Using the Number Line

Quarter Wit, Quarter WisdomBy now, you know that we like to discuss visual approaches to problems.  A visual tool that we have used before for solving inequality and modulus questions is the number line. The number line is also useful in helping us solve many number properties questions.

A few things to keep in mind when dealing with number line:

  1. x < y (in other words, x is less than y) implies x is to the left of y on the number line. x and y could be in any region i.e. negative or positive but x must be to the left of y in any case.
  2. ‘x – y > 0’ (in other words, x – y is positive) implies x is to the right of y on the number line. Again, x and y could be in any region of the number line but x will be to the right of y i.e. x will be greater than y in any case.

The importance of these points is not apparent without a couple of questions.

Question 1: If a, b, and c are positive integers, is b between a and c?

Statement 1: b is 3 greater than a, and b is 5 less than c.

Statement 2: c is 5 greater than b, and c is 8 greater than a.

Solution: You might be tempted to use algebra with equations such as b = a + 3, b = c – 5 etc. But the question ‘is b between a and c’ should remind you of the number line. If we can figure out the relative position of ‘a’, ‘b’ and ‘c’ on the number line, we can say whether ‘b’ is between ‘a’ and ‘c’. Many of these ‘is this number less than that number’ questions can be easily done using the number line.

The question ‘Is b between a and c?’ essentially means ‘does b lay between a and c on the number line?’

Statement 1: b is 3 greater than a, and b is 5 less than c.

This means ‘b’ is 3 steps to the right of ‘a’ but 5 steps to the left of ‘c’ on the number line. It must lay between ‘a’ and ‘c’.

This statement alone is sufficient to answer the question.

Statement 2: c is 5 greater than b, and c is 8 greater than a.

‘c’ is 5 steps to the right of ‘b’ which means ‘b’ is 5 steps to the left of ‘c’. ‘c’ is 8 steps to the right of ‘a’ which means ‘a’ is 8 steps to the left of ‘c’. ‘a’ is further to the left of ‘c’ than ‘b’. So ‘b’ must be between ‘a’ and ‘c’.

This statement alone is sufficient to answer the question too.

Hence the answer is (D).

Working with equations would have been far too cumbersome. Don’t take my word for it; try it on your own.

Let’s look at another question based on the same concepts.

Question 2: The points A, B, C and D are on a number line, not necessarily in this order. If the distance between A and B is 18 and the distance between C and D is 8, what is the distance between B and D?

Statement 1: The distance between C and A is the same as the distance between C and B.

Statement 2: A is to the left of D on the number line.

Solution: This question specifically mentions number line.

We are given that distance between A and B is 18. We don’t know how to place A and B on the number line yet:

We don’t know in which region they lay. We can make a similar diagram for C and D. Note that we don’t know how to place these points. All we know is the relative distance between them. We also don’t know which one lays to the left and which one lays to the right.

Statement 1: The distance between C and A is the same as the distance between C and B.

Since distance between C and A is the same as distance between C and B, C must lay in the center of A and B. There are still many different ways of placing B and D so the distance between B and D is not known yet.

This statement alone is not sufficient.

Statement 2: A is to the left of D on the number line.

If the only constraint is that A is to the left of D, there are many different ways of placing A relative to D.

The distance between B and D will be different in different cases. This statement alone is not sufficient.

Let’s consider both the statements together.  C is in the middle of A and B and A is to the left of D. There are still two different cases possible.

The distance between B and D will be different in the two cases. Hence, we still cannot say what the distance between the two points is.

Answer (E)

Number line is a very simple yet powerful visualization tool. Try to use it in various questions for a simpler, more intuitive solution.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

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