 348 views
Migrated from GO Mechanical 1 year ago by Arjun

3 Answers

+2 votes

Given

$X*Y*Z = 192$,

$Z=4,$

$P = \left ( \frac{X+Y}{2} \right )$


Solution

$X*Y*Z = 192$

$\Rightarrow$ $X*Y*4 = 192$

$\Rightarrow$ $X*Y = 48$

Now $48$ should be broken into product of 2 numbers such that they are close to each other because if numbers are close to each other then their average will be minimum..

For eg:- $48$ can be $4*12$, $1*48$, $6*8$, $16*3$ but here we will select $6*8$ because $6$ and $8$ are closest to each other with a difference of $2$.

$\Rightarrow$ $X*Y = 6*8$

$X=6, Y=8$

$P = \left ( \frac{X+Y}{2} \right ) = \left ( \frac{8+6}{2} \right )= 7$

$\therefore$ Option B. $7$ is the correct answer.

by
+2 votes

It is given that :- $X*Y*Z= 192$ , $Z=4$. So, $X*Y=48$

Now, $P=\frac{X+Y}{2}=\frac{X+\frac{48}{X}}{2}=\frac{1}{2}\left ( X+\frac{48}{X} \right )$

Now, to find the minimum value of $P,$ Differentiate $P$ with respect to $X$

$\frac{\mathrm{d} P}{\mathrm{d} X} = \frac{1}{2}\left ( 1 -\frac{48}{X^{2}} \right )$

Now, $\frac{\mathrm{d} P}{\mathrm{d} X} = 0$

So, $X= \pm 4\sqrt{3}$

Now, at $X= +4\sqrt{3}$,  $\frac{\mathrm{d}^{2}P }{\mathrm{d} X^{2}} > 0$

So, $P$ will be minimum at $X= +4\sqrt{3}$

So, Minimum value of $P$ will be $\frac{1}{2}\left ( 4\sqrt{3} + \frac{48}{4\sqrt{3}} \right ) = 2\sqrt{3} + \frac{6}{\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3} =6.928….$(An Irrational value)

No options are matching.

For this question, Marks were given to all.

by
0
in question x y z are integer so how x= 4*1.73

i think 7 is correct answer . x and y is 8,6
0
Thanks. I did mistake. I marked it as wrong answer and requesting moderators and admins to please remove it as best answer.
0 votes
xyz=192

z=4

xy=192/4

xy=48

p=x+y/2

we all know that least sum of x and y is 14 (x=8 , y=6)

which means p=7

but there comes the problem(p!=7)

in question it is said that x and y are integers

so we can take -1 and -48 whose product is 48 and average is -24.5

and -24.5<7
by
Answer:

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
52,314 questions
60,435 answers
201,774 comments
95,251 users