Let's take an example. Let the fractions be 2/5 and 3/7. Now take the number 35. 2/5 of 35=14 and 3/7 of 35=15. Now multiply 14 by 15=210. So we have 2/5*35*3/7*35=14*15=210. But (2/5*3/7)*35*35 must also be 210, because the order doesn't matter in multiplication. 35*35=1225. If we multiply the fractions together we get 2/5*3/7=6/35 if we multiply the numerators and denominators. What is 6/35*1225? We know that 35 goes into 1225 35 times so 6/35*1225=6*35=210. Therefore, in this example, we had to multiply the numerators and denominators otherwise we would not have the right answer.
Now take the fractions a/b and c/d and the number n=b*d. a/b times n=a*d and c/d times n=b*c. Multiply these two numbers: (a*d)*(b*c). This is like multiplying 14 by 15 in the example. So (a/b)*(c/d)*n*n=((a/b)*n)*((c/d)*n)=(a*c)/(b*d)*n*n=(a*c)*n. This is like the example: (2/5)*(3/7)*35*35=((2/5)*35)*((3/7*35)=14*15=(2*3)/(5*7)*35*35=6/35*35*35=6*35=210.