# Ratio of Speeds of Trains

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?

• Related to this question. – Bill Wallis Aug 24 '18 at 10:21
• Trying to memorize formulas in order to solve math problems will not get you far. The times the trains needed to reach their destinations were $t+9$ and $t+16$ respectively, $t$ is the time when they met. You know that the distance they traveled was the same, can you figure out the result? – tst Aug 24 '18 at 10:23
• To expand on what @tst is saying, if you tried to remember formulas for every question like this, there would be a lot of formulas, and trying to decide which one is appropriate in each case would be a nightmare. It's much better to know a few, really important formulas (like $\text{speed} = \frac{\text{distance}}{\text{time}}$) and then derive the formula you need for any other problem using those, use the new formula you just derived, then forget about it. – Arthur Aug 24 '18 at 10:43
• @tst I totally agree. That's why I asked the question here to understand how the formula actually works! – code_master5 Aug 25 '18 at 13:01

You need to introduce a third point, where they meet: $M$. Then we know the following things:
1. The time $T_1$ spent on $PM$ is equal to the time $T_2$ spent on $QM$
2. The time $T_1$ spent on $QM$ is equal to $9$ hours
3. The time $T_2$ spent on $PM$ is equal to $16$ hours
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 $\frac{\text{speed}_1}{\text{speed}_2}$.