# Number Line

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

For all math problems, read the problem carefully.  Identify your bottom line, and assess your options for reaching that bottom line.  Select the most efficient method to work the problem, and attack the problem.  Your last step is to loop back and make sure that the answer addresses the bottom line. On the line above, if AB < BC < CD < DE, which of the following must be true?

(A) AC < CD
(B) AC < CE
(C) AD < CE
(D) AD < DE
(E) BD < DE

You must decide which of the answer choices is true.  This is a “which of the following” question, and it would be really difficult to predict an answer choice, so start with answer choice (E).

(E) BD < DE

Normally you cannot depend on a drawing that has the words "Note: Figure not drawn to scale" underneath it, but this particular line follows the rule that each line segment is longer than the last.  Once you have ascertained that the image matches the information that you have been given, you can use the image to draw conclusions.  You can look at the line provided and see that this does not have to be true:  BD actually looks longer than DE.  You can also think of BD as BC + CD.  You don’t have any information to compare BC + CD and find out whether it is less than DE.  One way of proving this is to imagine that AB is 1, BC is 2, CD is 3 and DE is 4.  That fits AB < BC < CD < DE.  However, 2 + 3 is not less than 4.  Eliminate (E).

(D) AD < DE

Look back at the line.  You can clearly see that AD is longer than DE.  Eliminate (D).

(C) AD < CE

This one is not obvious from a glance at the line.  Think of AD as AB + BC + CD and CE as CD + DE.  Both equations share CD, but do we know that AB + BC is smaller than DE?  No.  Think about what would happen if AB had a high value.  Suppose that AB = 10, BC = 11, CD = 12 and DE = 13.  In that case AD would be 33 while CE would only be 25, and this answer choice would be false.  Eliminate (C).

(B) AC < CE

Think of AC as AB + BC and CE as CD + DE.  Compare the two.  You know that AB must be smaller than CE and BC must be smaller than DE.  What you really have is: small number + small number < big number + big number.  Is that always true?  Yes.  You do not have to check the last answer choice.

The correct answer is (B).

On sat.collegeboard.org, 70% of the responses were correct.

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