Problem Solving

Hi Guys,

This is another question that I had difficulty solving on the prep test today:

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

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

I thought that if you know at least one length of the 5 pieces: 140. And with this being the middle, you can have two pieces larger and two smaller. Furthermore, the smaller pieces have to be very small in the sense that it needs to bring the average all the way down to 124. This is my rational behind it. Therefore, if I needed to guess, it would be either A or B. But I am having a hard time getting to the right solution (B).

Thanks Again!!!

Priester wrote:Hi Guys,

This is another question that I had difficulty solving on the prep test today:

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

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

On questions where we want to maximize one item, we need to minimize everything else.

As you pointed out, we know that the middle length is 140cm and we can use the average formula (sum = avg * # of terms) to calculate that the 5 pieces add up to 5 * 124 = 620cm.

So, our set is {w, x, 140, y, z} and we want to maximize w.

Let's start by making y and z as small as possible. If our set were:

{w, x, 140, 140, 140}, the median would be 140. That's certainly the smallest we can make y and z, so that part is covered.

For w and x, we have 620 - 420 = 200cm remaining. We know that w is the smallest number in the set and to maximize it we want to set w=x. So, 200/2 = 100cm for each of w and x.

Our final set is:

{100, 100, 140, 140, 140}: Choose (B).

Stuart your awesome!! After I did the post, I spent some time going through possibilities and was able to come to the solution. Thanks again though for providing such a quick and accurate response. Your help has been much appreciated.