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M03-16

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The average (arithmetic mean) of 4 different positive integers is 125 and the largest of these integers is 150, what is the least possible value of the smallest of the 4 integers?

A. 1
B. 2
C. 12
D. 53
E. 100
[Reveal] Spoiler: OA

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Official Solution:

The average (arithmetic mean) of 4 different positive integers is 125 and the largest of these integers is 150, what is the least possible value of the smallest of the 4 integers?

A. 1
B. 2
C. 12
D. 53
E. 100


Suppose the 4 integers in ascending order are \(x\), \(y\), \(z\) and 150.

Since the average of 4 different positive integers is 125 then their sum is \(125*4=500\). So, \(x+y+z+150=500\), which simplifies to \(x+y+z=350\).

We want to minimize \(x\), so we should maximize \(y\) and \(z\), since given that all integers are distinct then \(y_{\text{max}}=148\) and \(z_{\text{max}}=149\). Therefore \(x+148+149=350\) and \(x=53\).


Answer: D
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M03-16 [#permalink]

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New post 22 May 2015, 04:19
Why cant be 1 ;

1+2+347=350
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New post 03 Sep 2016, 12:14
Yes exactly my question.
If we are asked to find the least possible value, then why we are not considering 1?
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Re: M03-16 [#permalink]

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New post 03 Sep 2016, 14:02
Notice that each element/integer of the set must be DIFFERENT.

So if we have the largest to be150, we have to maximise the other two elements to 49 & 48 to determine the least possible value of the smallest, which will result in 53 (D) as indicated in official solution given by Bunuel.

Hope this helps.

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New post 08 Jan 2017, 04:25
X is not 1 because:

1) All the integers are different so x,y, and z cannot take 150 as their value.
2) Simple rule to minimize a number is to maximize other numbers. Simple!! :)

Let's suppose x=1
y=148 (as given)
z=149 (as given in the solution)

1+148+149+150= 448 (But we found x+y+z+150= 500)

Therefore x=53 (as given in the solution)
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New post 27 Jul 2017, 10:16
I think this is a high-quality question and I agree with explanation.
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New post 31 Aug 2017, 20:43
Yes, but lets take the example wherein the other numbers are 118, 116,115 and 1? In this case, we have the sum of the other four numbers to be 350 and the least value of the smallest integer to be 1?

What am I missing here?
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New post 31 Aug 2017, 20:58
ttvenkat15 wrote:
Yes, but lets take the example wherein the other numbers are 118, 116,115 and 1? In this case, we have the sum of the other four numbers to be 350 and the least value of the smallest integer to be 1?

What am I missing here?


What is your set? {150, 118, 116, 115, 1}? But there are FOUR numbers in the set including 150, not 5!
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: M03-16   [#permalink] 31 Aug 2017, 20:58
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