Fish! (A maths puzzle and a proposed solution)

A fish is made up of three parts: head, tail and body.

The tail is equal in length to the head, plus one-quarter of the length of the body.
The body is equal to three-quarters of the entire length of the fish.
The head is 4cm in length.


How long is the entire fish?

SOLUTION

Let's pose this in the following way:

T = Tail
B = Body
H = Head
L = Total Length

Therefore:
4 + 4 + 1/4 B + 3/4 L = L

Which can be simplified to give:
8 + 1/4B + 3/4L = L

Look also at the relationship between the Tail, Body and Total Length. The body, so we're told, is 3/4 of the Total Length and the fractional part of the tail is 1/4 of the length of the body. This means that we must be able to describe the fractional part of the tail in terms of the Total Length:

1/4 x 3/4 = 3/16

So the fractional part of the tail is 3/16 of the length of the fish!

Now we know that, we can work out what proportion of the fish the fractional part of the tail plus the body is:

3/16 + 3/4 = 3/16 + 12/16 = 15/16

So the fractional part of the tail and the body account for 15/16 of the Total Length, giving us:

8 + 15/16 L = L

What of the remaining 1/16 L? Well given the above, 1/16 L must be equal to 8 which means:

8 x 16 = Total Length = 128

So the fish must be 128cm in length! Lets try it:

1/16 L + 3/16 L + 12/16 L = 1/16 x 128 + 3/16 x 128 + 12/16 x 128 = 8 + 24 + 96 = 128