Henry Ernest Dudeney/Modern Puzzles/32 - Riding in the Wind/Solution

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