How does $(0.06 + 0.06 + 0.06)$ + $(0.02 = 0.06/3)$ help intuit $0.20/0.06$?

Question

[ Source : ] This passage$^1$ exemplifies Posner's pragmatic, one might at times say cynical, appraisal of human and judicial nature -- a perspective that is enlightening even if not always persuasive. But this particular passage also contains, surprisingly, a mathematical error. $\color{green}{\text{0.20 divided by 0.06 does *not* equal 0.33; in fact it equals 3.33.}}$ (Readers whose recollection of decimals is fading can test this by adding 0.06 + 0.06 + 0.06; the result is 0.18, and the remaining 0.02 is exactly one more third of 0.06.)

I obviously know, and ask not about, the green.

Can someone please explain the bolded sentence? How exactly does it help intuit the green?

$^1$ Richard A. Posner, How Judges Think (2008). p. 142 Middle.

Answer 1

The bold sentence means that when you add $0.06$ three times and add one-third of it to the previously obtained sum, you get $0.20$

$$0.06+0.06+0.06 + \text{one third} (0.06)=0.20$$

$$\implies 0.06 \times (3+\frac 13)=0.20$$

$$\implies 0.06 \times 3.33=0.20 $$

$$\implies \color{blue}{\frac{0.20}{0.06}=3.33}$$

Answer 2

Maybe as the reader can see that $0.20$ is a bit more than three times $0.06$, so the result is going to be a bit more than $3$, and so the result $0.33$ is obviously wrong. Then, having "bought" one truth from the author, they are more likely to "buy" another (a bit harder one) - that the result $3.33$ is right, even if they are unwilling to go through the exact math about it.

In other words, the author of the blog post makes sure, early in the mathematical argument, that the reader has the impression the author "knows what they are talking about".