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20 Apr 2020, 05:25
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D
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The average (arithmetic mean) of an increasing sequence of 9 consecutive integers is x. If next 3 consecutive integers are included in the sequence, what would be the average (arithmetic mean) of the new sequence?
A. 1.5
B. 3
C. x
D. x + 1.5
E. x + 3
A. 1.5
B. 3
C. x
D. x + 1.5
E. x + 3
20 Apr 2020, 05:38
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Bunuel
Method-1
CONCEPT: fo the terms in Arithmetic Progression (A.P.) [series of evenly spaced terms]
\(Mean = Median = \frac{(First-Term + Last-Term)}{2}\)
\(Mean = Median = \frac{(First-Term + Last-Term)}{2}\)
Average of 9 consecutive integers = Median = 5th Integer in ascending order = x
i.e. the Consecutive numbers become - - - - x (x+1) (x+2) (x+3) (x+4)
Next three integers will be (x+5) (x+6) (x+7)
i.e. Total Numbers in the series now = 9+3 = 12
Avera of new series = Median Average of first and last term = average of two middle terms = Average of 6th and 7th term = [((x+1) +(x+2)]/2 = x + 1.5
Answer: Option D
Method-2
After including next three terms, the average will be average of two middle terms
Since two middle terms (6th and 7th terms) are consecutive integers so their average will be p.5 form (decimal form)
i.e. Only Options A and D remain and remaining three options OUT
Since the value of x is uncertain therefore the mean can't be a fixed value it has to be x dependent therefore option A is out
Answer: Option D
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20 Apr 2020, 06:30
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Another way of solving this is with number plugging if you're not so keen on large formulas with unknowns.
Taking (1,2,3,4,5,6,7,8,9) as an example first set of numbers, the average is 5 (the middle number, since we have an odd number of consecutive integers) so x =5.
If we add 3 more numbers to the sequence, the new list will be (1,2,3,4,5,6,7,8,9,10,11,12)
We can calculate the average of this new set by adding them up in pairs starting from the extremities:
1 + 12 = 13
2 + 11 = 13
3 + 10 = 13
[..]
6 + 7 = 13
So the mean of the whole set is 13 x 6 / 12 = 6.5.
Since x = 5 we know that x+1.5 is the solution. Answer (D)
Taking (1,2,3,4,5,6,7,8,9) as an example first set of numbers, the average is 5 (the middle number, since we have an odd number of consecutive integers) so x =5.
If we add 3 more numbers to the sequence, the new list will be (1,2,3,4,5,6,7,8,9,10,11,12)
We can calculate the average of this new set by adding them up in pairs starting from the extremities:
1 + 12 = 13
2 + 11 = 13
3 + 10 = 13
[..]
6 + 7 = 13
So the mean of the whole set is 13 x 6 / 12 = 6.5.
Since x = 5 we know that x+1.5 is the solution. Answer (D)
