Why does $t = \frac ds$ and not $\frac sd$? (from $ s = \frac dt$)

Question

I'm stupid so I don't understand why I get

$t =\frac sd$

when I isolate $t$ (time) in the speed formula $s =\frac dt$ when its actually

$t =\frac ds$

I've gotten so many answers wrong because of my stupidity and I don't understand why it's wrong.

Here's what I do:

$s = \frac dt \to s = \frac dt \bullet\frac1d \to \frac 1d\bullet s = \frac1t \to \frac sd = t$

I'm just a stressed highschool student so please have some patients with me. I'm not particularly the best at math.

Answer 1

You are okay until the last step. So start like you would but not your last step:

$$s = \frac dt \\ \to s \bullet\frac1d = \frac dt \bullet\frac1d \\ \to \frac 1d\bullet s = \frac1t \\ \to \frac{s}{d} = \frac{1}{t}$$

From here, you solved for $1/t$, not for $t$. To solve for $t$, reciprocate both sides of the equal sign in the last step above: $$ \frac{s}{d} = \frac{1}{t} \\ \to \frac{d}{s} = t $$

Answer 2

When you multiply by $1/d$, you have done only for RHS. If you multiply also by $1/d$ the LHS, you have: $\frac{s}{d}=\frac{1}{t}$. So raising to the :$-1$ power you obtain: $t=\frac{d}{s}$?