median and mean

[lynncf]

five 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 shortes piece of wood?

Can someone help to provide explanation on how to solve this problem? thanks

[aamir.jariwala]

If 5 pieces of wood have an average length of 124cm then the sum of the 5 pieces of wood must be 620.

Now if the median of the set is 140cm then lets imagine the following set based on the property of the median being the middle term of a set if the set is arranged in ascending order of length of wood.

[x, y, 140, 140, 140]

In order to maximise the length of the smallest segment we assume in the above set that the 4th and 5th largest pieces of wood are also 140cm as this does not change the median and still keeps our set in ascending order.

Hence x + y = 620 - 420 = 200

Now x<=y to keep the ascending order of our set so in order to maximise x, x must be equal to y.

Hence 2x = 200 and x can be 100cm at the outside and the set looks as follows:

[100, 100, 140, 140, 140]

Hope this helps.

[Ben Ku]

Aamir's solution looks great. The key is to maximize the shortest piece of wood we wnatto minimize everything else. Let me know if you have additional questions. Thanks.