sa800 wrote:
Skywalker18 wrote:
Bunuel wrote:
What is the sum of odd integers from 35 to 85, inclusive?
A) 1,560
B) 1,500
C) 1,240
D) 1,120
E) 1,100
Number of odd integers = (85-35)/2 + 1
= 50/2 + 1
= 26
Sum of odd integers = (35+85)/2 * 26
= 60 * 26
= 1560
Answer A
How did you know to use that formula to find the number of odd numbers. what if it was 35 to 86? how do you know when to add the one vs not to add it
Hi
sa800 Thanks for your query.
To get a better understanding of ‘
when to add 1’ and ‘
when not to add 1’, let me show you a few examples related to the same concept. After looking at these examples, you will be able to see the relevance of adding 1.
In fact, you will see we always add 1, and it’s actually some other terms that need to be taken care of.
So, without much ado, let’s
LEARN THROUGH EXAMPLES!EXAMPLE 1:Find the number of odd integers from 34 and 85, inclusive.Observe that even though the smallest integer in the given range is 34, the smallest integer that we will consider is 35 (since 34 is even and 35 is the smallest odd integer). Let’s call 35 the first
meaningful number – this is the first number that satisfies what we want: an odd integer.
So, the list of odd integers in our range is
35, 37, 39, 41, …, and 85. (Here, 85 is included because the question mentions “inclusive”. We would also have included 34 had it been an odd integer!)
- In the list 35, 37, 39, …, 85, we have:
- First meaningful number = 35 and last meaningful number = 85.
- So, our question boils down to finding the number of odd integers starting from 35 and ending at 85. (Including BOTH)
- The formula we will use is this:
\(\frac{(Last meaningful number – first meaningful number)}{2} + 1\)
- Using this, the answer to this example is = \(\frac{(85 – 35)}{2} + 1\) = 26
EXAMPLE 2: Find the number of odd integers from 34 to 86, inclusive.This time also, the first
meaningful number = 35 (34, though the smallest, is not odd; the first odd number after 34 is 35.)
Similarly, the last
meaningful number = 85 (86, though the largest, is not odd; the largest odd number before 86 is 85.)
So, the list of odd integers in our range is again
35, 37, 39, 41, …, and 85. Hence, the number of odd integers between 34 and 86 is the same as in Example 1 (Ans = 26).
IMPORTANT: Observe how we added 1 in both examples. What varied was not the 1 but the elements that became our first and last meaningful elements.EXAMPLE 3: Find the number of odd integers from 33 to 86, inclusive.This time, the smallest number in the range is also the smallest
meaningful number! This is because 33 itself is the smallest odd integer in the range.
But even though 86 is the largest integer in the given range, it is not the largest odd integer. The largest odd integer in the range is 85 and thus, this is the largest
meaningful number for us.
So, the list of odd integers in our range is
33, 35, 37, …, and 85.- Using the formula we discussed, the answer to this example is = \(\frac{(85 – 33)}{2} + 1\) = 27.
TAKEAWAY: In each example, we began by finding the first and the last term of the required list from within the given range. For convenience, we called these terms “meaningful” terms. And after this, it was always the same formula! 😊
Hope this helps!
Best,
Aditi Gupta
Quant expert,
e-GMAT _________________