Henry Ernest Dudeney/Puzzles and Curious Problems/23 - A Deal in Cucumbers/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $23$

A Deal in Cucumbers

"How much to you pay for these cucumbers?" someone asked.

The reply: "I pay as many shillings for six dozen cucumbers of that size as I get cucumbers for $32 \shillings$"

What was the price per cucumber?

Solution

$8 \oldpence$

Proof

Let $P$ be the price per cucumber in pence.

Let $A$ be the price per $6$ dozen cucumbers in shillings.

Let $B$ be the number of cucumbers you can buy for $32 \shillings$.

We have:

\(\text {(1)}: \quad\)

\(\ds 6 \times 12 P\)

\(=\)

\(\ds 12 A\)

"I pay as many shillings for six dozen cucumbers of that size ...

\(\text {(2)}: \quad\)

\(\ds A\)

\(=\)

\(\ds \dfrac {32 \times 12} P\)

... as I get cucumbers for $32 \shillings$"

\(\ds \leadsto \ \ \)

\(\ds A\)

\(=\)

\(\ds 6 P\)

simplifying $(1)$

\(\ds \leadsto \ \ \)

\(\ds 6 P\)

\(=\)

\(\ds \dfrac {32 \times 12} P\)

substituting for $A$ in $(2)$

\(\ds \leadsto \ \ \)

\(\ds P^2\)

\(=\)

\(\ds 64\)

simpifying

\(\ds \leadsto \ \ \)

\(\ds P\)

\(=\)

\(\ds 8\)

as $P$ must be positive

Hence the solution.

$\blacksquare$

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $23$. -- A Deal in Cucumbers