A telephonic communication service is working at a carrier frequency of \[10GHz\]. Only $10\% $ of it is utilized for transmission. How many telephonic channels can be transmitted simultaneously if each channel requires a bandwidth of \[5KHz\]?
a) $2 \times {10^5}$
b) $2 \times {10^6}$
c) $2 \times {10^3}$
d) $2 \times {10^4}$

Hint: Just remember the conversions of the terms giga, mega, kilo and the decimal values of these terms. No other unit of a particular term is given, so dimensional conversion is not required to solve this problem.

Complete step by step answer:
 Let’s define all the terms given in the question before we for the calculation and further steps.
In the question it is given that the telephonic communication is working at a carrier frequency which is given by, ${f_c} = 10GHz$
We need to convert this value into the terms of $Hz$
That is ${f_c} = 10GHz = 10 \times {10^9}Hz$
We get, \[{f_c} = {10^{10}}Hz\]
In the question, it is given that; only $10\% $ of the carrier frequency is used for transmission.
That is, the amount of used frequency will be equal to:
\[{f_u} = \dfrac{{10}}{{100}} \times {10^{10}}Hz\]
After cancelling we get, \[{f_u} = 10 \times {10^8}Hz\]
Now we need to find how many channels can be transmitted and it is given that each channel require a band width of \[5KHz\]
Let $n$ be the number of channel that can be transmitted
Number of channels, $n$ is given by;
$n = \dfrac{{{f_u}}}{{5KHz}}$

\[ \Rightarrow n = \dfrac{{10 \times {{10}^8}}}{{5 \times {{10}^3}}}\]
$ \Rightarrow n = 2 \times {10^5}$
That is the number telephonic channels each require a band width of \[5KHz\] can be transmitted by using $10\% $ of carrier frequency is equal to $2 \times {10^5}$
So the final answer is option (a)

Note: The carrier frequency is used to carry a weak informal signal for the transmitting purposes, so that the informal signal can travel the required distance. Amplification and frequency modulation is used for these purposes. The energy in the carrier component is a waste of transmitter power because the carrier itself is not useful in transmitting the information. Therefore, the carrier is not transmitted in many modern modulation methods.