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

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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What is the difference between the sum of all even integers between 51  [#permalink]

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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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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Senior Manager Status: Manager Joined: 27 Oct 2018 Posts: 451 Location: Egypt Concentration: Strategy, International Business GPA: 3.67 WE: Pharmaceuticals (Health Care) What is the difference between the sum of all even integers between 51 [#permalink] ### Show Tags 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 _________________ 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. GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 937 Re: What is the difference between the sum of all even integers between 51 [#permalink] ### Show Tags 09 Jan 2019, 09:00 2 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. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Manager Joined: 16 Oct 2011 Posts: 93 GMAT 1: 620 Q37 V38 GMAT 2: 640 Q36 V41 GMAT 3: 650 Q42 V38 GMAT 4: 650 Q44 V36 GMAT 5: 570 Q31 V38 Re: What is the difference between the sum of all even integers between 51 [#permalink] ### Show Tags 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) Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7590 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the difference between the sum of all even integers between 51 [#permalink] ### Show Tags 11 Jan 2019, 04:48 1 => $$( 52 + 54 + … + 100 ) – ( 2 + 4 + … + 50 ) = ( 52 – 2 ) + ( 54 – 4 ) + … + ( 100 – 50 ) = 50 + 50 + … + 50 = 50 * 25 = 1250.$$ Therefore, the answer is A. Answer: A _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: What is the difference between the sum of all even integers between 51  [#permalink]

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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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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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