Henry Ernest Dudeney/Modern Puzzles/32 - Riding in the Wind/Solution
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Modern Puzzles by Henry Ernest Dudeney: $32$
- Riding in the Wind
- A man on a bicycle rode a mile in $3$ minutes with the wind at his back,
- but it took him $4$ minutes to return against the wind.
- How long would it take him to ride a mile if there was no wind?
Solution
- $3 \tfrac 3 7$ minutes.
Proof
Let $v_w$ miles per minute be the speed of the wind.
Let $v_m$ miles per minute be the speed of the man without the wind.
Let $t$ minutes be the time taken to ride $1$ mile with no wind.
We have:
\(\text {(1)}: \quad\) | \(\ds v_m + v_w\) | \(=\) | \(\ds \dfrac 1 3\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds v_m - v_w\) | \(=\) | \(\ds \dfrac 1 4\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds v_m\) | \(=\) | \(\ds \dfrac 1 t\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 12 \paren {v_m + v_w}\) | \(=\) | \(\ds 4\) | $(1) \times 12$ | ||||||||||
\(\ds 12 \paren {v_m - v_w}\) | \(=\) | \(\ds 3\) | $(2) \times 12$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 24 v_m\) | \(=\) | \(\ds 7\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds v_m\) | \(=\) | \(\ds \dfrac 7 {24}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds t\) | \(=\) | \(\ds \dfrac {24} 7\) | substituting from $(3)$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 3 \tfrac 3 7\) |
$\blacksquare$
Sources
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $32$. -- Riding in the Wind
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $60$. Riding in the Wind