Henry Ernest Dudeney/Modern Puzzles/188 - Monkey and Pulley/Solution

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Modern Puzzles by Henry Ernest Dudeney: $188$

Monkey and Pulley
A rope is passed over a pulley.
It has a weight at one end and a monkey at the other.
There is the same length of rope on either side and equilibrium is maintained.
The rope weighs four ounces per foot.
The age of the monkey and the age of the monkey's mother total four years.
The weight of the monkey is as many pounds as the monkey's mother is years old.
The monkey's mother is twice as old as the monkey was
when the monkey's mother was half as old as the monkey will be
when the monkey is three times as old as the monkey's mother was
when the monkey's mother was three times as old as the monkey.
The weight of the rope and the weight at the end was half as much again as the difference in weight
between the weight of the weight and the weight and the weight of the monkey.
Now, what was the length of the rope?


Solution

$5$ feet.


Proof

First recall that there are $16$ ounces to the pound.

All weights will be discussed hereforth in ounces.


Let $l$ feet be the length of the rope.

Let $m$ years be the age of the monkey.

Let $n$ years be the age of the monkey's mother.

Let $w$ be the weight of the rope.

Let $x$ be the weight of the monkey.

Let $y$ be the weight of the weight.


Let $a$ be the number of years ago when the monkey's mother was $3$ times as old as the monkey.

Let $b$ be the number of years ahead when the monkey will be $3$ times as old as the monkey's mother was at $a$.

Let $c$ be the number of years ago when the monkey's mother was half as old as the monkey will be at $b$.


Now we proceed.

\(\text {(1)}: \quad\) \(\ds w\) \(=\) \(\ds 4 l\) The rope weighs four ounces per foot.
\(\text {(2)}: \quad\) \(\ds x + \dfrac w 2\) \(=\) \(\ds y + \dfrac w 2\) It has a weight at one end and a monkey at the other ... and equilibrium is maintained.
\(\text {(3)}: \quad\) \(\ds m + n\) \(=\) \(\ds 4\) The age of the monkey and the age of the monkey's mother total four years.
\(\text {(4)}: \quad\) \(\ds x\) \(=\) \(\ds 16 n\) The weight of the monkey is as many pounds as the monkey's mother is years old.
\(\text {(5)}: \quad\) \(\ds n\) \(=\) \(\ds 2 \paren {m - c}\) The monkey's mother is twice as old as the monkey was ...
\(\text {(6)}: \quad\) \(\ds n - c\) \(=\) \(\ds \dfrac 1 2 \paren {m + b}\) ... when the monkey's mother was half as old as the monkey will be ...
\(\text {(7)}: \quad\) \(\ds m + b\) \(=\) \(\ds 3 \paren {n - a}\) ... when the monkey is three times as old as the monkey's mother was ...
\(\text {(8)}: \quad\) \(\ds n - a\) \(=\) \(\ds 3 \paren {m - a}\) ... when the monkey's mother was three times as old as the monkey.
\(\text {(9)}: \quad\) \(\ds w + y\) \(=\) \(\ds \paren {y + x} - x + \dfrac 1 2 \paren {\paren {y + x} - y}\) The weight of the rope and the weight at the end was half as much again as the difference ... of the weight and the weight of the monkey.


Starting near the bottom and working our way up, we whittle down the variables:

\(\ds \paren {4 - m} - a\) \(=\) \(\ds 3 \paren {m - a}\) from $(3)$ into $(8)$
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds 2 m - 2\) simplifying
\(\ds \leadsto \ \ \) \(\ds m + b\) \(=\) \(\ds 3 \paren {\paren {4 - m} - \paren {2 m - 2} }\) from $(3)$ and above into $(7)$
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds 18 - 10 m\) simplifying
\(\ds \leadsto \ \ \) \(\ds \paren {4 - m} - c\) \(=\) \(\ds \dfrac {m + \paren {18 - 10 m} } 2\) from $(3)$ and above into $(6)$
\(\ds \leadsto \ \ \) \(\ds c\) \(=\) \(\ds \dfrac {7 m} 2 - 5\) simplifying
\(\ds \leadsto \ \ \) \(\ds 4 - m\) \(=\) \(\ds 2 \paren {m - \paren {\dfrac {7 m} 2 - 5} }\) from $(3)$ and above into $(5)$
\(\ds \leadsto \ \ \) \(\ds m\) \(=\) \(\ds \dfrac 3 2\) simplifying
\(\ds \leadsto \ \ \) \(\ds n\) \(=\) \(\ds 4 - \dfrac 3 2\) into $(3)$
\(\ds \) \(=\) \(\ds \dfrac 5 2\) simplifying
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds 16 \times \dfrac 5 2\) into $(4)$
\(\ds \) \(=\) \(\ds 40\) simplifying
\(\ds \) \(=\) \(\ds y\) simplifying $(2)$
\(\ds \leadsto \ \ \) \(\ds w + x\) \(=\) \(\ds \paren {x + x} - x + \dfrac 1 2 \paren {\paren {x + x} - x}\) substituting for $y$ into $(9)$
\(\ds \leadsto \ \ \) \(\ds w\) \(=\) \(\ds \dfrac x 2\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {40} 2\) simplifying
\(\ds \) \(=\) \(\ds 20\) simplifying
\(\ds \leadsto \ \ \) \(\ds l\) \(=\) \(\ds \dfrac {20} 5\) substituting for $w$ in $(1)$
\(\ds \) \(=\) \(\ds 5\) simplifying

$\blacksquare$


Sources

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