A IMO.

sanju09 wrote: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.



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given: Set {a,b,c} a=b OR b=c, Is range=mode ?
st(1) med=range --> a=2, b=a=2, c=4 --> med=2, range=2, range=mode ... Sufficient
st(2) smallest number can be 0 or -ve, or two large numbers Not Sufficient

A

This is a "yes/no" question. To answer, let's see what we need. For "mode" - we need to know which number occurs the most often. For range, we need to know the largest and the smallest number.

[1] Let's look at an example that makes this answer "yes": {0,2,2}

Here the range is 2-0 , or 2, the mode is 2, and the median is 2.

If we try to change the numbers around to get a "no" answer, and make the statement insufficient, it will change the median unless the mode is the median as well.

sanju09 wrote: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.

Hi there!

Beautiful problem, sanju.

From the question stem, we have only 3 possible scenarios... let us called them (I), (II) and (III) as follows:

(I) {a,a,a} with mode a and with focus on the simplified question (on this particular case): a = 0 ?

(II) {a,a,b} (where a<b) with mode a and with focus.... case): a = b-a, i.e., b = 2a ?

(III) {a,b,b} (where a<b) with mode b and with focus... case): b= b-a, i.e., a = 0 ?


Now let us discuss each statement alone in each of the 3 scenarios/simplified questions shown above!


(1) SUFFICIENT:

(I) From this sttm we get a=0, answering in the affirmative;
(II) From this sttm we get a=b-a, again in the affirmative;
(III) From this sttm we get b=b-a, again in the affirmative;


(2) INSUFFICIENT:

(I) From this sttm we get a=2a, therefore a=0, answering in the affirmative; (*)
(II) From this sttm we get b = 2a, again answering in the affirmative;
(III) From this sttm we get b = 2a, but here we have the possibility of answering in the negative: take a=1 and b=2;

Best Regards,
Fábio.

Post-Mortem:

Obs.1: the question stem should use the word "list" instead of "set", for technical reasons.
Obs.2: there is nothing wrong here, when there is only one number in the list, the largest and the smallest values coincide!

Night reader wrote:

sanju09 wrote: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.



https://www.platinumgmat.com

given: Set {a,b,c} a=b OR b=c, Is range=mode ?
st(1) med=range --> a=2, b=a=2, c=4 --> med=2, range=2, range=mode ... Sufficient
st(2) smallest number can be 0 or -ve, or two large numbers Not Sufficient

A

Please let me know if i am missing something here.....

from the question the xteristic of the numbers was not mentioned. so we can assume anything other than work from the two statements

3 numbers as a,b,c in ascending order.
Stmt :
Range = Median , therefore range = c-a ; median = b

..... b =c-a , there will be more one results from this since the numbers are not unique. INSUFF

Stmt2
c = 2a

so we will have ..a ,b, 2a .... Range = a, median is b ...... INSUFF

1&2 ....
From 1 range = median , so b = a as well

now we have : a, a, 2a ...

mode = 2 ; median = a . ; values of a can +ve or - ve ... ( -1, 2, ,2,3,5,,4,6,8.......)


I will go with E

My thought process.

numbers : (a) a a b or
(b) b a a depending on which one is bigger

1. median = range

(a) a = b - a or b = 2a

therefore a,a,2a hence mode equals range

(b) a = a - b or b = 0

therefore 0, a,a hence mode equals range

sufficent by itself

2. the largest value is twice the smallest value

(a) a ,a, 2a hence mode equals median

(b) b, 2b, 2b hence mode equals median

sufficent by itself.

either statement sufficent by itself Answer option: 4

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