Problem Solving

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

We have 7 rope lengths.
If the median is 84, then the lengths (arranged in ascending order) look like this: {_, _, _, 84, _, _, _}

The length of the longest piece of rope is 14 cm more than 4 times the length of the shortest piece of rope.
Let x = length of SHORTEST piece.
This means that 4x+14 = length of LONGEST piece.
So, we now have: {x, _, _, 84, _, _, 4x+14}

Our task is the MAXIMIZE the length of the longest piece.
To do this, we need to MINIMIZE the other lengths.
So, we'll make the 2nd and 3rd lengths have length x as well (since x is the shortest possible length)
We get: {x, x, x, 84, _, _, 4x+14}

Since 84 is the middle-most length, the 2 remaining lengths must be greater than or equal to 84.
So, the shortest lengths there are 84.
So, we get: {x, x, x, 84, 84, 84, 4x+14}

Now what?

At this point, we can use the fact that the average length is 68 cm.
There's a nice rule (that applies to MANY statistics questions) that says:
the sum of n numbers = (n)(mean of the numbers)
So, if the mean of the 7 numbers is 68, then the sum of the 7 numbers = (7)(68) = 476

So, we now now that x+x+x+84+84+84+(4x+14) = 476
Simplify to get: 7x + 266 = 476
7x = 210
x=30

If x=30, then 4x+14 = 134
So, the longest piece will be 134 cm long.

Answer = D

Cheers,
Brent

vaibhav101 wrote: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

We need to first recognize that we are working with a maximum problem. This means that of the 7 pieces of rope, we must make the lengths of 6 of those pieces as small as we possibly can, within the confines of the given information, and doing so will maximize the length of the seventh piece.

We are first given that 7 pieces of rope have an average (arithmetic mean) length of 68 centimeters. From this we can determine the sum.

sum = average x quantity

sum = 68 x 7 = 476

Next we are given that the median length of a piece of rope is 84 centimeters. Thus, when we arrange the pieces of rope from least to greatest length, the middle length (the fourth piece) will have a length of 84 centimeters. We also must keep in mind that we can have pieces of rope of the same length. Let's first label our 7 pieces of rope with variables or numbers, starting with the shortest piece and moving to the longest piece. We can let x equal the shortest piece of rope and m equal the longest piece of rope.

piece 1: x

piece 2: x

piece 3: x

piece 4: 84

piece 5: 84

piece 6: 84

piece 7: m

Notice that the median (the fourth rope) is 84 cm long. Thus, pieces 5 and 6 are either equal to the median or they are greater than the median. In keeping with our goal of minimizing the lengths of the first 6 pieces, we will assign 84 to pieces 5 and 6 to make them as short as possible. Similarly, we have assigned a length of x to pieces 1, 2, and 3.

We can plug these variables into our sum equation:

x + x + x + 84 + 84 + 84 + m = 476

3x + 252 + m = 476

3x + m = 224

We were also given that the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope. So, we can say:

m = 14 + 4x

We can now plug 14 + 4x in for m in the equation 3x + m = 224. So, we have:

3x + 14 + 4x = 224

7x = 210

x = 30

Thus, the longest piece of rope is 4(30) + 14 = 134 centimeters.

Answer: D