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Apr 2808:00 AM PDT
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08 Dec 2021, 09:45
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B
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Difficulty:
85%
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Question Stats:
52% (02:45) correct
48%
(02:40)
wrong
based on 249
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In the data set, {a, 0, 1, 6, 9, 10, 11}, consisting of seven distinct integers, one of the numbers is the mean as well as the median. How many values of ‘a’ are possible?
A. 1
B. 2
C. 3
D. 4
E. 5
Are You Up For the Challenge: 700 Level Questions
A. 1
B. 2
C. 3
D. 4
E. 5
Are You Up For the Challenge: 700 Level Questions
08 Dec 2021, 22:20
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The set has odd number of integers, which means the median has to be a part of the set.
Case 1 :
{a, 0, 1, 6, 9, 10, 11}
We know the sum / difference in the numbers should cancel out with respect to mean.
{a, -6, -5, 0, +3, +4, +5}
We see that the right side of the mean sums to +12, while the left side of the mean sums to -11. To balance and create a set where the mean is 6, a = 6-1 = 5
{0, 1, 5, 6, 9, 10, 11}
Due to its value, the position of 'a' should be between 1 and 6.
Case 2:
Lets assume a is the mean.
We know that the set consists of only distinct integer, hence the possible values of a include either 7 or 8
{0, 1, 6, a, 9, 10, 11}
Lets take a = 7
{0, 1, 6, 7, 9, 10, 11}
To balance the set around mean -
{-7, -6, -1, 0, +2, +3, +4}
We see that that the set is not balanced. Hence a cannot be 7.
Lets take a = 8
{0, 1, 6, 8, 9, 10, 11}
To balance the set around mean -
{-8, -7, -2, 0, +1, +2, +3}
We see that that the set is not balanced. Hence a cannot be 8.
Case 3:
Let's assume a to be at the right of 9
{0, 1, 6, 9, 10, 11, a}
To balance the set around mean -
{-9, -8, -3, 0, +1, +2, a}
Thus the value of 'a' should be 19 + 17 = 26. Also note that a can only be placed to the right of 11 due to its value.
Therefore, we can have only 2 sets possible.
IMO - B
Case 1 :
{a, 0, 1, 6, 9, 10, 11}
We know the sum / difference in the numbers should cancel out with respect to mean.
{a, -6, -5, 0, +3, +4, +5}
We see that the right side of the mean sums to +12, while the left side of the mean sums to -11. To balance and create a set where the mean is 6, a = 6-1 = 5
{0, 1, 5, 6, 9, 10, 11}
Due to its value, the position of 'a' should be between 1 and 6.
Case 2:
Lets assume a is the mean.
We know that the set consists of only distinct integer, hence the possible values of a include either 7 or 8
{0, 1, 6, a, 9, 10, 11}
Lets take a = 7
{0, 1, 6, 7, 9, 10, 11}
To balance the set around mean -
{-7, -6, -1, 0, +2, +3, +4}
We see that that the set is not balanced. Hence a cannot be 7.
Lets take a = 8
{0, 1, 6, 8, 9, 10, 11}
To balance the set around mean -
{-8, -7, -2, 0, +1, +2, +3}
We see that that the set is not balanced. Hence a cannot be 8.
Case 3:
Let's assume a to be at the right of 9
{0, 1, 6, 9, 10, 11, a}
To balance the set around mean -
{-9, -8, -3, 0, +1, +2, a}
Thus the value of 'a' should be 19 + 17 = 26. Also note that a can only be placed to the right of 11 due to its value.
Therefore, we can have only 2 sets possible.
IMO - B
08 Dec 2024, 07:30
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We can directly answer the question, as the required value a is the mean and median of the seven-numbered set. It can take up values between 7 and 9, considering distinct numbers, the number of values is 2(ie 8 and 9).
