Simple elementary school ratio question In a group the ratio of men to women is 5:3
In the same group, the ratio of children to adults is 1:2
What is the ratio of men:women:children?
Through simple trial/error/obviousness, one can see that it is 5:3:4
I can’t wrap my head around a sound algebraic way to solve this.
Please help.  I have a young girl that would love to know the “mathematical” way to solve.
Thanks
 A: 5 men : 3 women: (5+3) adults : (5+3)/2=4 children
The ratio of men to adults is 5:8, which corresponds to 4 kids.
A: Another a bit more "algebraic" way of solving this problem is as follows:
Let $n$ denote the number of people in the group. So, we have
\begin{eqnarray} n
& \stackrel{children : adults\; = 1:2}{=} &  \overbrace{\frac{2}{3}n}^{adults} + \overbrace{\frac 13 n}^{children} \\
& \stackrel{men : women\; = 5:3}{=} &  \overbrace{\frac{5}{8}\cdot\frac 23 n}^{men} + \overbrace{\frac{3}{8}\cdot\frac 23 n}^{women} + \overbrace{\frac 13 n}^{children} \\
& = &  \overbrace{\frac{\color{blue}{5}}{12}n}^{men} + \overbrace{\frac{\color{blue}{3}}{12}n}^{women} + \overbrace{\frac{\color{blue}{4}}{12} n}^{children} \\
\end{eqnarray}
A: Express both ratios in terms of $x$ and $y$ as follows:
$$m:w=5:3=5x:3x \Rightarrow \color{red}{m+w=8x}\\
c:a=1:2=y:2y \Rightarrow 2y=a=\color{red}{m+w=8x} \Rightarrow c=y=4x$$
Hence:
$$m:w:c=5x:3x:4x=5:3:4.$$
A: I'm not sure exactly what you want.  We know that there are $5k$ men and $3k$ women, for some value of $k$.  We also know that there are half as many children as adults, so there are $4k$ children.  Therefore, the desired ratio is as you say.
I don't know that this is any more "algebraic" than what you did. 
A: Let's use middle-school mathematics:
Denote $A, C, M,W$ the number of adults, children, men,women respectively.
We know that $\quad \dfrac M5=\dfrac W3$ and that $\enspace \dfrac A2 =C$.  By the properties of proportionalities, we have
$$\frac M5=\frac W3=\frac{M+W}{5+3}=\frac{A}8=\frac C 4,$$
i.e. $[M:W:C]=[5:3:4]$.
