Henry Ernest Dudeney/Modern Puzzles/41 - The Damaged Engine/Solution
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Modern Puzzles by Henry Ernest Dudeney: $41$
- The Damaged Engine
- We were going by train from Anglechester to Clinkerton, and an hour after starting some accident happened to the engine.
- We had to continue the journey at $\tfrac 3 5$ of the former speed, and it made us $2$ hours late at Clinkerton,
- Can you tell from that statement just how far it is from Anglechester to Clinkerton?
Solution
- $200$ miles, of which the first $50$ miles were at $50$ miles per hour, and the rest at $30$ miles per hour.
Proof
Let $d$ miles be the distance from Anglechester to Clinkerton.
Let $v$ miles per hour be the speed of the train when working properly.
Let $d_1$ miles be the distance from Anglechester to where the train malfunctioned.
Let $t$ hours be the time the train would normally take to travel the journey.
We have:
| \(\text {(1)}: \quad\) | \(\ds d\) | \(=\) | \(\ds v t\) | |||||||||||
| \(\text {(1')}: \quad\) | \(\ds d_1\) | \(=\) | \(\ds v\) | an hour after starting some accident happened to the engine. | ||||||||||
| \(\text {(2)}: \quad\) | \(\ds \dfrac {d_1} v + \dfrac {d - d_1} {v \times \tfrac 3 5}\) | \(=\) | \(\ds t + 2\) | We had to continue ... $2$ hours late at Clinkerton | ||||||||||
| \(\text {(3)}: \quad\) | \(\ds \dfrac {d_1 + 50} v + \dfrac {d - d_1 - 50} {v \times \tfrac 3 5}\) | \(=\) | \(\ds t + 1 \tfrac 1 3\) | the driver said ... $40$ minutes sooner | ||||||||||
| \(\text {(4)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 5 d - 2 d_1\) | \(=\) | \(\ds 3 v t + 6 v\) | simplifying $(2)$ | |||||||||
| \(\text {(5)}: \quad\) | \(\ds 5 d - 2 d_1 - 100\) | \(=\) | \(\ds 3 v t + 4 v\) | simplifying $(3)$ | ||||||||||
| \(\text {(6)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 100\) | \(=\) | \(\ds 2 v\) | $(4) - (5)$ | |||||||||
| \(\text {(7)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 5 d - 2 d_1\) | \(=\) | \(\ds 150 t + 300\) | substituting for $v$ in $(5)$ or $(6)$ and simplifying | |||||||||
| \(\text {(8)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 2 d - 2 d_1\) | \(=\) | \(\ds 300\) | substituting for $t$ in $(7)$ and simplifying | |||||||||
| \(\text {(9)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds 2 d - 100\) | \(=\) | \(\ds 300\) | substituting for $d_1 = v$ in $(8)$ | |||||||||
| \(\text {(10)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds d\) | \(=\) | \(\ds 200\) |
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $41$. -- The Damaged Engine
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $69$. The Damaged Engine