Seven pieces of rope have an average (arithmetic mean) length of 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? GMAT Explanation, Video Solution, and Additional Practice!
Seven pieces of rope have an average (arithmetic mean) length of 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?
(A) 82
(B) 118
(C) 120
(D) 134
(E) 152
Here’s another tough one that we see a lot in GMAT tutoring sessions. Video solution below! The biggest mistake I see on this one: creating 7 variables for the 7 pieces of rope. If you’re ever working a GMAT question and feel the need to create 7 variable you might want to re-think your approach. I don’t think I’ve ever seen a question that required more than 5 variables. I’m sure there’s some exception out there but generally, again, be careful about creating an army of variables.
Setup
Table of Contents
Instead of variables I’d plot out seven slots. Median is a spacial measurement so getting the numbers in a row is important. Then go ahead and pop the median, 84, in the middle slot. We know that the seven pieces of rope have an average length of 68 so the sum of the 7 divided pieces by 7 is 68. You can go ahead and divide the slots by 7 and set them equal to the average, 68.
Now you can also pick a variable for either the biggest or smallest piece of rope. I’d define the smallest as “S” and then derive the biggest as 4S + 14. Great. Now we’ve got 3 of 7 slots filled in. What to do with the rest? Well, we’re looking the maximize the length of the biggest piece of rope. So what does that mean in terms of the rest of the rope? Minimize it!
Great – let’s make the rest of the pieces 0!!! Wait a second. Can you do that? Especially if the median is 84? Nope! We’ve got some constraints. Let’s use them. So, again, what’s the smallest piece of rope? S! Yes. OK. So to the left of the median what’s the smallest we can make those pieces? 0! Hmmm. What’s the smallest piece of rope again? S! Fantastic. OK. So keeping in mind that the absolute smallest piece of rope is S what is the smallest we can make those two other pieces to the left of the median? S! YESSSS.
OK so now we need to figure out how small we can make the pieces of rope to the right of the median. So, how small can they be? S! Hmmm. Think constraints. Are there any constraints? Any difference being positioned to the right of the median? Oh. 84? Right! Why? Because to the right of the median must be equal to or greater than the median. Perfect. OK.
Solve
So now we have an equation with one variable and can solve. But before we do that let’s give the equation a quick look to see if we can simplify anything. I’d group like terms. Once you do that you might see a bunch of things divisible by 7. That makes the arithmetic easier. Solve for S and then plug that into 4S + 14 and you end up with the maximum possible length 134. D.
Video Solution: Seven pieces of rope have an average
Additional Challenging Word Problem Max/Min Example Questions
Here’s are two very similar ones:
A certain city with population of 132,000 is to be divided into 11 voting districts
Challenging Word Problem Max/Min GMAT Question of the Day
Here’s another challenging GMAT Max/Min question that is a little more of a puzzle but still tests the same concept:
for a certain race 3 teams were allowed to enter 3 members each
And here are more statistics examples:
Here’s a GMAT statistics challenge (median/range) which will teach you how to work with a whole bunch of unknowns. Also a good one to work on maximizing a value.
Here’s a mini-statistics puzzle from the GMAT Official Guide. It has a very simple solution. Still, it’s surprising how often it comes up in tutoring sessions.
Middle of the road GMAT statistics question. Be ready to interpret a chart/graph. What isn’t tough in practice can quickly become a rabbit hole on test day.
Excellent practice to sharpen your thinking on GMAT statistics concepts. It’s a problem solving question but having this down will also help you on DS statistics questions.
Middle difficulty DS statistics range question from GMAT question of the day
And here’s an extra challenging GMAT statistics range question from GMAT question of the day that also includes a bunch of algebra and number properties.
Good luck and happy studies!