Seven pieces of rope have an average

Seven pieces of rope have an average (arithmetic mean) length if 68 centimeters and a median length of 84 centimeters. If the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope?

4 Explanations

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Cydney Seigerman, Magoosh Tutor

Hi Jay :)

You're correct that technically we could list other lengths between the minimum and the median. However, in this problem, we want to maximize the length of the longest piece of rope. To accomplish this goal, we have to make the other pieces of rope as short as possible. In order to make the other pieces of rope as short as possible, then, we want the greatest number of pieces of rope with the minimum length, 68 cm. We know that the median is 84 cm, which means that once we reach the median, the pieces of rope have to be at least 84 cm long. Before we get to the median, however, the pieces of rope can be any length between 68 and 84 cm. We choose 68cm for all of these pieces since that's the shortest length they can be and this allows us to maximize the length of the longest piece of rope.

I hope this helps!

Feb 6, 2016 • Comment

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Siddharth Sahasrabudhe

The same question can be solved as:
Let length of smallest rope:X
Length of largest rope: 4X+14
Length of 4th rope - 84 (median)
Let Length of rest of the ropes (other than the smallest, largest and the median) : Y
we are given the avg: 68
The sum of the lengths: 68*7 = 476
(X+4x+14+84+Y) = 476
5X+Y=378
X=(378-Y)/5
Hence X must be divisible by 5, i.e. 378-Y must be multiple of 5
Check the answers, 134 satisfies the above equation.
134-14=120; 120/4=30 ->30 is divisible by 5.

Answer choice D

Jun 24, 2015 • Comment

Jonathan , Magoosh Tutor

Nice innovative approach! Be careful, though, the question does not specify that the lengths of the rope must be integers, though the correct answer does indeed give integers.

Jun 29, 2015 • Reply

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AYAKA SOUZA

Can I take the approach by looking for the number that is divisible by 4 from the answer choices based on the assumption that the length of the ropes are integers, and I still get the correct answer choice D?

Mar 15, 2015 • Comment

Jonathan , Magoosh Tutor

Hi Ayaka,
The short answer to your question no -- we definitely cannot assume that all pieces of rope are integer lengths. That's a dangerous assumption to make.

On the other hand, it is indeed an interesting observation that when we subtract 14 from the answer choices, only one choice is divisible by 4. I might say, if you're stuck on the question, you could make a tactical guess based on this observation.

However, to go back to the first point, you cannot assume integers for something like lengths. We can only assume integers if 1) the problem tells us we are dealing with integers only or 2) we are dealing with things that can't have fraction parts, such as people or dogs or chairs. I hope that helps.

Mar 29, 2015 • Reply

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Mike McGarry, Magoosh Tutor

Dec 28, 2013 • Comment

JAY SETHI

why here everything is taken as X before the median 84 and 84 after the median 84? Before the median 84 not just identical nos can come any numbers can come.

Jan 23, 2016 • Reply