Why do both multiplying by 3 and dividing by 1/3 make the triangle bigger

In short, dividing by 1/3 = multiplying by 3.

Look at this example. . . 

$${\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{2}} = {\mathtt{8}}$$

$${\frac{{\mathtt{4}}}{\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)}} = {\mathtt{8}}$$  

$${\frac{{\mathtt{4}}}{{\mathtt{0.5}}}} = {\mathtt{8}}$$

See how there are 8 0.5's in 4?

Look at this also . . .

$$\frac{1}{2}\times{\frac{4}{5}=\frac{4}{10}=\frac{2}{5}$$

$${\frac{\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)}{\left({\frac{{\mathtt{5}}}{{\mathtt{4}}}}\right)}}$$=$$\frac{1}{2}\times{\frac{4}{5}=\frac{4}{10}=\frac{2}{5}$$

Instead of dividing by a fraction we flip the denomonator and numerator and then multiply by the fraction. Notice how 3 can be written as 3/1, which when it is flipped is 1/3.

Also when you are dividing by a third, you are seeing how many thirds go into a number, so it makes sense that there are 3 times more thirds in a number than 1's, because 1/3 goes into 1 3 times. 1/1 can be written as 3/3.

I may have overcomplicated this/been confusing but I hope I helped :) 

Thanks Zac, I am going to put this one in the wrap.

I like the answers that you present.