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# The arithmetic mean of the even integers from 200 to 300

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Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3306
Location: India
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The arithmetic mean of the even integers from 200 to 300  [#permalink]

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22 Dec 2017, 06:00
00:00

Difficulty:

35% (medium)

Question Stats:

73% (01:26) correct 27% (02:09) wrong based on 64 sessions

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The arithmetic mean of the even integers from 200 to 300 (both inclusive) is greater by what number than the arithmetic mean of the odd integers in the same range?

A. -2
B. -1
C. 0
D. 1
E. 2

Source: Experts Global

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Re: The arithmetic mean of the even integers from 200 to 300  [#permalink]

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22 Dec 2017, 06:33

300-200 + 1 = 101
299-201 + 1 = 99
101 - 99 = 2
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Joined: 27 May 2014
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Re: The arithmetic mean of the even integers from 200 to 300  [#permalink]

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22 Dec 2017, 06:56
2
pushpitkc wrote:
The arithmetic mean of the even integers from 200 to 300(both inclusive) is greater by what number
than the arithmetic mean of the odd integers in the same range?

A. -2
B. -1
C. 0
D. 1
E. 2

Source: Experts Global

Lets consider a smaller range:
(1) 200 to 204
even numbers: 200, 202, 204; Avg = 202
Odd numbers: 201, 203; Avg = (201+203)/2 = 202
Difference = 0

(1) 200 to 210
even numbers: 200, 202, 204, 206, 208, 210; Avg = (204+206)/2 = 205
Odd numbers: 201, 203, 205, 207, 209 Avg = (201+203+205+207+209)/5 = 205
Difference = 0

Ans: C.
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Re: The arithmetic mean of the even integers from 200 to 300  [#permalink]

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22 Dec 2017, 06:57
Writing down the first few elements of the two sequences can help us understand what we're dealing with.
This is a Logical approach.

The even numbers between 200 to 300 are 200, 202, 204..... 296, 298, 300.
As usual for sequences on the GMAT, the difference between each two terms is constant.
So, the sum of the first and last terms (200+300) equals the sum of the second and second-last terms (202+298) and so on.
Then the average of this sequence is just (200+300)/2=250.

Similarly, the odd sequence is 201, 203, 205....295, 297, 299.
Once again, the sum of the first and last (201+299) is the same as the sum of the second and second-last (203 + 297) and so on.
Then the average of this sequence is (201+299)/2=250.

Therefore the difference between averages is 0.
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The arithmetic mean of the even integers from 200 to 300  [#permalink]

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22 Dec 2017, 08:54
pushpitkc wrote:
The arithmetic mean of the even integers from 200 to 300(both inclusive) is greater by what number
than the arithmetic mean of the odd integers in the same range?

A. -2
B. -1
C. 0
D. 1
E. 2

Source: Experts Global

Evenly spaced sequence

The arithmetic mean of an arithmetic sequence is
$$\frac{FirstTerm + LastTerm}{2}$$

Even integers, arithmetic mean:
$$\frac{200+300}{2} = 250$$

Odd integers, arithmetic mean:
$$\frac{201 + 299}{2} = 250$$

Difference: 0

Small sample
As posters above demonstrate, to use a small sample is often shrewd.
Yet another way to take a small sample:

Replicate the pattern of the numbers in the prompt
• begin and end with even numbers
• if you're not sure, use integers that end in the same units digit (here, 0), and
• if you're going to do all the math, choose small numbers

The arithmetic mean of the even integers from 0 to 10, inclusive
$$0, 2, 4, 6, 8, 10$$, where $$A =\frac{S}{n} = \frac{30}{6} = 5$$

Arithmetic mean of odd integers from 0 to 10, inclusive
$$1, 3, 5, 7, 9$$, where $$A=\frac{S}{n} = \frac{25}{5} = 5$$
Difference: 0

The arithmetic mean of the even integers from 200 to 300 &nbs [#permalink] 22 Dec 2017, 08:54
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