Simple elementary school ratio question

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

Answer 1

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}

Answer 2

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.

Answer 3

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.

Answer 4

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. $[M:W:C]=[5:3:4]$.

Answer 5

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: $$m:w:c=5x:3x:4x=5:3:4.$$