$\frac{840}{x}$ and $\frac{960}{x}$ are both positive integers, and so is $x$. What is the largest possible value of $x$?

To find the largest possible value of $x$ such that both $\frac{840}{x}$ and $\frac{960}{x}$ yield positive integers, $x$ must be the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of $840$ and $960$.
Using the difference method:

$$\text{Difference}=960-840=120$$

Now, verify if $120$ divides both numbers completely:

$$\frac{840}{120}=7$$

$$\frac{960}{120}=8$$

Since $120$ divides both numbers without leaving a remainder, the HCF is $120$. Thus, the largest possible value of $x$ is $120$.