Henry Ernest Dudeney/Modern Puzzles/188 - Monkey and Pulley/Solution
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
- The weight of the rope and the weight at the end was half as much again as the difference in weight
- 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
- 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $188$. -- Monkey and Pulley
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $246$. Monkey and Pulley