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Big Fish

Stage: 4 Short Challenge Level: Challenge Level:1

Let the weights in kilograms of the tail, head and body of the fish be $t$, $h$ and $b$ respectively.

We are told that:

$t = 9$
$h = t + \frac{1}{3} b$
$b = h+t$

$h = 9 + \frac{1}{3} b$
and $b = h+9$.

Therefore $b = 9 + \frac{1}{3} b + 9$,
$\frac{2}{3} b = 18$,
$\frac{1}{3} b = 9$,
$b = 27$.

Hence $h+t = 27$, and the whole fish weighed $54$kg.

Since $b = h+t$
$2b = h+t+b$
$2b = 9 + \frac{1}{3} b + 9+ b$ 
$2b = \frac{4}{3} b + 18$
$\frac{2}{3} b = 18$,
$b = 27$
$h+t+b = 54$kg.

Jonah tackled this problem by setting up a sequence that converged on the solution.
Here is one way to write out Jonah's method:

We know that

Initially, let $b_0=9$

Substituting each term to get the next term in the sequence, you get the first few terms:

If we generate the terms in a spreadsheet, we see that the sequences converge to $b=27$, $h=18$ Jonah's sequence 

This problem is taken from the UKMT Mathematical Challenges.