Are these numbers $3\,\!:27\,\!:\,:9\,\!:9$ in proportion?

Hint: Proportion is actually the operator or sign which represents that the two ratios are equal. For example $a\,\!:b\,\!:\,:c\,\!:d$or we can also write it as $\dfrac{a}{b}=\dfrac{c}{d}$ known as ratio form, both forms are right. Now as in question it is asked whether the given is in proportion or not. So we just need to find the ratio and if they are equal, then they are in proportion, if the ratios are not equal then they are not in proportion.

Complete step by step answer:
So, moving further with the question i.e. $3\,\!:27\,\!:\,:9\,\!:9$, asked whether it is in proportion or not. So we will check it through the ratio, if the ratio will be equal then they are in proportion or not. So writing the above given in the ratio form, we will get;
$\dfrac{3}{27}=\dfrac{9}{9}$
Now reducing this ratio form into simpler ratio, we will get
$\dfrac{1}{9}=\dfrac{1}{1}$
As we can say it directly by seeing that $\dfrac{1}{9}$and $\dfrac{1}{1}$are not equal. So $\dfrac{3}{27}$and $\dfrac{9}{9}$ are also not equal. So we can say that $3\,\!:27\,\!:\,:9\,\!:9$ are not in proportion.
Hence, $3\,\!:27\,\!:\,:9\,\!:9$ are not in proportion.

Note: In order to find the proportion, first always reduce them into ratio form, as it seems to be simple for us to solve. Now never forget to reduce them to a simpler ratio, then only you will get the right answer, about whether they are equal or not.