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A certain list consists of 3 different numbers. Does the median of the

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Bunuel

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09 Nov 2021, 01:09

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BeepBoop

Bunuel

A certain list consists of 3 different numbers. Does the median of the 3 numbers equal the average (arithmetic mean) of the 3 numbers?

Say the three numbers are x, y, and x, where x < y < z. The median would be y and the average would be (x + y + z)/3. So, the question asks whether y = (x + y + z)/3, or whether 2y = x + z.

(1) The range of the 3 numbers is equal to twice the difference between the greatest number and the median --> the range is the difference between the largest and the smallest numbers of the set, so in our case z - x. We are given that z - x = 2(z - y) --> 2y = x + z. Sufficient.

(2) The sum of the 3 numbers is equal to 3 times one of the numbers --> the sum cannot be 3 times smallest numbers or 3 times largest number, thus x + y + z = 3y --> x + z = 2y. Sufficient.

Answer: D.

Hope it's clear.

Hi Bunuel !

Thanks a lot for the explanation, it's concise and to the point - and these help a lot!

Could I ask one thing, though? In the second statement, you mention that the sum cannot be three times the smallest or largest number.
Why not? Is there a rule for series/sets that forbids this, or is this some logical deduction?

All the best,

Gil

We have that x < y < z.

The sum = x + y + z.

Obviously 3x < x + y + z < 3z because x < y < z (x + x + < x + y + z < z + z + z).

Does this make sense?

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09 Nov 2021, 01:14

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Bunuel

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Bunuel

We have that x < y < z.

The sum = x + y + z.

Obviously 3x < x + y + z < 3z because x < y < z (x + x + < x + y + z < z + z + z).

Does this make sense?

Oh wow when you put it like that it does. Thanks a lot!

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27 Dec 2022, 12:39

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karovd

A certain list consists of 3 different numbers. Does the median of the 3 numbers equal the average (arithmetic mean) of the 3 numbers?

(1) The range of the 3 numbers is equal to twice the difference between the greatest number and the median.
(2) The sum of the 3 numbers is equal to 3 times one of the numbers.

Hi BrentGMATPrepNow, question asked Does the median of the 3 numbers equal the average (arithmetic mean) of the 3 numbers? So it means are these numbers evenly spaced set?
Does this mean we should only assing even numbers here e.g 8 ,10 ,12 if number testing? Thought assign 1,2,3 work too but is this still correct though? Thanks Brent

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Kimberly77

karovd

Since we know a list consists of three different numbers, then we can reward the target question as "Are the three numbers equally spaced?"
Don't forget that, when testing numbers for each statement, the values must satisfy the information in each statement.

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1. Median = Mean if the set is symmterical around median. Symmetrical means lets say for a set a,b,c where b is median if a-b=c-b then Median = Mean
A-b = c-b
2b-c-a......(1)

Once we solve Statement A we get to the exact same equation so A is sufficient and quite direct

2. B is what seems indirect and requires 2 layers of thinking and has two methods to solve

a)Translation Method
What is mean?
Avg = Sum of 3 values in set/3
What is statement Telling us?
The sum of three values in set is = 3 times the value of one number in set

Avg = Sum of 3 values / 3
3*Avg = Sum of 3 values

Now interpret slowly Sum of 3 values which is our RHS (Right hand side of equation ) IS 3 times value of one number in set
Hence Avg = One number in set
Avg cannot be lowest or highest value and has to be middle number in an odd set (odd number of terms)

b)Logical Method
Lets say we couldn't translate properly. What do we now? The obvious doubt is Sum is 3 times of which number - smallest, middle or largest

Lets say three nos are a,b,c and a<b<c

Can sum be 3a??
Sum = a+b+c >3a Why??? Because we know that the three nos. are different and the two numbers are BIGGER than a so there is no way that the sum of 3 nos could be equal to three times the smallest number. [In which case would the sum be 3a? If all nos were same which violates our given info]

Can sum be 3c??
Sum = a+b+c <3c Why??? Because we know that the three nos. are different and the two numbers are SMALLER than c so there is no way that the sum of 3 nos could be equal to three times the largest number. [In which case would the sum be 3c? If all nos were same which violates our given info]

Hence only one option remains sum is equal to 3 times median. Once you transform this equation we get back to 2b = a+c which is exactly what we had to prove!

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