SOLUTION: The head of a fish is 9 inches long. The body length of the fish between the head and the tail is equal to the head plus the tail. The tail length is equal to the head plus one hal

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Question 622920: The head of a fish is 9 inches long. The body length of the fish between the head and the tail is equal to the head plus the tail. The tail length is equal to the head plus one half of the body length between the head and the tail. This is enough information to work out that the total length of the fish is what? (Pencil and paper may be used.)
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Let t be the length of the tail, in inches.
Let b be the length of the body (between the head and the tail), in inches.
We could name a variable for the length of the head, but we already know it is 9 inches, so I would not bother.
"The body length of the fish between the head and the tail is equal to the head plus the tail" can be translated as
b=9%2Bt .
"The tail length is equal to the head plus one half of the body length between the head and the tail" can be translated as
t=9%2B%281%2F2%29b <--> t=9%2B0.5b (or 2t=18%2Bb if we multiply both sides of the equal sign times 2).
(I listed 3 equivalent versions of that equation, but the additional versions are optional, just in case you would rather not work with the first one).
With those two equations we have a system of linear equations:
system%28b=9%2Bt%2Ct=9%2B%281%2F2%29b%29 or system%28b=9%2Bt%2Ct=9%2B0.5b%29 or system%28b=9%2Bt%2C2t=18%2Bb%29 .
(You can chose which version of the second equation you want to work with).
A system of linear equations could have one solution, no solution, or an infinite number of solutions.
This system has one solution, and it can be easily found by substitution.
If you substitute the expression 9%2Bt for b in the second equation, you can find t, and then you can plug t into b=9%2Bt to find b .
--> t=9%2B9%2F2%2B%281%2F2%29%2At --> t=27%2F2%2B%281%2F2%29t
Subtracting %281%2F2%29t from both sides of the equal sign, we get
t=27%2F2%2B%281%2F2%29t --> t-%281%2F2%29t=27%2F2%2B%281%2F2%29t-%281%2F2%29t%29 --> %281-1%2F2%29t=27%2F2 --> %281%2F2%29t=27%2F2
Multiplying both sides times 2, we get
%281%2F2%29t=27%2F2 --> 2%2A%281%2F2%29t=2%2A%2827%2F2%29 --> highlight%28t=27%29
b=9%2Bt --> b=9%2B27 --> highlight%28b=36%29
Now we can add fish part lengths (in inches) to get the length of the whole fish (in inches):
9%2B36%2B27=72
The head length is 9 inches.
The body length is 36 inches, and
the tail length is 27 inches.
The whole fish is highlight%2872%29 inches long.

Verifying:
The head length is 9 inches is true because the problem says that.
The body length is equal to the head plus the tail is true, because the length of the head plus the length of the tail is
9%2B27=36.
The tail length is equal to the head plus one half of the body length is true because
9%2B%281%2F2%29%2A36=9%2B18=27.