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# What is the ratio of the sum of the odd positive integers between 1 an

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Math Expert
Joined: 02 Sep 2009
Posts: 57279
What is the ratio of the sum of the odd positive integers between 1 an  [#permalink]

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15 May 2018, 02:49
00:00

Difficulty:

55% (hard)

Question Stats:

60% (02:39) correct 40% (02:20) wrong based on 45 sessions

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What is the ratio of the sum of the odd positive integers between 1 and 100, inclusive, and the sum of the even positive integers between 100 and 150, inclusive?

(A) 2 to 3

(B) 5 to 7

(C) 10 to 13

(D) 53 to 60

(E) 202 to 251

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Posts: 503
Location: India
GMAT 1: 700 Q49 V36
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Re: What is the ratio of the sum of the odd positive integers between 1 an  [#permalink]

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15 May 2018, 03:08
5
First, lets find out sum of all odd integers between 1 and 100

Total # of odd integers between 1 and 100

= $$\frac{99 - 1}{2}+1$$
= 50

Sum of all odd integers between 1 and 100
= total # of terms * average (First term + Last term /2)
$$=50 * \frac{1+99}{2}$$
$$=50 * 50$$

Now, lets find out sum of all even integers between 100 and 150

Total # of even integers between 100 and 150

= $$\frac{150 - 100}{2}+1$$
= 26

Sum of all even integers between 100 and 150
= total # of terms * average (First term + Last term /2)
$$= 26 * \frac{100+150}{2}$$
$$=26 * 125$$

There ratio
$$=\frac{50 * 50}{26 * 125}$$
$$= \frac{10}{13}$$

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Posts: 3018
Re: What is the ratio of the sum of the odd positive integers between 1 an  [#permalink]

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15 May 2018, 03:12

Solution

To find:
• Ratio between the sum of the odd positive integers between 1 and 100, inclusive, and the sum of the even positive integers between 100 and 150, inclusive

Approach and Working:

• From 1 to 100, inclusive, there are total 50 odd positive integers, which form an arithmetic progression
o 1st term of the AP = 1
o Common difference of the AP = 2
o Number of terms of the AP = 50
o Hence, sum of the AP = $$\frac{50}{2}$$ (1 + 99) = $$25 * 100$$ = 2500

• From 100 to 150, inclusive, there are 26 even positive integers, which form an arithmetic progression
o 1st term of the AP = 100
o Common difference of the AP = 2
o Number of terms of the AP = 26
o Hence, sum of the AP = $$\frac{26}{2}$$ (100 + 150) = $$13 * 250$$ = 3250

• Therefore, the required ratio = 2500 : 3250 = 250 : 325 = $$25 * 10 : 25 * 13$$ = 10 : 13
Hence, the correct answer is option C.

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Re: What is the ratio of the sum of the odd positive integers between 1 an  [#permalink]

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11 Aug 2019, 06:01
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Re: What is the ratio of the sum of the odd positive integers between 1 an   [#permalink] 11 Aug 2019, 06:01
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