The average of 7 distinct integers is 12, and the least of these integ

Question

The average of 7 distinct integers is 12, and the least of these integers is -15. What is the maximum possible value of the largest integer in the set?

A. 84
B. 158
C. 159
D. 160
E. 161

GMATinsight · Accepted Answer

Answer: Option C

Step-by-Step Video solution by  GMATinsight

https://www.youtube.com/watch?v=fNbVZbRA2uA

BrentGMATPrepNow · Answer

The average of 7 distinct integers is 12, and the least of these integers is -15  
If we let a, b,... to g represent the seven integers in  ascending order , we can write:  \frac{a+b+c+d+e+f+g}{7}=12

Multiply both sides of the equation by  7  to get:  a+b+c+d+e+f+g = 84 
Since the smallest number is -15, we can write:  (-15)+b+c+d+e+f+g = 84

What is the maximum possible value of the largest integer in the set?  
In order to maximize the largest integer (aka g), we must MINIMIZE the other values in the sum. 
Since each integer must be  distinct , the second smallest integer must be -14, and then -13, and so on.
We get:  (-15)+(-14)+(-13)+(-12)+(-11)+(-10)+g = 84 
Simplify:  (-75)+g = 84 
Add  75  to both sides:  g = 159

Answer: C

kungfury42 · Answer

We know the minimum score is -15 and that all scores are distinct integers.

Therefore, in order to maximize the highest score, we must minimize all the other 5 scores keeping in mind the above constraints.

Therefore, lowest 6 scores are = -15, -14, -13, -12, -11, -10
Their sum is = 3*(-11) + 3*(-14) = -33 - 42 = -75

Average across 7 tests is 12
Or the sum across 7 tests is = 12*7 = 84

Therefore, maximum score in a test = 84-(-75) = 84+75 = 159

Hence, C.

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BrushMyQuant · Answer

Given that The average (arithmetic mean) of seven distinct integers is 12, and the least of these integers is –15 and we need to find the maximum possible value of the greatest of these integers

============================================================

Theory
 ‣‣‣ Sum = Mean * Total Number of values.  
============================================================

=> Sum = 12 * 7 = 84

Least number is -15, so to find the maximum value of the greatest number we need to make all other numbers as small as possible.

Smallest number is -15, so other integers can be taken close to this
=> Set is -15, -14, -13, -12, -11, -10 and the maximum number = 84 - (-15 -14 -13 -12 -11 -10) = 84 + 75 = 159

So,  Answer will be C .
Hope it helps!

Watch the following video to Learn the Basics of Statistics

https://www.youtube.com/watch?v=Cvy_HHw4KIs

Abhishek009 · Answer

-15 -14 - 13 - 12 - 11 - 10 + x = 84

Or,  x  = 159 , Answer must be (C)

egmat · Answer

Great question! Let's break this down step by step.

Step 1: Find the total sum. 
If the average of  7  integers is  12 , then their total sum =  7  ×  12  =  84 .

Step 2: Think about the strategy. 
We want to make ONE number as large as possible. Since all  7  numbers must add up to  84 , we need to make the OTHER  6  numbers as SMALL as possible. The smaller they are, the more "room" is left for the largest number.

Step 3: Minimize the other  6  numbers. 
The smallest number is  -15 . Now, the integers must all be DISTINCT (different from each other). So we pick the smallest possible  distinct integers  starting from  -15 :

-15, -14, -13, -12, -11, -10

Their sum =  -15  +  -14  +  -13  +  -12  +  -11  +  -10  =  -75

Step 4: Find the largest number. 
Since all  7  numbers must sum to  84 :

Largest number =  84  - ( -75 ) =  84  +  75  =  159

Answer: C

Common mistake : Some students pick  -15, -14, -13, -12, -11, -10  but accidentally forget the negative signs when adding, or they try using  0, 1, 2, 3, 4, 5  as the smallest values — forgetting that we should go as negative as possible to leave the most room for the largest integer.

Key principle:  To maximize one value in a fixed-sum set of  distinct integers , minimize all the others by making them  consecutive integers  starting from the smallest allowed value.

The average of 7 distinct integers is 12, and the least of these integ

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The average of 7 distinct integers is 12, and the least of these integ

Prep Toolkit

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