Hi
jabhatta2 Thanks for your query.
As far as I can understand from your query, you are struggling with two main process skills – translation and visualization - ‘
TRANSLATION’ of the question stem and then ‘
VISUALIZATION’ of numbers on a number line.
Let me first help you with the translation, so that we at least understand the question completely. Then, we will move forward to a number line representation.
TRANSLATION:Question stem: “How many integers n are there such that r < n < s?”.
Let’s try to translate this sentence piece-by-piece into mathematical terms. This will ensure that we don’t miss even a single word.
- “How many integers n”
- Integers n – This itself makes it clear that ‘n’ is an integer.
- This is all you need to know about ‘n’. (Spend time here and see why you were confused about the nature of n. Build the correct skills right here!)
- “are there such that r < n < s”
- This part clarifies that we have to consider only those INTEGRAL (integers) values of ‘n’ which are STRICTLY less than ‘s’ and STRICTLY more than ‘r’.
So, from the question stem, we understand that we need to find the
NUMBER of integers (n) that lie between r and s, exclusive (r < n < s).
Now, let’s try to understand it’s visualization on a number line.
VISUALIZATION:This is the number line representation you shared:
The main thing you need to understand is that in every basic representation of a number line, the tick represents the exact integer written below it.
Let’s try to understand that by just considering the red portion above as an example. Note that the starting mark represents ‘r’, and the end represents ‘s’.
Red portion: Since the red line starts from some point between integers 1 and 2, we are considering ‘r’ to be something between 1 and 2. Say r = 1.5.
Similarly, since the red line ends at some point between 6 and 7, we are considering ‘s’ to be something between 6 and 7. Say 6.5.
Then, all the tick marks that you see between r and s are the integer values that n can take. If you count the ticks under the red line from the number line, you will find exactly 5 ticks. This means that there are exactly 5 integers between 1.5 and 6.5.
You can easily verify that by actually listing all the integers between 1.5 and 6.5. These are: 2, 3, 4, 5, and 6. See how it is exactly 5 as we got by counting the ticks also!
Now, just try going to ThatDudeKnows’ explanation again. You should get it! 😊
Hope this helps!
Best,
Aditi Gupta
Quant expert,
e-GMAT