GRE Newsletter Signup

Enter your email address below and subscribe to our newsletter

Absolute value GRE comparison question

Quantitative Comparison – Absolute Values

A very interesting question testing concepts in absolute values, number line, and the meaning of arithmetic mean of a set of numbers. A good question to understand the meaning of the absolute values of the difference between two numbers.


p, q, and r are three points on the real number line where q = \frac{2pr}{p+r}.

Quantity A: \left | \frac{1}{p} - \frac{1}{r}\right |

Quantity B: \left | \frac{1}{q} - \frac{1}{p}\right |

Correct Answer

Choice A – Quantity A is greater

Explanatory Answer

Given: q = \frac{2pr}{p + r}

Let’s make sense of this information.
Cross multiplying, we get q(p + r) = 2pr
i.e., qp + qr = 2pr

Divide the equation by pqr
\frac{1}{r} + \frac{1}{p} = \frac{2}{q}

Divide the equation by 2
\frac{\frac{1}{r} + \frac{1}{p}}{2} = \frac{1}{q}

What can we infer from the above equation?

The same inference that we can draw if someone told us that z = \frac{x + y}{2}
The inference we will draw is that z is the arithmetic mean (average) of x and y.

Using the same logic, we can infer that \frac{1}{q} is the arithmetic mean (average) of \frac{1}{p} and \frac{1}{r}.

A few key insights from this inference

If z is the average of x and y, it is evident that z will lie between x and y on the number line.

By the same logic, \frac{1}{q} will lie between \frac{1}{p} and \frac{1}{r} on the number line.

What are we trying to compare?

\left | \frac{1}{p} - \frac{1}{r}\right | and \left | \frac{1}{q} - \frac{1}{p}\right |

If x and y are two points on the number line, what does |x – y| measure?

|x – y| measures the distance between the two points x and y.

Extending that logic, \left | \frac{1}{p} - \frac{1}{r}\right | measures the distance between points \frac{1}{p} and \frac{1}{r}

Similarly, quantity B measures the distance between points \frac{1}{p} and \frac{1}{q}

We deduced that \frac{1}{q} lies between \frac{1}{p} and \frac{1}{r}.

Therefore, the distance between \frac{1}{p} and \frac{1}{r} (quantity A) will be greater than the distance between \frac{1}{p} and \frac{1}{q}

Hence, Quantity A is greater than Quantity B

There’s more where this question came from

GRE Algebra Practice Questions

Share your love

Leave a Reply

Your email address will not be published. Required fields are marked *

Stay informed about all things GRE, subscribe now!