Apologies blink005, if I am getting this wrong. But how did you get this?
The minimum possible values of L4 and L5 could be 140, hence the L1+L2 = 620 - 420 = 200.
Below is step by step analysis of this question. Hope it helps.5 peices of wood have an average length of 124 inches and a median of 140 inches. What is the MAX possible length of the shortest piece of wood?
5 peices of wood have an average length of 124 inches --> total length = 124*5=620. Also median = 140.If a set has odd number of terms the median of a set is the middle number when arranged in ascending or descending order;
If a set has even number of terms the median of a set is the average of the two middle terms when arranged in ascending or descending order.
As we have odd # of pieces then 3rd largest piece \(x_3=median=140\).
So if we consider the pieces in ascending order of their lengths we would have \(x_1+x_2+140+x_4+x_5=620\). Question:
what is the MAX possible length of the shortest piece of wood? Or \(max(x_1)=?\)General rule for such kind of problems:
to maximize one quantity, minimize the others;
to minimize one quantity, maximize the others.
So to maximize \(x_1\) we should minimize \(x_2\), \(x_4\) and \(x_5\). Min length of the second largest piece of wood, \(x_2\) could be equal to \(x_1\) and the min lengths of \(x_4\) and \(x_5\) could be equal to 140 --> \(x_1+x_1+140+140+140=620\) --> \(x_1=100\).
Why couldn't x4 and x5 be bigger than 140 and thus making x1 and x2 even smaller?