The arithmetic mean of the even integers from 200 to 300

Question

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

DavidTutorexamPAL · Accepted Answer

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.
(C) is our answer.

CanCanuck90 · Answer

is the answer E?

300-200 + 1 = 101
299-201 + 1 = 99
101 - 99 = 2

saswata4s · Answer

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.

generis · Answer

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

Answer C

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

Answer C

Logo · Answer

Let's cut out wasted time on arithmetic here. This can simply be done with thought.

The amount of even integers will be two more integers than the odd range. Additionally, the even range will have an amount of 2 more than the odd range (201-200 & 300-299)

Therefore,

avg = sum / amount

no change = X+2 / X+2

MathRevolution · Answer

Even integers from 200 to 300 inclusive : 200 , 202, 204, .... 300 = 50 terms.

Artihemtic Mean :  \frac{200 + 300 }{ 2}  = 250 ( As they form an A.P.)

Odd integers from 200 to 300 inclusive : 201 , 203, 205, .... 299 = 50 terms.

Artihemtic Mean :  \frac{201 + 299 }{ 2}  = 250 ( As they form an A.P.)

Difference : 250 - 250 = 0

Answer C

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The arithmetic mean of the even integers from 200 to 300

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The arithmetic mean of the even integers from 200 to 300

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