Question

A physical quantity is measured and the result is expressed as $\;nu$ where $u$ is the unit and $n$ is the numerical value. If the result is expressed in various units then,
A. $n\propto size\;of\;u$
B. $n\propto {{u}^{2}}$
C. $n\propto \sqrt{u}$
D. $n\propto \dfrac{1}{u}$

Answer 1

To solve the question, we can understand the relation between the unit and the numerical values by considering certain examples of any quantity in lower and higher units. We can see from a general case that whenever we consider higher units, the numerical value will be comparatively lower.

Complete step by step answer:
Here, we are given a physical quantity which is expressed as $nu\;$, where, $n$ is a numerical value which expresses the value of the quantity which we also know as magnitude and $u$ is the unit corresponding to the magnitude or the numerical value.

Let us consider an example as $100\;cm$. Here, $\;100$ is the numerical value and $\;cm$ is the unit which signifies centimeters. Now, let us consider various examples to understand the relation between unit and numerical value. Let us consider a quantity of length $5000\;millimeters$. Now, we know that we can convert millimeters to a higher unit centimeters from the relation
$10\,mm=1\,cm$
Hence, the quantity can be expressed in centimeters as $500\;centimeters$.

Similarly we can convert centimeters to a higher unit meters using the relation
$100\,cm=1\,m$
Hence, the quantity can be expressed in meters as $5\;meters$. From the above values, we can understand that as we go from smaller units to higher units, the numerical value keeps on decreasing. Hence, with the increase in units, the numerical value decreases. Hence, the relation is inverse, which can be expressed as
$Numerical\;Value\propto \dfrac{1}{Unit}$

Hence, the correct answer is option D.

Note: Here, we have considered the example for going on higher units from smaller units. We can understand the relation by going from higher to lower units also. For example, a substance of $1\;kilogram$ can be expressed as $1000\;grams$ . Hence, we can see that, with decrease in the unit, the numerical value increases. Hence, the relation between numerical value and unit is inverse.