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Is median of a, b and c equal to their average (arithmetic m

by Max@Math Revolution » Thu Apr 26, 2018 12:11 am

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[GMAT math practice question]

Is median of a, b and c equal to their average (arithmetic mean)?

1) a ≤ b ≤ c
2) b = ( a + c ) / 2
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Thu Apr 26, 2018 6:38 am

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Max@Math Revolution wrote:Is median of a, b and c equal to their average (arithmetic mean)?

1) a ≤ b ≤ c
2) b = ( a + c ) / 2
Target question: Is the median of a, b and c equal to their average (arithmetic mean)?

Statement 1: a ≤ b ≤ c
Let' TEST some values.
There are several values of a, b and C that satisfy statement 1. Here are two:
Case a: a = 1, b = 1 and c = 1. The median = 1 = mean. In this case, the answer to the target question is YES, the median and mean ARE equal
Case b: a = 1, b = 2 and c = 9. The median = 2 and the mean = 4. In this case, the answer to the target question is NO, the median and mean are NOT equal
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b = (a + c )/2
Notice that (a + c )/2 represents the AVERAGE of a and c. So, b is the average of a and c.
IMPORTANT: The average of 2 different values is HALFWAY between those two values.
For example, the average of 4 and 10 is 7, and 7 is halfway between 4 and 10.
Also notice that 4, 7, and 10 are EQUALLY spaced numbers.
Likewise, we can conclude that a, b and c are EQUALLY spaced numbers.

There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
For example, in each of the following sets, the mean and median are equal:
{7, 9, 11, 13, 15}
{-1, 4, 9, 14}
{3, 4, 5, 6}

So, if a, b and c are EQUALLY spaced numbers, then their mean = their median
This means the answer to the target question is YES, the median and mean ARE equal
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Sun Apr 29, 2018 5:21 pm

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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 3 variables 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) & 2)
b is the median of a, b and c since a ≤ b ≤ c.
b is the average of a, b and c since b = ( a + c ) / 2.
Thus, the median and the average of a, b and c are the same.

Both conditions together are sufficient.

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)
If a = 1, b = 2, and c = 3, the average and the median are 5.
If a = 1, b = 2, and c = 6, the average is 3 and the median is 2. The average and median are different.
Thus, condition 1) is not sufficient.

Condition 2) :
b is the median of a, b and c since b = ( a + c ) / 2 => a + c = 2b.
The average of a, b and c is ( a + b + c ) / 3 = ( 2b + b ) / 3 = 3b/3 = b.
Thus, the average and the median are the same.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

In cases where 3 or more additional equations are required, such as for original conditions with "3 variables", or "4 variables and 1 equation", or "5 variables and 2 equations", conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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