First, try to get rid of any factors in front of y. If 4/9 of something equals 16, then 4 units of something will be 9 times larger, that is, 16*9, right? So, we multiplied both parts by 9 to obtain equation 4y=16*9=144. Now, by the same logic, if 4 units of something is 144, then one unit will be 4 times less than 144. You now can complete the problem yourself, I will leave this last calculation for you to do.
Overall, we multiplied both parts by 9 and divided by 4. In other words, we multiplied both parts by 9/4 or divided both parts by 4/9--these are just two different ways to say the same thing! In general, if you have equation a*x=b and a≠0, then you divide both parts by a to get x=b/a, whatever the number a is. 4/9 is as good a number for the coefficient in front of x as any other number. So we divide 16 by 4/9, that is 16:4/9. In order to divide a number by a simple fraction p/q, one can multiply by the reciprocal fraction, q/p. So 16:4/9 becomes 16*9/4. The rest shall be easy.
Hope it helps.