TITLE: Child's homework equation related to algebra
QUESTION [5 upvotes]: My grandson's homework.... There are 23 fish. Guppies (G) are 3 more than Z-fish (Z). There are 2 times the Z-fish as Goldfish (GF). How many of each? .... I can see there are 8 Z-fish (Z), 11 Guppies (G) & 4 Goldfish (GF)....  HOWEVER, When I try to prove the equation it's wrong.. Z + (Z+3) +(Z/2) = 23... equation? So Z + (Z/2) = 20/2 (=10).... not correct..so.. answer should be 12 (if Z=8, G =11, GF=4 ). help?  I don't know what I'm doing wrong.  I can do a simple one if two unknowns but can't do this 3 unknowns with the fraction.

REPLY [2 votes]: The initial system given by the problem is:
$$\left\{\begin{align*}
&G+Z+GC=23\\
&G=Z+3\\
&Z=2GF
\end{align*}\right.$$
The second and third equations make it easy to express $G$ and $GF$ in terms of $Z$: we already have $G=Z+3$, and from the third equation we get $GF=\frac12Z$. Substitute these into the first equation, and you’ll have an equation with just the single unknown $Z$:
$$(Z+3)+Z+\frac12Z=23\;.$$
I expect that both you and your grandson can handle that without much trouble, but I’ll go ahead and finish it off. Simplifying, we have
$$\begin{align*}
&\frac52Z+3=23\;,\\\\
&\frac52Z=20\;,\text{ and}\\\\
&Z=\frac25\cdot20=8\;,
\end{align*}$$
so $G=Z+3=11$ and $GF=\frac12Z=4$. And as a check, $11+8+4=23$.