Henry Ernest Dudeney/Puzzles and Curious Problems/299 - The Weight of the Fish/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $299$

The Weight of the Fish
A man caught a fish.
The tail weighed $9$ ounces.
The head weighed as much as the tail and half the body,
and the body weighed as much as the head and tail together.
What is the weight of the fish?


Solution

$72$ ounces, or $4 \tfrac 1 2$ pounds.


Proof

Let $t$, $h$ and $b$ denote the weights in ounces of tail, head and body respectively.

We have:

\(\text {(1)}: \quad\) \(\ds t\) \(=\) \(\ds 9\) The tail weighed $9$ ounces.
\(\text {(2)}: \quad\) \(\ds h\) \(=\) \(\ds t + \dfrac b 2\) The head weighed as much as the tail and half the body,
\(\text {(3)}: \quad\) \(\ds b\) \(=\) \(\ds h + t\) and the body weighed as much as the head and tail together.
\(\ds \leadsto \ \ \) \(\ds h\) \(=\) \(\ds 9 + \dfrac {h + 9} 2\) substituting from $(1)$ and $(3)$ into $(2)$
\(\ds \leadsto \ \ \) \(\ds h\) \(=\) \(\ds 27\) simplifying
\(\ds \leadsto \ \ \) \(\ds b\) \(=\) \(\ds 27 + 9 = 36\) substituting into $(3)$
\(\ds \leadsto \ \ \) \(\ds h + b + t\) \(=\) \(\ds 27 + 9 + 36 = 72\) totting them up

The weight in pounds can be calculated from the conversion factor of $16$ ounces to the pound.

$\blacksquare$


Sources

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