Interpolation and decimation I have a sinusoid in continuous time, with a frequency of 18kHz, it is sampled ideally with a continuous to discrete convertor, with a frequency of 27kHz. After that, we change the sampling speed in discrete time, using interpolator system and decimate system,so that result frequency equivalent in discrete time is 13.5KHz. 

The interpolation factors and decimation factors are:

a) I=3 and D=2.

b) I = 1 and D=2

c) I= 3 and D=4

d) I = 4 and D = 3

I see this example on the internet and I don't know what is the correct one. Can you please help me to know what is the true answer and how it is calculated?
It seems to me that the only way in which one of those solutions is possible is without aliasing. Probably there is a typo error, and the actual sampling rate is greater than 36 kHz. In such a case, for example, if the sampling rate was actually 37 kHz, we can interpolate with a factor I = 4, and then decimate with a factor D = 3 (in that order):
$$
f' = f / I = 18.0~kHz / 4 = 4.5~kHz; \\
f'' = f' \times D = 4.5~kHz \times 3 = 13.5~kHz
$$
Where $f'$ is the frequency component after interpolation, and $f''$ is the frequency component after decimation. Therefore, my final solution is (d) if, and ONLY if, there is a typo and the sampling rate ($f_s$) is actually greater than the Nyquist frequency: $$f_s &gt; f_{Nyquist} = 2\times f = 2\times 18.0~kHz = 36.0~kHz$$.