Problem Solving

5 pieces of wood have an average length of 124 centimeters and a median length of 140 centimeters. What is the maximum possible length, in centimeters, of the shortest piece of wood?

A) 90

B) 100

C) 110

D) 130

E)140

OA B

in this we need to keep in mind that avg is 124 and median is 140 and we need to maximize the shortest length
in order to do that we need to minimize the longest wood and the minimum value it can hold is 140
so there will be 3 woods of 140 each ; tot=420;left=620-420=200 ; therefore the possible length will be 200/2=100

The average formula has 3 parts: average, sum, and number of values. A question that tells you 2 out of 3 is really telling you about the 3rd part, so always keep that in mind.

5 pieces avg 124 means that their sum is 620. Median is the length of the middle piece = 140, so the other pieces must add up to 620-140=480

In general to maximize a value when the total is fixed, you ought to minimize every other value. So we'll make the other values as small as possible. The smallest possible values of the two longest pieces is 140 (the median). If one of them is shorter than the median, then the median would no longer be the median!

So the top 3 pieces are: 140 140 140. They add up to 420. The bottom 2 pieces must add up to 620-420=200.

To maximize the shortest piece, we must make the 2nd shortest as short as possible. This would be 100. If the 2nd shortest were any shorter, then it would become the shortest!

So the two shortest pieces are 100, 100.

The 5 pieces are: 100 100 140 140 140. By minimizing 2-4, we guarantee that 100 is the maximum possible length of the first (shortest) piece. The answer is B.

A detailed solution and video explanation can be seen; this is GMATPrep question 1169

Good luck,
-Patrick

well,it seems all easy after seeing the explanation ....but i must confess that i was stumped or was not able to get to answer under 2 minutes....

what may be difficulty level of this question?

The solution I linked to above shows question difficulty. This is likely an upper 600 level question.

selango wrote:5 pieces of wood have an average length of 124 centimeters and a median length of 140 centimeters. What is the maximum possible length, in centimeters, of the shortest piece of wood?
A) 90 B) 100 C) 110 D) 130 E)140
OA B

Since the average is 124, the shortest piece cannot be 130 or 140.
The total of 5 pieces = 124*5 = 620
If median is 140, the total of the three biggest piece including the median i.e. 140 is >= 140*3 = 420
and the total of smallest two piece will =< 200 (620 - 420)
So it is not possible that the smallest is 110cm when total of two smallest piece is =< 200,
but both can be 100cm which is the maximum length possible for the shortest piece.

Hey Patrick,

Thanks for your detailed replies!Pretty helpful!!

One thing I didn't get about the 1st and 2nd pieces...I know how we reached that they should sum up to 200, and Im aware that second should be taller than the first, but why 100-100? why not 90-110 for example?

Thank you!

The problem states that the average length of the 5 pieces of wood is 124. Remember this about averages:

Whenever you're given an average, figure out the sum.

The sum of the lengths of the 5 pieces is 5 * 124 = 620.

Since the median of the 5 pieces has to be 140, the middle piece likely will be 140 centimeters.

Let's call the 5 pieces, from shortest to longest: shortest, x, 140, y, z.

The question asks for the maximum possible length of the shortest piece of wood.

Whenever a PS question asks for a specific amount, consider trying out the answers in order to determine which is correct.

Since we want the shortest piece to be as long as it can be, we should start with 140, the biggest answer choice:'

Using 140 for the shortest piece, the 5 pieces will be 140, x, 140, y, z.
This means x = 140, so the 5 pieces will be 140, 140, 140, y, z.
This means the sum of the 3 shortest pieces will be 140+140+140 = 420, so the sum of y and z will have to be 620 - 420 = 200.
This doesn't work because y and z each have to be at least 140, and 140 + 140 = 280.
Darn!

The next largest answer choice is 100. Let's try it:

Using 100 for the shortest piece, the 5 pieces will be 100, x, 140, y, z.
If we make y and z each 140 (the shortest each can be), the 5 pieces will be 100, x, 140, 140, 140.
This means the sum of the 4 known pieces will be 100 + 140 + 140 + 140 = 520, making x = 620-520 = 100
So the 5 pieces will be 100, 100, 140, 140, 140.
This works! The sum is 100+100+140+140+140 = 620, and the median piece is 140.

No reason to try the other answer choices because they're smaller than answer choice D, and we need the biggest answer that will work.

The correct answer is D.