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.

Question

p, q, and r are three points on the real number line where q = .

Quantity A:

Quantity B:

Explanatory Answer

Given: q =

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
+ =

Divide the equation by 2
=

What can we infer from the above equation?

The same inference that we can draw if someone told us that z =
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 is the arithmetic mean (average) of and .

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, will lie between and on the number line.

What are we trying to compare?

and

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, measures the distance between points and

Similarly, quantity B measures the distance between points and

We deduced that lies between and .

Therefore, the distance between and (quantity A) will be greater than the distance between and

Hence, Quantity A is greater than Quantity B