Henry Ernest Dudeney/Puzzles and Curious Problems/76 - How Far Was It/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $76$

How Far Was It?

"The steamer," remarked one of our officers home from the East, "was able to go twenty miles an hour down-stream,

but could only do fifteen miles an hour upstream.

So, of course, she took five hours longer in coming up than in going down."

One could not resist working out mentally the distance from point to point.

What was it?

Solution

$300$ miles.

Proof

Let $D$ be the distance travelled.

Let $T$ be the time taken to travel $D$ downstream.

We have:

\(\ds D\)

\(=\)

\(\ds 20 T\)

The steamer was able to go twenty miles an hour down-stream

\(\ds D\)

\(=\)

\(\ds 15 \paren {T + 5}\)

but could only do fifteen miles an hour upstream. So, of course, she took five hours longer in coming up than in going down.

\(\ds \leadsto \ \ \)

\(\ds 5 T\)

\(=\)

\(\ds 75\)

eliminating $D$ and simplifying

\(\ds \leadsto \ \ \)

\(\ds T\)

\(=\)

\(\ds 15\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds D\)

\(=\)

\(\ds 20 \times 15\)

substituting for $T$

\(\ds \)

\(=\)

\(\ds 300\)

simplifying

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $76$. -- How Far Was It?
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $90$. How Far Was It?