Find number grams of the solute in a dissolution given its density and Molarity Have a dissolution of $\ce{HNO_3}$ in water. The density is
  $1.42\ce{g/mL}$ with a concentration of $\ce{16M}$. What is the
  concentration in % mass/mass?


Alright. So the % mass/mass is expressed as

$$\frac{\text{grams of } \ce{HNO_3}}{\text{grams of the dissolution}}\cdot100$$

So I need to find the grams of $\ce{HNO_3}$ first. They tell me that the Molarity is

$$16 = \frac{\text{moles of } \ce{HNO_3}}{\text{liters of dissolution}}$$

$\ce{HNO_3}$ has a molar mass of $63$. So to find out how many moles of $\ce{HNO_3}$ are in the dissolution I need to divide the grams of $\ce{HNO_3}$ by $63$... But how am I supposed to find out the grams of $\ce{HNO_3}$?

I was told that the numerator is $63$. But... $63$ is the molar mass of $\ce{HNO_3}$, not the number of moles of $\ce{HNO_3}$ in the dissolution.

Anyway, let's suppose that it is indeed $63$. Therefore

$$16 = \frac{63}{\text{liters of dissolution}} \implies \text{liters of dissolution} = 3.9$$

So there are $3.9$ liters of dissolution? I'm not sure - I feel like I got this all wrong. How do you know there are $63$ moles of $\ce{HNO_3}$?
Using 1L of solution, you have 16M of HNO3, each mole of which weighs 63g. Therefore, in 1L you'd have 63 X 16 = 1008g. Mass of 1L is given as 1420g from the density. 

%mass/mass = 1008/1420 = 70% HNO3