Source: Official Guide Revised GRE 2nd Ed. Part 9; Section 5; #7

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# Section 5 #7

Quantity A : The sum of the odd integers from 1 to 199 Quantity B : The sum of the even integers from 2 to 198 Quantity A is greater., Quantity B is greater., The two quantities are equal., The relationship cannot be determined from the information given.

### 4 Explanations

1

Wahid El Chaar

I found the following method more intuitive:
- Quantity A: Adding 199 to 1 gives 200, then adding 3 and 197 also gives 200... that means if we keep pairing the remaining first and last terms, we will end up with 200 multiplied by the number of pairs.
Now, how to find the number of odd pairs between 1 and 199? Recall that there are exactly 200 integers between 1 to 200 inclusive, 100 of which are odd and 100 are even. So Quantity A only contains 99 of the even integers, but still has 100 odd integers. Therefore there are 100 terms in the set of all odd integers between 1 and 199. Which means there are 50 pairs between 1 and 199. Using this information, the sum must be 200*50

Quantity B: The same logic could be applied, there are 99/2 = 49.5 pairs of even numbers. 200*49.5

So Quantity A: 200*50
and Quantity B: 200*49.5

Divide both quantities by 200, and the question becomes a simple comparison of 50 and 49.5... Obviously the answer is (A)!

Nov 5, 2016 • Comment

Sam Kinsman

Yes, good job Wahid! :)

Rachael Nickerson

Sam, (of whoever is still at Magoosh) will the above solution provided by Wahid work every time? It was the solution I came to haphazardly, and want to be sure it didn't just work out specifically for this problem. Thanks!

Sam Kinsman

Hi Rachel,

Yes! This method will work, as long as the set of numbers you are adding together is evenly spaced. For example, if you are adding together all of the multiples of 4 between 4 and 24 (that is, 4, 8, 12, 16, 20, 24), you can use this approach. 4 + 24 = 28, and 8 + 20 = 28, and 12 + 16 = 28, so we can do 28 * 3.

If the set of numbers you are adding together is not evenly spaced (for example, 2, 4, 8, 16, 32), then the method won't work.

P P

I don't get why the even numbers is 49.5, I see it as being the same as the odd numbers

Sam Kinsman

Good question! For quantity B, we have all the even numbers from 2 to 198. We can think of this as all the even numbers from 1 to 198 (it's OK to add 1, since 1 is not even). So the question is, how many even numbers are there from 1 to 198? Well, there are 198 numbers in total. And half of them are even. So we have 198 / 2 = 99 odd numbers.

Now, we want to divide those 99 odd numbers into pairs. Each pair has 2 numbers. So 99/2 = 49.5 pairs.

2

Prithiviraj Damodaran

We can solve this much more quickly and intuitively by looking at the sequence in a different way.

Simple sequence, lets compare sum of odds between 1 to 6 and sum of evens between 1 to 6.

Odds: 1 to 6

* Manual workout, 1+3+5 = 9
* Shortcut: Total number of odds between 1 and 6 is 3, say n = 3, Formula of "sum of odds" is n * n, hence 3 *3 = 9

Evens: 1 to 6

* Manual workout, 2+4+6 = 12
* Shortcut: Total number of evens between 1 and 6 is 3, say n = 3, Formula of "sum of evens" is n * (n+1), hence 3 * 4 = 12

Applying the same principle to the above question
A. Sum of odds between 1 to 199: n = 100, n * n = 10000
B. Sum of evens between 2 to 198: n = 99, n * (n+1) = 9900

Apr 7, 2016 • Comment

Cydney Seigerman, Magoosh Tutor

Thanks for sharing your solution! :)

3

Tanvi Sengupta

Hi Chris, can we do it using a representative series, such as the sum of Odd numbers between 1 to 9 and even numbers between 2 and 8 ?

Sep 27, 2015 • Comment

Cydney Seigerman, Magoosh Tutor

Hi Tanvi! Great question. Yes, in this case, the method you've suggested would work great! As Chris explains in the video, for each "inning," B (the sum of even numbers) has 1 more than A (the sum of odd numbers). If we were to apply this logic to the sum of the series of odd numbers 1 to 9 and even numbers 2 to 8, we would see that from 2 to 8, there are 4 terms. So, after 4 terms, B would be 4 greater than A. However, since the series of odd numbers from 1 to 9 has 1 more term than the series of even numbers from 1 to 8, in the end, A would be 9-4=5 greater than B :)
Hope this helps!

7

Chris Lele

Dec 8, 2012 • Comment