How to count

GRE Quant Problem Solving
1

Hi everyone, this must be a stupid question. I am usually really bad at counting.

See this example, (x,y) are both positive integers, and the sum of x+y is less than or equal to 20, how many of those pairs in total?

I am bad at adding one or subtracting one when counting numbers, say, how many consecutive integer numbers are between 5 to 200, it should be 200-5+1 right? I guess that is why the above question complicates? I am doing like (1,19)…(19, 1), and then like forming a triangle in the first quartdrum, and then I don’t know how to count.

2

So in the above question you’ve correctly identified the extreme pairs, that is, (1, 19) and (19, 1). How many numbers are between 1 and 19, both inclusive? 1 + 19 - 1 = 19, which is the answer.

3

Not really…There are other pairs, say (2, 18), which is not a straight line between (19,1) and (1, 19).
I don’t exactly remember one of the choices, but certainly much bigger than 20. This question was found at GRE or GMAT forum.

4

That does not matter: we consider both (2,18) and (18,2).

5

BUT (1,1),(1,2)…(1,19) already have 19 pairs right. then, (2, 1), (2,2)…(2,18) are another 18 pairs, so on and so forth. How can there only be 19 pairs???

6

it is less than or equal to 20, not equal to 20.

7

Sorry, I didn’t realise that.

OK then, the answer is not 19. But even then, we can do it similar to the way described at - GregMat

Also,

is a good way too. Hint: (3,1), (3,2) … (3,17) has 17 pairs. You’ll get a sequence.

8

do you know what is the summing formula, like the first term adds the second term,etc.

9

You mean 1 + 2 + 3 + ... + n = \frac{n(n + 1)}{2}?

10

oh… so in this case, for the sum of x+y for (x,y) positive integer less than or equal to 200, will that be (199)*(199+1)/2=19900?

11

Yes.