Quarter Wit, Quarter Wisdom: Analyzing a 700+ Quant Question

Quarter Wit, Quarter WisdomThis week, let’s look at some more properties of exponents and roots. Using a high level data sufficiency question, we will see how a number x is related to ?x and to x^3.

Question: Is x > y?

Statement 1: ?x > y
Statement 2: x^3 > y

It is one of those gorgeous questions that seem very simple at first but surprise you later.

The question asks us whether x is greater than y, but the statements tell us the relation between ?x, x^3 and y. If we know that ?x is greater than y, when can we say that x is certainly greater than y too? If we know that x is greater than ?x, then we can say for sure that x is greater than y too. Is x always greater than ?x? No. Look at the diagram given below.



?x is not defined for negative values of x so let’s ignore the section to the left of 0. When the value of x lies between 0 and 1, x is less than ?x (for example: when ?x = 1/2, x = 1/4). When the value of x is greater than 1, x is greater than ?x (for example: when ?x = 2, x = 4).

Similarly, let’s look at the relation between x^3 and x. If we know that x^3 is greater than y, when can we say that x is certainly greater than y too? If we know that x is greater than x^3, then we can say for sure that x is greater than y too.  Is x always greater than x^3? No. Look at the graph below.



When the value of x lies between -1 and 0 or in the region greater than 1, x is less than x^3 (for example: when x = 2, x^3 = 8). When the value of x lies in the region less than -1 or between 0 and 1, x is greater than x^3 (for example: when x = 1/2, x^3 = 1/8)

Let’s look at the statements now

Statement 1: ?x > y

Since ?x is not defined for negative x, we get that x >= 0. As we saw in the first graph above, for some values, x is greater than ?x, for others, x is less than ?x. When x is less than ?x, x may not be greater than y. So this statement alone is not sufficient.

Statement 2: x^3 > y

As we saw in the second graph above, for some values, x is greater than x^3, for others, x is less than x^3. When x is less than x^3, x may not be greater than y. So this statement alone is not sufficient.

Using both the statements together, we know that x >= 0.
When x lies between 0 and 1, we know that x >= x^3. Since statement (2) says that x^3 > y, we can say that x > y.
When x is greater than 1, we know that x > ?x. Since statement (1) says that ?x > y, we can deduce that x > y.
Therefore, for all possible values of x, we can say that x > y. Together the statements are sufficient. Answer (C).

It is important to understand these relations. This concept is very useful, especially for GMAT Algebra!

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 in Detroit, and regularly participates in content development projects such as this blog!