# GMAT Quantitative — The Number Line Problem

Problems involving Number Lines are a regular feature on the Quantitative section of the GMAT® . The best way to solve these problems is by using a combination of plugging values and visualizing them on a Number Line instead of trying to solve them algebraically. The GMAT® problem below highlights how to do the same.

DIRECTIONS: Each data sufficiency problem below consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you are to option

A if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
B if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
C if BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question asked;
E if statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

If x is a positive number less than 10, is z greater than average of x and 10?

(1) z is closer to 10 than to x on the number line
(2) z = 5x

The problem states that x lies between 0 and 10. The question is asking whether z is greater than the average of x and 10.

The biggest mistake is to translate the question into an algebraic format is z > (x+10)/2. Instead it is better to proceed to the first statement that states that z is closer to 10 that it is to x on the number line.

What does this mean when represented on a number line?

The mid-point between x and 10 will be equidistant from both x and 10. If z if closer to 10 than to x on the number line it means that it will be to the right of the mid-point between x and 10. Since the mid-point between x and 10 is nothing but the average of x and 10, z will be to the right of it, which means that z is greater than the average of x and 10. Hence, the question can be answered with the first Statement (1) alone.

We then move to the next statement that says that z = 5x. The best way to go about this is to plug values of for x and then see if the question can be answered. We know that x lies between 0 and 10.

If x = 2 then z = 10 and the average of x and 10 is 6, so z is greater than the average of x and 10; similarly z will be greater than the average for any value of x greater than 2.

If x = 1 then z = 5 and the average of x and 10 is 5.5, so z is less than the average of x and 10.

So from Statement (2) we cannot say whether z is greater than the average or not. Hence the answer is (A).

So when presented with a question that is based on Number Line you have to evaluate each statement by either representing it on a number line or by plugging values. Usually visualizing information on a Number Line simplifies the problem and helps you reach the answer faster.

If you had gone through a previous post, you will see that visualizing information on the number line can take you to the solution much faster even when the problem has nothing to do with numbers!