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# If the elements of set X are a, b, c and d, is the average (arithmetic

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 6639
GMAT 1: 760 Q51 V42
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If the elements of set X are a, b, c and d, is the average (arithmetic  [#permalink]

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12 Apr 2018, 00:57
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Difficulty:

25% (medium)

Question Stats:

73% (01:25) correct 27% (01:48) wrong based on 77 sessions

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[GMAT math practice question]

If the elements of set X are a, b, c and d, is the average (arithmetic mean) of a, b, c, and d contained in set X?

1) The average (arithmetic mean) of every pair of elements of set X is 10.
2) The average (arithmetic mean) of every three elements chosen from set X is 10.

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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If the elements of set X are a, b, c and d, is the average (arithmetic [#permalink] ### Show Tags 15 Apr 2018, 17:40 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Since we have 4 variables (a, b, c and d) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first. Conditions 1) and 2): Condition 1) gives rise to the following equations: $$\frac{( a + b )}{2} = 10 => a + b = 20$$ $$\frac{( a + c )}{2} = 10 => a + c = 20$$ $$\frac{( c + d )}{2} = 10 => c + d = 20$$ Condition 2) gives rise to the following equations: $$\frac{( a + b + c )}{3} = 10 => a + b + c = 30$$ $$\frac{( a + b + d )}{3} = 10 => a + b + d = 30$$ $$\frac{( a + c + d )}{3} = 10 => a + c + d = 30$$ $$\frac{( b + c + d )}{3} = 10 => b + c + d = 30$$ Combining these equations yields $$a = b = c = d = 10.$$ Therefore, the average is $$10$$, which is an element of set X, and conditions 1) and 2) are sufficient, when taken together. Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B. Condition 1) Condition 1) gives rise to the following equations: $$\frac{( a + b )}{2} = 10 => a + b = 20$$ $$\frac{( a + c )}{2} = 10 => a + c = 20$$ $$\frac{( c + d )}{2} = 10 => c + d = 20$$ Combining these equations yields $$a = b = c = d = 10.$$ Condition 2) Condition 2) gives rise to the following equations: $$\frac{( a + b + c )}{3} = 10 => a + b + c = 30$$ $$\frac{( a + b + d )}{3} = 10 => a + b + d = 30$$ $$\frac{( a + c + d )}{3} = 10 => a + c + d = 30$$ $$\frac{( b + c + d )}{3} = 10 => b + c + d = 30$$ Combining these equations yields $$a = b = c = d = 10.$$ Therefore, the answer is D. Answer: D Since conditions 1) and 2) are equivalent, D is the answer by Tip 1) of the VA method. Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If the elements of set X are a, b, c and d, is the average (arithmetic  [#permalink]

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24 Apr 2018, 19:48
I)if avg of any pair is same it simply means the values are equal.
So mean is value actually.
Eg 10,10,10,10

II)if avg of any three is same ,it also means that all values are same.

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Re: If the elements of set X are a, b, c and d, is the average (arithmetic &nbs [#permalink] 24 Apr 2018, 19:48
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