The volume of a child's model plane is $1200~\rm{cm}^3$ & that of a full size plane is $4050~\rm{m}^3.$ Find the scale of the model in the form $1:n.$ 
The volume of a child's model plane is $1200~\mathrm{cm}^3.$ The volume of a full size plane is $4050~\mathrm{m}^3.$ Find the scale of the model in the form $1:n.$ 

I thought of first converting the $4050~\mathrm{m}^3$ to $\mathrm{cm}^3,$ giving me $4050000000~\mathrm{cm}^3.$ Then, I divided the number by $1200$: reversing the order of conversion. So, my final answer was $1: 3375000.$ However, the answer in the mark scheme is $150.$
 A: Start with simpler shapes and simpler ratios to get the idea.
Say there is a real cube measuring $1\times 1\times 1$ meters, and I have a model of it in the scale $1:2$. Then my model will measure $\frac12\times\frac12\times\frac12$ meters. What is the ratio of volumes between my model cube and the real cube? What if my model was $1:3$?
See if you can use these simple examples to figure out the general pattern that links the model scale to the ratio of volumes. Then, from your ratio of volumes, deduce the scale.
A: Your idea was almost correct, but remember that the scaling will be in terms of length, so you should take the cube root of your answer of 3375000.
A: Ok, your volume ratio is to be converted to length ratio by taking cuberoot.
Looking at it same way... the scaling is for linear/length dimension. So,
$$ n=(4050/1200)^{1/3}=1.5 $$
if we ignore the units. To take units into consideration this should be multiplied by scale factor
$$ \dfrac{meter}{centimeter}= 100$$
giving $$ n=150. $$
