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Math Revolution GMAT Instructor V
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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Difficulty:   45% (medium)

Question Stats: 69% (02:09) correct 31% (01:38) wrong based on 79 sessions

### HideShow timer Statistics [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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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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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.
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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

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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1
=>

$$( 52 + 54 + … + 100 ) – ( 2 + 4 + … + 50 ) = ( 52 – 2 ) + ( 54 – 4 ) + … + ( 100 – 50 ) = 50 + 50 + … + 50 = 50 * 25 = 1250.$$

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

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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

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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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

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If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: What is the difference between the sum of all even integers between 51   [#permalink] 21 Jan 2019, 18:47
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