If the range of the set containing the numbers x, y, and z

Question

If the range of the set containing the numbers x, y, and z is 8, what is the value of the smallest number in the set?

(1) The average of the set containing the numbers x, y, z, and 8 is 12.5.
(2) The mean and the median of the set containing the numbers x, y, and z are equal.

Bunuel · Accepted Answer

If the range of the set containing the numbers x, y, and z is 8, what is the value of the smallest number in the set?

Let the numbers in ascending order be {x, y, z} (it really doesn't matter how we group them).

The range of a set is the difference between the largest and smallest elements in the set.
Given:  z-x=8 . Question:  x=?

(1) The average of the set containing the numbers x, y, z, and 8 is 12.5 -->  x+y+z+8=4*12.5=50  -->  x+y+z=42 : 2 distinct linear equations 3 with unknowns. Not sufficient.

(2) The mean and the median of the set containing the numbers x, y, and z are equal -->  mean=\frac{x+y+z}{3}=y=median  -->  x+z=2y : 2 distinct linear equations with 3 unknowns. Not sufficient.

(1)+(2) 3 distinct linear equations with 3 unknowns --> we can solve for  x . Sufficient.

Answer: C.

mohan514 · Answer

hello sir..

here when you have calculated the mean .. as per the data in 1st option mean should include 8 in the set and its value is 12.5

but you have excluded it...
can you just let me know where i am goin wrong

please explain me about the relation between mean and median.....

Bunuel · Answer

I don't understand what you mean.

(1) says: The average of the set containing the numbers  x, y, z, and 8  is 12.5. So,  \frac{x+y+z+8}{4}=12.5  -->  x+y+z+8=4*12.5=50  -->  x+y+z=42 .

adineo · Answer

I had a doubt regarding option A:

In that we are given that x + y+ z = 42 and also that range is 8.

Now lets say smallest number is x and largest is z then equation is (x) + (x+n) + (x+8). Where 0<n<8 .

isnt this constraint enough to give us the required answer? Since we are just looking for the smallest number, for finding the middle number we will need another equation.

Bunuel · Answer

From (1) we can have many options, for example: {10, 14, 18} or {11, 12, 19}, therefore this statement is not sufficient.

Hope it's clear.

thefibonacci · Answer

C.

did it like this

1) x+y+z+8 = 50
=> x+y+z = 42 
and |x-z| = 8 
so, sets can be: {8,16,16}, {9,15,17}, {10,14,18}, {11,13,19}, and {12,12,20} 
A is insufficient.

2) (x+y+z)/3 = y
=> x+z = 2y 
and |x-z| = 8 
B is insufficient.

(1)+(2) 
among the sets in (1) the sets which have x+z = 2y is {10,14,18} 
hence C.

Sarjaria84 · Answer

Bunuel   VeritasKarishma

If the question statement would have specified to find the least possible value of the smallest number then the answer would be choice 'A', right ?

And the value of the smallest number in that case would be 0.

Please correct me if I am wrong.

Thanks
Saurabh

KarishmaB · Answer

Saurabh,

Note that minimum/maximum values of variables may not make sense in DS questions. 
Say the question is what is the smallest value of integer x?
Stmnt 1: x is a positive integer.
Smallest value of x is 1.
Stmtn 2: x > 10
Smallest value of x is 11. As you get more information, your answer may be different. 
Hence, the DS question may not make a lot of sense.

Additionally, in our original question, using stmnt 1 alone, the smallest possible value of x is not 0.

Range of x, y, z = 8
Say z is the greatest number and x the smallest. 
x = z - 8 and y must lie between x and z.

Mean of x, y, z, 8 = 12.5
Note that 8 is already 4.5 less than 12.5. So there needs to be atleast one number greater than 12.5
If x needs the smallest value, z needs to be greatest but z cannot be 8 more than x. We need to maximise positive deviation so y = z

x + (x + 8) + (x + 8) = 42
x = 8.667

bgurveen · Answer

since you are able to find the least value of x as 8.667 shouldn't the answer be A?

Akshay1298 · Answer

Range(x,y,z) = 8;

Statement 1:

/4 = 12.5
x+y+z+8 = 50
x+y+z = 42 ( Insufficient  to find the smallest number)

Statement 2:

Mean = Median
Example 1:
Consider x=2(smallest), y=6, z=10;
Example 2:
Consider x=4(smallest), y=8, z=12;

Both of the examples has the same  Range. 
Therefore two of the smallest integers have different values.( Insufficient )

By combining both the statements,
From  Statement 1 ,

x+y+z = 42
 /3 = 14

Since Mean = median from  Statement 2 , Consider the value y as median which states that   y = 14 ,

Therefore using the value y=14, it is  sufficient  to find out the smallest value(  x=10  ), and the largest value(  z=18  ).

bumpbot · Answer

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If the range of the set containing the numbers x, y, and z

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