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\'manpreet singh: Master | Next Rank: 500 Posts; Posts: 271; Joined: Tue May 22, 2012 3:22 amFollowed by: 3 members; Thanked: 7 times

by \'manpreet singh » Sat Jul 07, 2012 10:23 pm

Q.1)For any positive integer n, the sum of the first
n positive integers equalsn(n+1)/2 . What is the sum
of all the even integers between 99 and 301 ?
(A) 10,100
(B) 20,200
(C) 22,650
(D) 40,200
(E) 45,150

Q.2)This year Henry will save a certain amount of his
income, and he will spend the rest. Next year Henry
will have no income, but for each dollar that he saves
this year, he will have 1 + r dollars available to spend.
In terms of r, what fraction of his income should Henry
save this year so that next year the amount he has
available to spend will be equal to half the amount that
he spends this year?

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Source: Beat The GMAT — Problem Solving | Sat Jul 07, 2012 10:23 pm

eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Sat Jul 07, 2012 10:28 pm

'manpreet singh wrote:Q.1)For any positive integer n, the sum of the first
n positive integers equalsn(n+1)/2 . What is the sum
of all the even integers between 99 and 301 ?

For an evenly spaced consecutive set,

Sum = number of terms * (average of first and last term)
We have first term as 100, last as 300, and even integers means the difference is 2.
Number of terms = (last - first)/(difference) + 1 = (300-100)/2 + 1 = 101.
Average of first and last = (100+300)/2 = 200.
Hence Sum = 101*200 = 20200.

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eagleeye: Legendary Member; Posts: 520; Joined: Sat Apr 28, 2012 9:12 pmFollowed by: 49 members; Thanked: 339 times; GMAT Score:770

by eagleeye » Sat Jul 07, 2012 10:39 pm

'manpreet singh wrote:
Q.2)This year Henry will save a certain amount of his
income, and he will spend the rest. Next year Henry
will have no income, but for each dollar that he saves
this year, he will have 1 + r dollars available to spend.
In terms of r, what fraction of his income should Henry
save this year so that next year the amount he has
available to spend will be equal to half the amount that
he spends this year?

Let the fraction saved this year be x. Therefore the fraction spent = 1-x.
Next year the money he has to spend = (1+r) for each dollar saved = x*(1+r)
This should be equal to half the money spent this year => x*(1+r) = (1-x)/2
=> 2x+2rx = 1-x => x(2+1+2r) = 1 => x = 1/(3+2r).

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Jul 08, 2012 2:54 am

'manpreet singh wrote:Q.1)For any positive integer n, the sum of the first
n positive integers equalsn(n+1)/2 . What is the sum
of all the even integers between 99 and 301 ?
(A) 10,100
(B) 20,200
(C) 22,650
(D) 40,200
(E) 45,150

I posted an explanation here:

https://www.beatthegmat.com/for-any-posi ... tml#361604

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sun Jul 08, 2012 2:56 am

'manpreet singh wrote: 2)This year Henry will save a certain amount of his
income, and he will spend the rest. Next year Henry
will have no income, but for each dollar that he saves
this year, he will have 1 + r dollars available to spend.
In terms of r, what fraction of his income should Henry
save this year so that next year the amount he has
available to spend will be equal to half the amount that
he spends this year?

I posted a solution here:

https://www.beatthegmat.com/official-gui ... 61258.html

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\'manpreet singh: Master | Next Rank: 500 Posts; Posts: 271; Joined: Tue May 22, 2012 3:22 amFollowed by: 3 members; Thanked: 7 times

by \'manpreet singh » Tue Jul 10, 2012 8:04 am

Thanks Eagle and Mitch .
But I feel it is tough to immediately take the algebraic route.
So I think the plug and chuck method is the best for this case.

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