Henry Ernest Dudeney/Modern Puzzles/186 - The False Scales/Solution 1

Modern Puzzles by Henry Ernest Dudeney: $186$

The False Scales

A pudding, when placed into one of the pans of a balance, appeared to weigh $4$ ounces more than $\tfrac 9 {11}$ of its true weight,

but when placed into the other pan it appeared to weigh $3$ pounds more than in the first pan.

What was its true weight?

Solution

Suppose that the false balance has imbalanced pans.

That is, the pans are of different weight.

Then the true weight of the pudding is $154$ ounces.

Proof

Let $W$ ounces be the true weight of the pudding.

We are given that:

in one pan the apparent weight was $\dfrac {9 W} {11} + 4$

in the other pan the apparent weight was $\dfrac {9 W} {11} + 52$.

Hence we have:

\(\ds W\)

\(=\)

\(\ds \dfrac {\paren {\tfrac {9 W} {11} + 4} + \paren {\tfrac {9 W} {11} + 52} } 2\)

True Weight from False Balance with Imbalanced Pans

\(\ds \leadsto \ \ \)

\(\ds 2 W\)

\(=\)

\(\ds \dfrac {18 W} {11} + 56\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds 22 W\)

\(=\)

\(\ds 18 W + 56 \times 11\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds 4 W\)

\(=\)

\(\ds 616\)

simplifying

\(\ds \leadsto \ \ \)

\(\ds W\)

\(=\)

\(\ds 154\)

simplifying

$\blacksquare$

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $186$. -- The False Scales
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $97$. The False Scales