For a set of 3 numbers, assuming there is only one mode,

Question

For a set of 3 numbers, assuming there is only one mode, does the mode equal the range?

(1) The median equals the range 
(2) The largest number is twice the value of the smallest number

kp1811 · Answer

The question stem says " assuming there is only one mode" so we need to consider sets where Mode is well defined  
A
stmnt 1 - median is equal to range
lets consider a set { 0,2,2}. here we have median is equal to range. mode  is also equal to range
lets consider the set { 2,2,4} here we median is equal to range but mode is equal to range. hence suff

stmnt2 - lets consider a set { 2,2,4}. here 4 = 2* 2(smallest number) and mode  is equal to range .
lets consider a set { 5,10,10} here 10 = 2* 5(smallest number) but mode  is not equal to range .
hence insuff

PS: What is the source of this question?

Bunuel · Answer

I think that above solution is correct and the answer is A, though there is one case missing.

Set can be of a form:
A. {X,Y,Y}
B. {X,X,Y}
OR
C. {XXX}

Basically telling us that there is only one mode, stem is saying that we  do not  have three distinct numbers in the set.

(1) The median equals the range:

A. {X,Y,Y} --> Y=Y-X --> X=0 --> Set: {0,Y,Y}. Mode=Y, Range=Y-0=Y, --> Y=Y. True.
B. {X,X,Y} --> X=Y-X --> 2X=Y --> Set: {X,X,2X}. Mode=X, Range=2X-X=X --> X=X. True.
C. {XXX} --> X=X-X --> X=0 --> Set: {0,0,0}. Mode=0, Range=0 --> 0=0. True.

Sufficient.

(2) The largest number is twice the value of the smallest number:

A. {X,Y,Y} --> Y=2X --> Set: {X,2X,2X}. Mode=2X, Range=2X-X=X, --> 2X equals to X only if X=0 (set: {0,0,0}), but we don't know that. Not always true. For example we can have set: {1,2,2} Mode=2, but range=1, 2#1.
B. {X,X,Y} --> Y=2X --> Set: {X,X,2X}. Mode=X, Range=2X-X=X --> X=X. True.
C. {XXX} --> X=2X --> X=0 --> Set: {0,0,0}. Mode=0, Range=0 --> 0=0. True.

Two different answers (cases B and C always true, A not always).

Not sufficient.

Answer: A.

xcusemeplz2009 · Answer

thanks bunuel for the soln..

xcusemeplz2009 · Answer

thanks for the expln
OA is  A

source-https://www.takegmat.com/

rxs0005 · Answer

For a set of 3 numbers, assuming there is only one mode, does the mode equal the range?

s1 The median equals the range

s2 The largest number is twice the value of the smallest number

can some one list all the properties of range / SD / median

for EG: SD can never be negative or SD = 0 if all memberes of sets are =

thanks

Bunuel · Answer

Merging similar topics. For SD please check: math-standard-deviation-87905.html ps-questions-about-standard-deviation-85897.html ds-questions-about-standard-deviation-85896.html Hope it helps.

mainhoon · Answer

Bunuel
Can you please explain this about mode? I believe it is the number that occurs most frequently in the set? So in this case if all 3 numbers were different then would the mode be 0? Mode 1 means a number repeats more than once? Can there be mode 2 or 3?

Posted from my mobile device

Bunuel · Answer

The mode is the number that occurs the most frequently in a data set. For example mode of the set {2, 3, 4, 4} is 4.

Set can have more than one mode, for example set {2, 2, 3, 3, 5} has 2 modes 2 and 3.

If every number in the set occurs an equal number of times, then the set has no mode. For example set {1, 2, 3} has no mode.

IanStewart · Answer

We don't really need to consider cases for Statement 1. We have one mode, so at least two of our elements are equal. So our set is {a, a, b}, where b could be anything. Whether b is greater than, equal to, or less than a, a is going to be the median, so median=mode. If from S1, median=range, then mode=range.

S2 is not sufficient, as seen above; your set can be {a, a, 2a} or {a, 2a, 2a}, and the answer might be yes or no respectively.

AnkitK · Answer

For a set of 3 numbers, assuming there is only one mode, does the mode equal the range?

(1) The median equals the range
(2) The largest number is twice the value of the smallest number

zuberahmed · Answer

Very nice question. I did it wrong and understood and very important concept. Thanks

Kudos.

vivesomnium · Answer

mode is the number which occurs most frequently in a set of numbers.

lets say mode is x. there have to be at least two instances of x. 
set can be x, x, y
or z,x,x
median equals range:Range= y - x = x --> y = 2x. Mode x is equal to range y-x which is equal to x
or x - z = x --> z--> 0 thus range is x which is equal to the median/mode
if all three numbers are same : set is x,x,x range is zero; but it can be equal to median only if x is zero, or the set is 0,0,0 here range= median = mode =0
1 is sufficient

2. largest number is twice smallest number. For there to be a single mode either the largest number comes twice, or the smallest number.
n,n,2n ---> Range=n, median is n and so is mode hence satsified
n,2n,2n ---> Range is n , median is 2n so is mode but mode is not equal to median or range. thus this is not giving one result.. not sufficient.

TomB · Answer

I got D as answer.

Let a,b,c is 2,2,4. As per statement 1which is true. then mode(2) equals to range 4-2=2.sufficient.

statement 2 is also implies same meaning but with different wording. By taking 2,2,4 as a,,b,c values , we can conclude that mode equals to range. Sufficient. The answer is D

cledgard · Answer

However, by taking 2, 4, 4 as values Mode is 4 and range is 2, so statement 2 is insufficient.

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