Henry Ernest Dudeney/Puzzles and Curious Problems/142 - Longfellow's Bees/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $142$

Longfellow's Bees

If one-fifth of a hive of bees flew to the ladambra flower,

one-third flew to the slandbara,

three times the difference of these two numbers flew to an arbour,

and one bee continued to fly about, attracted on each side by the fragrant ketaki and the malati,

what was the number of bees?

Solution

There were $15$ bees in the hive.

Proof

Let $n$ be the number of bees.

We have:

\(\ds \dfrac n 5 + \dfrac n 3 + 3 \paren {\dfrac n 3 - \dfrac n 5} + 1\)

\(=\)

\(\ds n\)

from the problem definition

\(\ds \leadsto \ \ \)

\(\ds 3 n + 5 n + 15 n - 9 n + 15\)

\(=\)

\(\ds 15 n\)

multiplying by $15$ to clear the fractions

\(\ds \leadsto \ \ \)

\(\ds 15\)

\(=\)

\(\ds n\)

simplifying

$\blacksquare$

Historical Note

In the words of Henry Ernest Dudeney:

When Longfellow was Professor of Modern Languages at College Harvard College he was accustomed to amuse himself by giving more or less simple arithmetical puzzles to the students.

Here is an example:

and indeed so follows the puzzle posed here.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $142$. -- Longfellow's Bees
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $214$. Longfellow's Bees