An audio signal consists of two distinct sounds: one a human speech signal in the frequency band of 200 Hz to 2700 Hz, while the other is a high frequency music signal in the frequency band of 10200 Hz to 15200 Hz. The ratio of the AM signal band width required to send both the signals together to the AM signal band width required to send just the human speech is:

We recall the basic fact for double-sideband amplitude modulation (ordinary AM):

For a modulating (base-band) signal of bandwidth $$B_m$$, the transmitted AM signal occupies a total bandwidth

$$B_{\text{AM}} = 2B_m.$$

This happens because every frequency that exists in the modulating signal appears once in the upper side-band and once in the lower side-band, so the total spread is doubled.

Now we examine the two required situations one by one.

1. Only the human speech signal

The speech signal extends from $$200\;{\rm Hz}$$ up to $$2700\;{\rm Hz}.$$

Its own bandwidth is simply the difference between the upper and lower limits:

$$B_{m(\text{speech})}=2700-200=2500\;{\rm Hz}.$$

Therefore the AM bandwidth needed to transmit only the speech is

$$B_{\text{AM(speech)}} = 2B_{m(\text{speech})}=2\times2500=5000\;{\rm Hz}.$$

2. Both the speech and the music signals together

Besides the speech band we also have a music band that runs from $$10200\;{\rm Hz}$$ to $$15200\;{\rm Hz}.$$

The two bands do not overlap and there is a wide gap between $$2700\;{\rm Hz}$$ and $$10200\;{\rm Hz}.$$ Because bandwidth counts only the frequencies that are actually present, the total base-band bandwidth is the sum of the individual occupied stretches:

$$\begin{aligned} B_{m(\text{total})} &=(2700-200) + (15200-10200)\\[4pt] &=2500 + 5000\\[4pt] &=7500\;{\rm Hz}. \end{aligned}$$

Using the same AM formula, the transmission bandwidth required for the combined signal is

$$B_{\text{AM(total)}} = 2B_{m(\text{total})}=2\times7500=15000\;{\rm Hz}.$$

3. Required ratio

The question asks for

$$\frac{\text{AM bandwidth for speech + music}}{\text{AM bandwidth for speech only}} =\frac{B_{\text{AM(total)}}}{B_{\text{AM(speech)}}} =\frac{15000}{5000}=3.$$

Hence, the correct answer is Option A.