Answer :
Let A be the number of ants and B be the number of beetles. It is given that ¾ of the number of ants was equal to ⅓ of the number of beetles.
So, we can write,
[tex]\begin{gathered} \frac{3}{4}A=\frac{1}{3}B \\ \frac{A}{B}=\frac{4}{3\times3} \\ \text{ }\frac{A}{B}\text{ }=\frac{4}{9} \end{gathered}[/tex]Now, let A=4x and B=9x, where x is a constant.
There was a total of 442 ants and beetles. So,
A+B=442.
Hence, we can write
[tex]\begin{gathered} A+B=4x+9x \\ \text{ }A+B\text{ =13x} \\ \text{ 442=13x} \\ \frac{442}{13}=x \\ 34=x \end{gathered}[/tex]Since the number of ants is 4x, we can write
[tex]4x=4\times34=136[/tex]Therefore, the number of ants is 136.