Question

Anand travelled 300 km by train and 200 km by taxi. It took him 5 hours and 30 minutes. However, if he travels 260 km by train and 240 km by taxi, he takes 6 minutes more. The speed of the train is:

• A. 100 km/h
• B. 120 km/h
• C. 80 km/h
• D. 110 km/h

Hint:

Form two equations in 1/speed for train and taxi from the two travel scenarios, then eliminate one variable.

Answer 1

The Correct Option is A

Let $p = 1/x$ and $q = 1/y$, where $x$ is the train's speed and $y$ is the taxi's speed. The two conditions become a linear system in $p$ and $q$:

1. Write the linear system. $300p + 200q = 5.5$ ... (i), and $260p + 240q = 5.6$ ... (ii).
2. Eliminate $q$. Multiply (i) by $6$ and (ii) by $5$ to make the $q$-coefficients equal ($1200$ in both): $1800p + 1200q = 33$ and $1300p + 1200q = 28$.
3. Subtract. $(1800p - 1300p) = 33 - 28 \Rightarrow 500p = 5 \Rightarrow p = 0.01$.
4. Recover $x$. Since $p = 1/x = 0.01$, we get $x = 100$.
5. Cross-check with $q$. From (i): $300(0.01) + 200q = 5.5 \Rightarrow 3 + 200q = 5.5 \Rightarrow q = 0.0125 \Rightarrow y = 80$, which satisfies (ii) as well: $260(0.01) + 240(0.0125) = 2.6 + 3 = 5.6$.

So the train's speed is $100$ km/h, option A.