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What is the difference between the sum of all even integers between 51

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What is the difference between the sum of all even integers between 51  [#permalink]

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New post 09 Jan 2019, 01:22
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[Math Revolution GMAT math practice question]

What is the difference between the sum of all even integers between 51 and 100 (inclusive) and the sum of all even integers between 1 and 50 (inclusive)?

A. 1,250
B. 2,000
C. 2,500
D. 3,750
E. 5,000

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What is the difference between the sum of all even integers between 51  [#permalink]

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New post Updated on: 09 Jan 2019, 09:26
each set has 25 even consecutive numbers,
and the difference between each element is 50,
(2, 4, 6, ... , 50)
(52, 54, 56, ... , 100)
so the difference between the two sets is 25*50 = 1250
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Originally posted by Mahmoudfawzy83 on 09 Jan 2019, 02:28.
Last edited by Mahmoudfawzy83 on 09 Jan 2019, 09:26, edited 2 times in total.
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Re: What is the difference between the sum of all even integers between 51  [#permalink]

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New post 09 Jan 2019, 09:00
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the difference between the sum of all even integers between 51 and 100 (inclusive) and the sum of all even integers between 1 and 50 (inclusive)?

A. 1,250
B. 2,000
C. 2,500
D. 3,750
E. 5,000

\({S_A}\,\,\, = \,\,\,2 + 4 + 6 + \ldots + 50\)

\({S_B}\,\,\, = \,\,\,\left( {50 + 2} \right) + \left( {50 + 4} \right) + \left( {50 + 6} \right) + \ldots + \left( {50 + 50} \right)\,\,\, = \,\,\,{S_A} + 25 \cdot 50\)

\(?\,\,\, = \,\,\,{S_B} - {S_A}\,\,\, = \,\,\,25 \cdot 50\,\,\, = \,\,\,1250\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {\rm{A}} \right)\)


This solution follows the notations and rationale taught in the GMATH method.

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Fabio.
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Re: What is the difference between the sum of all even integers between 51  [#permalink]

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New post 09 Jan 2019, 10:43
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the difference between the sum of all even integers between 51 and 100 (inclusive) and the sum of all even integers between 1 and 50 (inclusive)?

A. 1,250
B. 2,000
C. 2,500
D. 3,750
E. 5,000


Sum of all evens (52-100) - sum of all evens (2-50)

We need to find the number of terms in both groups.

Number of terms (52-100) =(100-52)/2 +1 =25. Number of terms (2-50) =(50-2)/2+1 = 25

The sum of an evenly spaced set is S=(a1+an)/2 *n

So the value we are looking for is ((52+100)/2)(25) - ((50+2)/)(25) = 25(152/2-52/2) = 25(100/2)=25*50=1250 (A)
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Re: What is the difference between the sum of all even integers between 51  [#permalink]

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New post 11 Jan 2019, 04:48
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\(( 52 + 54 + … + 100 ) – ( 2 + 4 + … + 50 ) = ( 52 – 2 ) + ( 54 – 4 ) + … + ( 100 – 50 ) = 50 + 50 + … + 50 = 50 * 25 = 1250.\)

Therefore, the answer is A.
Answer: A
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Re: What is the difference between the sum of all even integers between 51  [#permalink]

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New post 11 Jan 2019, 05:23
1
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the difference between the sum of all even integers between 51 and 100 (inclusive) and the sum of all even integers between 1 and 50 (inclusive)?

A. 1,250
B. 2,000
C. 2,500
D. 3,750
E. 5,000



sn = n/2 * ( 2a+ (n-1) *d )

for 1-50
even terms = 25
25/2 * ( 2* 2+ ( 24*2))
= 650
for 51-100
even terms = 25
25/2 * ( 2*52+ 24 * 2 )
= 1900
1900-650
= 1250 IMO A
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Re: What is the difference between the sum of all even integers between 51  [#permalink]

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New post 21 Jan 2019, 18:47
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the difference between the sum of all even integers between 51 and 100 (inclusive) and the sum of all even integers between 1 and 50 (inclusive)?

A. 1,250
B. 2,000
C. 2,500
D. 3,750
E. 5,000


To find the sum of a set of numbers, we first find the arithmetic average of the set and then multiply by the number of items in the set.

The sum of the even integers from 51 to 100, which is really from 52 to 100, is:

sum = average x number of items = (100 + 52)/2 x [(100 - 52)/2 + 1]

sum = 76 x 25 = 1,900

The sum of the even integers from 1 to 50, which is really from 2 to 50 is:

sum = average x number of items = (2 + 50)/2 x [(50 - 2)/2 + 1]

26 x 25 = 650

Thus, the difference is 1,900 - 650 = 1,250.

Alternate solution:

(52 + 54 + … + 98 + 100) - (2 + 4 + … + 48 + 50)

= (52 - 2) + (54 - 4) + … + (98 - 48) + (100 - 50)

= 50 + 50 + … + 50 + 50 (Note: Since there are there are twenty-five pairings, there are twenty-five 50s.)

= 50 x 25

= 1,250

Answer: A
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Re: What is the difference between the sum of all even integers between 51   [#permalink] 21 Jan 2019, 18:47
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