GMAT Math

Hi all,
As I've been going though my GMAT quantitative review second edition, I came across a question which just doesn't make sense logically, making me think it could be a mistake. It is question #150 (if anyone has the book); but if not, I'll try to describe it:
Essentially there is a pentagon PQRST, where PQ=3, QR= 2 RS = 4, and ST = 5. It then asks which of the lengths 5, 10, and 15 could be the value of PT?
They split up the pentagon into three separate triangles: PQT; QRT; and RST.
It then says that because RS=4 and ST=5, RT must be < RS+ST (4+5) and therefore RT<9. So, okay, this makes sense. If RT is LESS THAN 9, its highest value should be 8 (as its not "less than or equal to - just less than). But, then it goes on to analyze triangle QRT (again, QR=2, but QT is unknown, and RT<9). The solution states "then RT+2 < 9+2 = 11, which then implies QT<11.". However, I don't understand why they are assuming that RT can equal 9. The solution already confirmed that RT is LESS THAN 9. So if RT is less than nine, then again, it's highest value can be 8. As such, QT should really be less than 8 + 2 (i.e., RT+QR), implying that QT is < 10, not 11 as the book states.
Going by the books solution, if RT is < 9, and QT is <11, then QT could, according to QT<11, be 10. But, if RT<9, and its highest value can therefore be 8, and QR = 2, then 8+2 =10 which would be = to QT (i.e., again according to the books logic that QT <11). This defies the law of a triangle in that the length of one side must be less than the sum of the other two sides, not equal to it. So am I correct in saying that the book is wrong?
I know this is a confusing question so hopefully someone has the actually text to refer to.
Thanks in advanced for nay help.

150. In pentagon PQRST, PQ = 3, QR = 2, RS = 4, and
ST = 5. Which of the lengths 5, 10, and 15 could be
the value of PT?
(A) 5 only
(B) 15 only
(C) 5 and 10 only
(D) 10 and 15 only
(E) 5, 10, and 15





Now in RST,


(5-4)<RT<(5+4)
so 1<RT<9

Now in QRT


for RT = 1.x(greater than 1) and 8.y(less than 9), but to make it easy we can take their extreme values 1 and 9 because if we find that some value is less/greater than 1 and 9 then it will be greater/less than 1.x and 8.y .

Also just don't assume that length will be an integer value. It can any value greater than 1 and less than 9.

we'll try to find the value of QT for both extremes of RT.
RT = 1
(2-1)<QT(2+1)
1<QT<3

RT = 9
7<QT<11

so now you know that RT will be either greater than 1 or less than 9, thus QT will be either less than 1 or 7, or greater than 3 or 11 , depending on the value of RT.

Also keep in mind that we have to find what is/are the possible value/s of PT.

so now in PQT

taking QT = 11(highest possible value)
(11-3)<QT<(11+3)
8<QT<14.

so even if you take max possible length of each side then to be there is no chance that PT will be equal to 15.

I hope it make it clear.

Kudos,
Hemant

I guess you are confusing < (less than) and <= (less then or equal to). Besides, the question did not say the length of the edges are integers. Thus if RT<9, doesn't mean RT<=8. It applies also to the length of QT. RT<9 <=> RT+2<9+2 <=> RT+2<11 which is not equivalent to RT+2<=10. So the books answer is correct.

When I did the question, my way was:

Given: PQ = 3, QR = 2
PR can be: > (3-2) & < (3+2). i.e, PR can be >1 && <5.

Step2:
Given: RS = 4, ST = 5,
So, RT can be >1 && < 9

Step3: PRT forms a traingle
So, PT can take values till less than 9+5 (14).

Answer: 5 & 10 are possible but 15 is not possible. The maximum side has < 14

hemant_rajput wrote:
so now in PQT

taking QT = 11(highest possible value)
(11-3)<QT<(11+3)
8<QT<14.

Thank you for the response, and for posting the question.
However, I'm still confused by your solution (not what the final answer is, as that is a given..but the steps you took). Specifically, you mentioned that QT's highest possible value is 11. But... If we've already established that RT is LESS THAN 9, thus RTs highest value is 8.999...(to infinity). So to say that QT's HIGHEST possible value is 11 would seem to me to be incorrect as 11 is > 8.999 + 2 = 10.999 (i.e., RT + QR). IN the end the .1 difference doesn't affect the final answer, but it's just the principle of how can the max value of QT be 11 when that value is greater than the max value of RT (i.e., 8.9) and QR (i.e., 2). RT cannot equal 9 becuase RS+ST=9, and so RT cannot equal RS+ST. So it has to be that it's maximum value is 8.999, and thus QTs maximum value is 10.999. Can you see where I'm getting confused with your solution?

Ninedot wrote:

hemant_rajput wrote:
so now in PQT

taking QT = 11(highest possible value)
(11-3)<QT<(11+3)
8<QT<14.

Thank you for the response, and for posting the question.
However, I'm still confused by your solution (not what the final answer is, as that is a given..but the steps you took). Specifically, you mentioned that QT's highest possible value is 11. But... If we've already established that RT is LESS THAN 9, thus RTs highest value is 8.999...(to infinity). So to say that QT's HIGHEST possible value is 11 would seem to me to be incorrect as 11 is > 8.999 + 2 = 10.999 (i.e., RT + QR). IN the end the .1 difference doesn't affect the final answer, but it's just the principle of how can the max value of QT be 11 when that value is greater than the max value of RT (i.e., 8.9) and QR (i.e., 2). RT cannot equal 9 becuase RS+ST=9, and so RT cannot equal RS+ST. So it has to be that it's maximum value is 8.999, and thus QTs maximum value is 10.999. Can you see where I'm getting confused with your solution?

I mentioned that QT's highest possible value is 11,which is not true, because I'm considering the extreme values. I've no information from the question about the precision of decimal value, so for my ease I chose and extreme value and see if I can solve using them, which we did. Assuming that that much precision won't be tested in real test, I can sometime take extreme values.

Kudos,
Hemant

Hemant you need make no assumptions for this. The triangle theorem says that any leg is strictly less than the sum of the other two. Strictly less.
So RT is less than 9, QT is less than 9+2, and PT is less than 9+2+3.
misterholmes

You can simplify this to a polygon. Any side must be shorter than the sum of all the other sides. 15 is too big.