M09-34

Official Solution:


Even when we consider both statements together we cannot have a definite answer. Consider two examples below:




Answer: E
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Hi, I Have a Question. I May be missing something but, when we take both statements together isn't it enough to answer the question because if AC + BC must = 10 and Also AC + AB must be More than 10 ? ( that is don't we need the same set of points to satisfy both statement, here we take 2 different set of points but the second set dose not satisfy both statements. Therefore I believe that because only the first statement satisfies both statements and it = 10 we should be able to say that it is not greater than 10. Please let me know if I am missing something. Thank you.

If A, B, and C are distinct points on the number line. Is the length of the line segment AB less than 10?


(1) The sum of the lengths of line segments AC and BC is 10

(2) The sum of the lengths of line segments AB and AC is more than 10




WHY doesn't the second set satisfy both statements?

BC = 2
AB = 6
AC = 6 + 2 = 8

(1) The sum of the lengths of line segments AC and BC is 10 --> AC + BC = 8 + 2 = 10.

(2) The sum of the lengths of line segments AB and AC is more than 10 --> AB + AC = 6 + 8 = 14 > 10.
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so since the question does not specify that the points are in the order a,b, then c.. one cannot make this assumption?

Yes, if the order is not specified you cannot assume that there is any particular one.
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Hi ,
Is there any other method to solve this question quickly ? i took 5 mins to solve this question and even then got it wrong because i got confused between all the possible sets / arrangements i had to assume for this question !

I think , from the explanation given above, It is not clear whether both statement together are sufficient or not.
So, I represent a case where AB = 10 (could be less than 10 too , as represented by bunuel).

A---------6---------C-------4--------B
here
AC + BC = 10
and
AB + AC > 10
So we cannot say for sure that AB is less than 10.

Hi, Can we also rule out both the statements immediately because neither the question stem nor the two statements tell us the positions of the order of the given points?

Hi,

what is the best way to solve questions like this ?

Thanks.

We have 6 ways to order A, B and C.

1- A B C
2- A C B
3- B A C
4- B C A
5- C A B
6- C B A

We can say 5 and 6 are flip of 1 and 3 respectively but let's not make things complex.

Statement 1 - If sum of the lengths of line segments AC and BC is 10, then AB should be 10 or less in cases 1,3 5 and 6 while equal to 10 in 2 and 4. (Assume points 2, 6 and 12 on coordinate line and then check cases 1-6 above to understand the point) - Not Sufficient.
Statement 2 - If the sum of the lengths of line segments AB and AC is more than 10, then AB could theoretically be less or greater than 10 in any of the 6 cases. AB are closest in case 1, 3, 5 and 6 but if sum of AB and AC is greater than 10, AB could be less equal or greater than 10 (example case 1 (order of points A,B,C on number line) above, points 2,6,9 (AC+BC = 11 i.e. > 10 but AB < 10), points 2,12,13 (AB + BC = 21 i.e. > 10 but AB = 10) or points 2,13,14 (AB + AC = 23 i.e. > 10 but AB > 11) all illustrate this point- Not Sufficient.

Combined, since 2 doesn't add much info to 1, answer should be E.

Hi, but considering the two statements together, doesn't it mean that C must lie between A and B and therefore AB is always 10 (AC+CB=10, from S1), and therefore AB cannot be less than 10? In which case both statements together are sufficient. Many thanks for your help!

The solution gives TWO possible cases



Both of them satisfy the statements and give different answers to the question.
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Bunuel could you check my work?

If we denote \(|A-B|=x\) and \(|B-C|=y\) the question becomes whether \(x<10\)

statement I: \(x+2y=10 \implies\) 2 variables, 1 equation: insufficient

statement II: \(2x+y>10 \implies\) 2 variables, 1 equation: insufficient

statement I & II: \(\frac{(10-x)}{2} +2x>10 \implies x>\frac{10}{3} \implies E\)
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I think this is a high-quality question.

Bunuel yashikaaggarwal when st 1 says The sum of the lengths of line segments AC and BC is 10
does this mean the points are in this order A and then C or B and then C or can it be in the opp order too (C then B)?

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