Henry Ernest Dudeney/Puzzles and Curious Problems/81 - An Irish Jaunt/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $81$

An Irish Jaunt
Colonel Crackham was going from Boghooley to Ballyfoyne, using Pat Doyle's horse and cart,
which moved at a steady rate, but more slowly than would normally be expected.
After they had been on the road for $20$ minutes, they had travelled half as far from Boghooley than it was to Pigtown.
They stopped for refreshment at Pigtown when they arrived there.
Five miles further on, it was half as far to Ballyfoyne as it was from Pigtown.
After another hour they had arrived in Ballyfoyne.
What is the distance from Boghooley to Ballyfoyne?


Solution

$10$ miles.


Proof

Let $d_1$ and $d_2$ miles be the distances from Boghooley to Pigtown and from Pigtown to Ballyfoyne respectively.

Let $v$ miles per hour be the speed of transport.

We have:

\(\ds \dfrac {d_1} 3\) \(=\) \(\ds v \times \dfrac 1 3\) After they had been on the road for $20$ minutes, they had travelled half as far from Boghooley than it was to Pigtown.
\(\ds 5 \times \dfrac 3 2\) \(=\) \(\ds d_2\) Five miles further on, it was half as far to Ballyfoyne as it was from Pigtown.
\(\ds \dfrac 5 2\) \(=\) \(\ds v \times 1\) After another hour they had arrived in Ballyfoyne.
\(\ds \leadsto \ \ \) \(\ds d_2\) \(=\) \(\ds 7 \tfrac 1 2\)
\(\ds v\) \(=\) \(\ds \dfrac 5 2\)
\(\ds d_1\) \(=\) \(\ds 2 \tfrac 1 2\)
\(\ds \leadsto \ \ \) \(\ds d_1 + d_2\) \(=\) \(\ds 10\)

$\blacksquare$


Sources

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