# Number Line

Read the following SAT test question and then select the correct answer.

Use the Knowsys Method for every question on the SAT.  The math method is always the same.  Read the question carefully, identify the bottom line, assess your options for solving the problem, select the most efficient method for solving the problem, and attack the problem.  Once you have finished your work, loop back to make sure that you solved for the bottom line.  Many problems have multiple steps, so you want to be sure that you are answering the question that was asked! Which of the following statements must be true of the lengths of the segments on line m above?

I.  AB + CD = AD
II.  AB + BC = AD – CD
III.  AC – AB = AD – CD

Bottom Line:  You must find out which of the statements above must be true.  You will need to mark each one true or false in order to find the correct answer.

Assess your Options.  You could plug in numbers for these spaces, but you are given no numerical information about the line.  You run the risk of finding answers that can be true rather than answers that must be true if you use this method.  Instead, keep the information abstract and use the line to evaluate the equations that you are given in order to see which ones work.

Attack the Problem:

I.  You are given AB + CD = AD.  You can tell from the line what AD must be equal to if you add up all the parts within AD.  From the line you can see that AD = AB + BC + CD.  Now plug that information into the given equation for AD.  Your new equation is AB + CD = AB + BC + CD.  One side has no BC while the other side has a BC.  You know that BC cannot be equal to zero because it is allotted a certain measure of space on the line.  This equation is not true.

II.  You are given AB + BC = AD – CD.  Look up at the line and see which parts of the line are still included if you take AD and subtract CD.  You are left with AB and BC.  Your new equation is AB + BC = AB + BC.  If you cannot get that information from looking at the line, think about this problem a little differently.  Substitute what you know about AD into the problem.  AD = AB + BC + CD.  Plug that in and your given equation is AB + BC = AB + BC + CD – CD.  The CD will cancel when you subtract it from itself, and you are left with AB + BC = AB + BC.  Will that always be true?  Yes, this equation is true.

III.  You are given AC – AB = AD – CD.  Look up at the line.  If you consider all of AC and then subtract out AB, what are you left with?  BC.  If you consider all of AD and then take out CD, what remains?  AB + BC.  Will BC = AB + BC?  Again, a space on a line cannot be equal to zero, so this equation is false.

Loop Back:  You broke down each of these equations and determined whether they were true or false, so look down at your answer choices.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III