Grades 5-6 Video Solutions 2015-Level 5&6 Video Solutions 2015 problem29

Level 5&6 Video Solutions 2015 problem29

Video Transcription

Question number 29. Four points lie on the line. The distances between them are, in increasing order, 2, 3, K, 11, 12, and 14. What is the value of K?  Well, let's draw this number line, and let's label four points on it. And it could be a number line. We really don't know the location of the first point, which I will label A.  So let's say that it is at 0. Then there is a point B, C, and D somewhere. If A is at 0, and D is farthest away from A, then let's make that distance a 14.  Okay, now what we have are some remaining values. If we assume that B is 2 away from A, then the distance between B and D is 12, which is good.  We're using our numbers up over here. And we have 3 and K remaining. So where is 3 and where is K? If 3 is the distance between B and C, then C is at 5.  And the distance between C and D, that would be a 9. The distance between C and A would be a 5. But those numbers do not match our increasing list of distances.  So B cannot be at 2. That is not possible. So let us move B to 3. In that case, let's make the distance here between B and C an 11. That is a possibility.  And where shall we put the number 12? Well, if C is at 12, then the distance between C and A is 12. The distance between B and C here is 9. So perhaps that is K. That's equal to 9.  And then we have a distance of 2 between C and D. So all the numbers are counted for. We have here a distance of 3, a distance of 9, that is equal to K, a distance of 11, and a distance of 2. So K could be equal to 9.  Are there any other ways to construct this number line? Let's try once again. Let's make pretty much the same number line. Let's put an A here at 0. Let's put a B, C somewhere, and D at 14.  If we now demand that B is 2 away from A and C is 3 away from D, write that over here, C is 3 away from D, then C is at 11.  The distances between B and C are again 9, and that would be the value K, the same value as before. And then we have a value of 11 separating A and C.  So again, we have 2, 3, K is 9, 11, and 14. Where is the value of 12? That is the distance between B and C. These are 12 apart. So we have all of our distances again, and two different possibilities.  So in either case, E is the answer to the question, but missing distance in increasing order, the letter of A has to be 9, so the answer is E.

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