Logical fallacy that suggests $3/9=3/10$

Question

I have seen that $3/9 = 1/3$ can be written as $0.3$. However, $0.3=3/10$. Does this mean that $3/9=3/10$, or am I confused?

Answer 1

The error is that $1/3\ne 0.3$. So you find

$$\frac39=0.333\dots > 0.3 = \frac3{10}$$

which makes sense, since you're dividing by less. That is, $9$ is smaller than $10$, so $3/9 > 3/10$.

Answer 2

The issue here is the difference between equality $=$ and approximation $\approx$. Here is what’s actually happening:

$$\frac39 = \frac13 = 0.33333\cdots \approx 0.3$$

The approximation comes when we round down.

Because we know that $0.3=3/10$, we can just replace $0.3$ with $3/10$ on the righthand side, giving us

$$\frac39 = \frac13 = 0.33333\cdots \approx \frac3{10}$$

We don’t need three names for the same value, so let’s take out the second two “equal to” terms:

$$\frac39 \approx \frac3{10}$$

Voilà! If you decide to use $=$ instead, that’s you’re prerogative, but in most instances it’s slightly wrong $\ddot\smile$