What is the sum of odd integers from 35 to 85, inclusive? (Problem with my approach?)

GRE Quant Problem Solving
1

Hey guys.
I came across this question in one of the Quant Practice Videos of Greg. Wondering what’s wrong in my approach:

  1. 35 to 85 has a total of 51 numbers

  2. Their midpoint is 60. Hence, we can write 35 as (60 - 25) and 85 as (60 + 25). The end result being that all numbers cancel out and 60 is added 51 times (If they were a sum of all integers)

  3. The sum of the 51 numbers is 3060. Since, we need the sum of odd integers and half of the integers are odd, I did 3060/2 = 1530.

Where did I go wrong?

2

We don’t have half odd and half even integers
There are 51 numbers, among which we have 26 odd and 25 even numbers
So, the answer would be \frac{26}{51} * 3060 = 1560

3

Is there a way of knowing that it’s 26 odd and 25 even instead of vice versa? (without counting manually that is)

4

Yeah, number series always exist in a pattern of o,e,o,e,o…
So if a series starts at o and ends at e, it has equal number of odd and even numbers:
1,2,3,4
If it starts at e and ends at o, it has equal number of odd and even numbers:
2,3,4,5
If it starts at o and ends at o, it has an extra odd number:
1,2,3,4,5
If it starts at e and ends at e, it has an extra even number:
2,3,4,5,6

Hope this helps

5

There is a formula you could memorize for question such as sum of all odds, evens, multiple of 4,5,6,7 from X to Y it’s:

((last - first)/d ) + 1, where d is the difference between n and n+1,

So for this question it’s:
85 - 35 / 2 = 25 + 1 = 26

Or, to find numbers of odd from 1 - 10:
take first and last value that fits the criteria, in this case, 1 and 9. We can’t take 10 because it is not an odd integer.
9 - 1 / 2 = 4 + 1 = 5

Or for multiple of 7 from 20 - 200 it’s:
196 - 21 / 7 = 25 + 1 = 26

But for this kind of question that involves multiple I generally do
200/7 = 28.57 ignore the decimal there is 28 multiple of 7
20/7 = 2.85 ignore the decimal there is 2 multiple of 7
28 - 2 = 26 multiple of 7 from 20 - 200

1 Like
6

This is a great strategy! I will note it down, my friend!

7

Thank you! Makes a lot of sense now.

8

Hello! what do you mean by the difference between n and n+1? What is n in this case?