The average of 7 integers is 89 and the median of these integers

Question

The average of 7 integers is 89 and the median of these integers is 105. If the largest integer is 20 more than thrice the smallest integer, which of the following can be the maximum possible value of the largest integer?

A) 105
B) 106
C) 120
D) 140
E) 164

A) 105
B) 106
C) 120
D) 140
E) 164

zlishz · Accepted Answer

Let the smallest number be s. Then, the largest number will be 3s + 20
Given, Avg is 89, so sum of 7 integers is =7*89 = 623
And the middle number would be the median =105
Now the numbers would look like,

s, b, c, 105, e, f ,  (3s + 20)

Now, if we want to maximize the underlined part above, we will need to minimize the bold part.
 b and c can take the value of s to be least 
e and f can take the value 105 to be least 
So, we can create the equation for the sum of the 7 integers:

s + s + s + 105 + 105+ 105+ (3s + 20) = 623

6s + 335= 623

6s = 288

s = 48

Therefore, the greatest possible value of the largest number is 3s+20 = 3 x 48 + 20 = 164.

Answer: E

kabirgandhi · Answer

The inherent contradiction in these questions, based on my understanding is this:

Let the set be A = {a,b,c,105,d,e,3a+20}

We know that sum of all these = 623 = a + b + c + 105 + d + e + 3a+20

Now, in order to maximise the largest term, we would have to minimise d and e, for sure, since they are being added together. The lowest value for these would be 105 each.

Similarly, since a, b, c are also being added to 3a+ 20 to get a 623, intuitively, we should minimise all these as well. However, if we minimise these, then wouldn't a's value, and therefore 3a+20 also be minimised?

I feel the trick to these questions lies in what we do with the part of the set that is less than the median.

Question: Should my aim in these questions be always to minimise values like a,b,c in order to maximise the largest integer, even though it is dependent on the first 3?

Bunuel   KarishmaB   WhitEngagePrep   GMATNinja

Bunuel · Answer

There’s no contradiction. We have seven integers: a, b, c, 105, d, e, and the largest one, 3a + 20. We want to maximize 3a + 20. To maximize one value when the sum of the values is fixed, we minimize the remaining values. Since a and 3a + 20 are tied together, we minimize the rest by setting d and e equal to the median (105) and b and c equal to a.

Then the sum of the integers is a + a + a + 105 + 105 + 105 + (3a + 20) = 89 * 7. Solving gives a = 48, and accordingly 3a + 20 = 164.

Answer: E.

egmat · Answer

Looking at your analysis, you've identified the exact crux of this problem! Your confusion about the apparent contradiction is completely understandable and shows sophisticated thinking.

Let me clarify the key principle that resolves this.

The Key Insight:

You're absolutely right that there's a dependency relationship, but you've got the optimization direction backwards!

Since the largest integer =  3a + 20 , when you increase  a , you automatically increase the largest integer. So to maximize the largest integer, you actually want to  maximize   a , not minimize it!

The Strategic Principle:

In these constraint-based optimization problems:

Identify what you're maximizing and its dependencies: If maximizing something that depends on a variable (like  3a + 20  depends on  a ), maximize that variable 
 Minimize everything else  within their constraints

Working Through Your Example:

Let's use your notation:  {a, b, c, 105, d, e, 3a+20} 
From  4a + b + c + d + e = 498 :

To maximize  a : 
 
 We need to minimize  b, c, d, e  
 BUT they have constraints:  b \geq a ,  c \geq b , and  d, e \geq 105

The optimal strategy: 
 
 Set  b = c = a  (minimum possible given they must be ≥  a ) 
 Set  d = e = 105  (minimum possible given they must be ≥ 105)

This gives us:  4a + a + a + 105 + 105 = 498   6a = 288   a = 48

Therefore, the largest integer =  3(48) + 20 = 164

Why Your Initial Instinct Was Backwards

You correctly identified that we need to minimize certain values, but missed that  we want  a  as large as possible  because the largest integer directly increases with  a . The values  b  and  c  should be minimized  relative to their constraints  (they can't be less than  a ), which means setting them equal to  a .

Answer:  E) 164
 Remember:  When maximizing a dependent variable, trace back to what it depends on and maximize that source variable within the constraints!

The average of 7 integers is 89 and the median of these integers

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The average of 7 integers is 89 and the median of these integers

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