Five pieces of wood have an average length (arithmetic mean)

[Harish Dorai]

Five pieces of wood have an average length (arithmetic mean) of 124 centimeters and a median length of 140 centimeters. What is the maximum length, in centimeters, of the shortest piece of wood?

A) 90
B) 100
C) 110
D) 130
E) 140

[anadi]

Average 124 means

a+b+c+d+e = 124*5 = 620

Assume a,b,c,d,e is the acending order, then since 140 is the median, c will be 140.

Smallest piece will be at its maximum length when pieced D and E are at their smallest possible (which will be 140 here because of the order) and the A and B are equal.

So,

a+a+140+140+140 = 620

2a = 620-420 = 200
a=100

Pieces will be

100 100 140 140 140.

[anadi]

B. 100

[Harish Dorai]

Awesome! That is the correct answer.

[pjain01]

is it not the case of we take four numbers as 140, i.e the series can also be
x,140,140,140,140
where X is the smallest integer.Even in this case median will not change at all(remain 140).
Am I missing something in question????

[RonPurewal]

is it not the case of we take four numbers as 140, i.e the series can also be
x,140,140,140,140
where X is the smallest integer.Even in this case median will not change at all(remain 140).
Am I missing something in question????

The average length has to be 124, according to the problem statement. (The average of 140, 140, 140, 140, and 140 is not 124.)

[sfbay]

I think what the person was asking is why not z, 140, 140, 140, 140.

5 x 124 = 620

4 x 140 = 560

thus z = 60

this would be the case if we were trying to find smallest size for z but since we are maximizing z need to have z+z +140 +140+140 =620 ..............i am sure Ron can clarify/articulate

[selvakumar.esra]

i would go with B.

Question: What is the maximum length, in centimeters, of the shortest piece of wood?

for easy understanding lets POE method.

Given Data: median is 140 mean is 124

so x,y,140,140+a,140+a+b (a and b could be any value including 0)
As mean is less than median, the shortest length should always be less than mean(124). So D ,E are OUT.

for option C, if 110 is the ahortest, then mean of 110,110,140,140,140 will not be 124.

So only option left is A and B. both can satisfy mean and median condition.
i,e 100,100,140,140,140 or 90,110,140,140,140
but the question is MAXIMUM values of shortest length which 100.

A) 90
B) 100
C) 110
D) 130 OUT
E) 140 OUT

[RonPurewal]

I think what the person was asking is why not z, 140, 140, 140, 140.

5 x 124 = 620

4 x 140 = 560

thus z = 60

this would be the case if we were trying to find smallest size for z but since we are maximizing z need to have z+z +140 +140+140 =620 ..............i am sure Ron can clarify/articulate

yep, you've got it.