The sum of n consecutive positive integers is 45. What is th

[rakeshd347]

The sum of n consecutive positive integers is 45. What is the value of n?

(1) n is even

(2) n < 9

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

Hi Everyone this problem is from Manhattan Practise test 4. This problem looked very simple to me but I am stuck with it. Despite looking at the explanation which is long and hard to understand, I couldn't understand what so ever. However while I was giving exam I thought that there is a formula for the sum of n consecutive positive integers which is n(n+1)/2 and this problem could be solved easily. But that doesn't apply to this problem. Can someone please help me with this.

OA is E.

Thanks
Rakesh

[messi10]

If there was a formula and it does not come to you after 5-10 seconds, then just start solving the question by logic. There is no point wasting any more time. Remember that your aim is to look for insufficiency i.e. you should try and look for ways to make the statement insufficient.

1) n is even

The smallest positive even number is two so lets start there. Let the consecutive numbers be x and x + 1

therefore, x + x + 1 = 45
=> 2x = 44
=> x = 22

So the numbers can be 22 and 23, i.e. n = 2 is a possibility

Try 4 consecutive numbers now (x, x+1, x+2, x+3):

x + x +1 + x + 2 +x + 3 = 45
=> 4x + 6 = 45
=> 4x = 39

This is not possible as x will be a non-integer value.

Try 6 consecutive numbers next. But here, instead of writing the whole equation, you can do it a lot more quickly if you realize that 6 consecutive numbers will be 6x plus the sum of 1,2,3,4 and 5.

which is 6x + 15 = 45, which gives x = 5

So the set can also be 5,6,7,8,9,10 i.e. n = 6

Since we have two possible answers for n, statement 1 is insufficient.

2) n < 9

You don't need to test anything here because we can see from our cases above that both 2 and 6 are less than 9. So statement 2 is also insufficient.

Taking the two statements together does not tell us anything new . So answer is E

Hope this helps

[RonPurewal]

If there was a formula and it does not come to you after 5-10 seconds, then just start solving the question by logic. There is no point wasting any more time. Remember that your aim is to look for insufficiency i.e. you should try and look for ways to make the statement insufficient.

well said!

1) n is even

The smallest positive even number is two so lets start there. Let the consecutive numbers be x and x + 1

therefore, x + x + 1 = 45
=> 2x = 44
=> x = 22

this works, of course, but it seems to me that we're bringing way too much artillery to the table by doing algebra here.
to me, it's easier (and faster) just to try pairs of numbers.
20 and 21? that's only 41... too small.
umm... 23 and 24? ok, that's 47... too big.
22 and 23? that's 45. just right.
that took me about five seconds.

you can do the same for the larger sets of numbers.

what's most important, of course, is that you be able to use either of these approaches with equal facility.

[RonPurewal]

I thought that there is a formula for the sum of n consecutive positive integers which is n(n+1)/2 and this problem could be solved easily.

that formula only works if you start counting with "1". it doesn't work if the first number is anything else.