Hannah finds purple, blue, and silver space rocks while exploring. $\frac{2}{3}$ of the rocks are purple and $\frac{3}{4}$ of the remainder are blue. If there are $3$ silver rocks, how many space rocks does Hannah have?

Step-1: Find fraction of blue rocks:

We know that $P\left(A\right)+P\left(\overline{A}\right)=1$, where $P\left(A\right)$ is the probability of event $A$ and $P\left(\overline{A}\right)$ is the probability of event not $A$.

It is given that $\frac{2}{3}$ of the rocks are purple.

$$\begin{gathered} \Rightarrow \mathrm{Remaining}\mathrm{rocks}=1-\frac{2}{3} \\ \therefore \mathrm{Remaining}\mathrm{rocks}=\frac{1}{3} \end{gathered}$$

It is also given that, $\frac{3}{4}$ of the remainder are blue.

$$\begin{gathered} \Rightarrow \mathrm{Blue}\mathrm{rocks}=\frac{3}{4}\times \frac{1}{3} \\ \therefore \mathrm{Blue}\mathrm{rocks}=\frac{1}{4} \end{gathered}$$

Step-2: Find fraction of silver rocks:

$\mathrm{Fraction}\mathrm{of}\mathrm{silver}\mathrm{rocks}=1-\left[\left(\mathrm{Fraction}\mathrm{of}\mathrm{purple}\mathrm{rocks}\right)+\left(\mathrm{Fraction}\mathrm{of}\mathrm{blue}\mathrm{rocks}\right)\right]$

Substituting $\frac{2}{3}$for fraction of purple rocks and $\frac{1}{4}$ for fraction of blue rocks, we have

$\begin{array}{rcl}& \Rightarrow & \mathrm{Fraction}\mathrm{of}\mathrm{silver}\mathrm{rocks}=1-\left[\frac{2}{3}+\frac{1}{4}\right]\\ & =& 1-\left[\frac{2\left(4\right)+3}{12}\right]\left[\mathrm{Taking}\mathrm{LCM}\right]\\ & =& 1-\left[\frac{8+3}{12}\right]\\ & =& 1-\frac{11}{12}\\ & =& \frac{12-11}{12}\left[\mathrm{Taking}\mathrm{LCM}\right]\\ \therefore \mathrm{Fraction}\mathrm{of}\mathrm{silver}\mathrm{rocks}& =& \frac{1}{12}\end{array}$

Step-3: Find total number of rocks:

Let $x$ be the total number of rocks Hannah have.

It is given that there are $3$ silver rocks.

Since the silver rocks are $\frac{1}{12}$ of $x$, then

$$\begin{gathered} \Rightarrow \frac{1}{12}\times \left(x\right)=3 \\ \Rightarrow x=3\times 12\left[\mathrm{On}\mathrm{cross}\mathrm{multiplication}\right] \\ \therefore x=36 \end{gathered}$$

Hence, Hannah have $36$ space rocks.