Recently, I came across a question:
A train $T_1$ from station $P$ to $Q$, and, the other train $T_2$ from station $Q$ to $P$, start simultaneously. After they meet, the trains reach their destinations after $9$ hours and $16$ hours respectively. The ratio of their speeds is ___.
(i) $2:3$
(ii) $4:3$
(iii) $6:7$
(iv) $9:16$
The only formula I remembered was $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \cdotp $$ I couldn't solve the question. It became more and more complex when I tried using this formula directly. Later I found out that the correct answer is (ii) $4:3$. The only explanation given was $\sqrt{16}:\sqrt{9}$. I can't understand how $$ \sqrt{\operatorname{Time}(T_2)} : \sqrt{\operatorname{Time}(T_1)} $$ is the right formula to solve such kind of questions?
You need to introduce a third point, where they meet: $M$. Then we know the following things:
Also note that $\text{speed} = \frac{\text{distance}}{\text{time}}$ becomes $\text{time} = \frac{\text{distance}}{\text{speed}}$ with some elementary manipulation. Use this to make equations out of the three above facts, using sensible variables, and note that what we're after is not $\text{speed}_1$ or $\text{speed}_2$, but $\frac{\text{speed}_1}{\text{speed}_2}$.
Alternatively, you can game the question, and note that train 1 is clearly faster, and only one option allows that.