The average (arithmetic mean) of four different positive...

Question

The average (arithmetic mean) of four different positive integers is 9. If the first of these integers in 3 times the second integer and the second integer is 2 less than the third integer, what is the least possible value of the fourth integer?

A) 5
B) 4
C) 3
D) 2
E) 1

A) 5
B) 4
C) 3
D) 2
E) 1

GyanOne · Answer

Let the second integer be x and the fourth be a.

Then  /4 = 9
=> 5x + 2 + a = 36
=> 5x + a = 34
=> a = 34 - 5x

From the above equation we can see that a is minimum when x is maximum, provided both are positive 
The maximum value that x can take in the above equation while still keeping a positive is x=6
This gives us a= 34 - 30 = 4

Therefore the minimum value that the fourth integer can have is 4. Option B.

law258 · Answer

Starting out...

/4 = 9

w=3x --> 3y-6
x = y-2

Plug back into main equation

3y-6+y-2+y+z=36
5y-8+z=36
5y+z=44

We know multiples of 5 give us either a unit digit of 0 or 5 --> In this case all of the values w,x,y,z can take on are positive, thus we are limited to a unit digit of 0, making z = 4 & y = 8

B is the correct answer.

gracie · Answer

a+b+c+d=36
a+b+c=5b+2
greatest possible value of 5b+2<36=32
36-32=4=least possible value of d
B

stonecold · Answer

Here is my solution to this one ->

Let the integers be-->
w1
w2
w3
w4

Mean = 9

Using -->

Mean = \frac{Sum}{#}

Hence sum(4)=9*4=36

As per Question-->
w1=3w3-6
w2=w3-2
w3
w4

Adding them all => 5w3-8+w4=36

Hence w4=44-5w3

To minimise w4 => w3=> 8

Hence w4 => 4

Hence => B

Abhishek009 · Answer

a + b + c + d = 36

(3c - 6) + (c - 2) + c + d = 36

Or, 5c + d = 44

Now, use Maximization/Minimization...

least possible value of the fourth integer ( ie, d ) is the maximum possible value of 5c

Here maximum possible value of 5c = 40 ; so d = 4
  
Hence, Correct answer will be (B) 4

JeffTargetTestPrep · Answer

We are given that the average of four different positive integers is 9. Thus, the sum of the integers is 4 x 9 = 36.

We can create the following equation in which a = the first integer, b = the second integer, c = the third integer, and d = the fourth integer:

a + b + c + d = 36

We are given that the first of these integers is 3 times the second integer. Thus:

a = 3b

We are also given that the second integer is 2 less than the third integer. Thus:

b = c - 2

b + 2 = c

We can substitute 3b for a and (b + 2) for c in the equation a + b + c + d = 36:

3b + b + b + 2 + d = 36

5b + d + 2 = 36

5b + d = 34

5b = 34 - d

b = (34 - d)/5

Since b is an integer, 34 - d must be a multiple of 5, and since the largest multiple of 5 less than 34 is 30, the smallest value for d is 4.

Answer: B

roadrunner · Answer

SUM ( a_1 ,..., a_5 ) = 36
 a_1  = 3 a_2 
 a_3  =  a_2  + 2

SUM ( a_1 ,..., a_5 ) = 3 a_2  +  a_2  +  a_2  + 2 +  a_4  = 36 --> 5 a_2  +  a_4  = 34 --> Min( a_4 ) = 4

KanishkM · Answer

If i can get the 3 unknowns in one variable and add the 4th unknown variable to it, i should be good

a= 3b = 3c -6
b =c-2
c =c
d= d

Now a+b+c+d = 36
d = 44-5c
c = 8

d = 4

B

Kushchokhani · Answer

First dash= 3x
Second dash= x
Third dash= x+2
To minimize 4th dash, we need to maximize first three dashes
Total of 4 dashes= 9*4=36
Maximum integer value of sum of first three dashes=3x+x+x+2=5x+2=32
So, least value of fourth dash=36-32=4
Hence, answer is B.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The average (arithmetic mean) of four different positive...

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The average (arithmetic mean) of four different positive...

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards