Let $n$ be the number of students in the original group. Write an expression for the total hours for the original group, and then add in the total hours from the 4 new students who each slept $8.5$ hours.

After the 4 students are added, the total hours is the original total plus the hours from the 4 new students, and the number of students is $n+4$. Set this new total divided by $n+4$ equal to the new mean, $6.8$.

Once you have an equation like $(\text{expression with } n)/(n+4) = 6.8$, clear the fraction, collect like terms, and solve for $n$. Check your solution by plugging back into the mean formula.

In Desmos, type y = (6.4x + 34)/(x + 4) on one line and y = 6.8 on another line. These represent the new average as a function of $x$ (the original number of students) and the constant new average of $6.8$.

Look for the point where the two graphs intersect. Click on that intersection point; the $x$-coordinate of this point is the number of students in the original group.

Let $n$ be the number of students in the original group.

• The original average (mean) is $6.4$ hours.
• Total hours for the original group is mean × number of students: $6.4n$.

So the original group, together, slept $6.4n$ hours per night.

Four more students are added, each sleeping $8.5$ hours.

• Total hours from these 4 students: $4 \times 8.5 = 34$ hours.
• New total hours: $6.4n + 34$.
• New number of students: $n + 4$.

The new average is $6.8$ hours, so we can write the equation

Multiply both sides by $n + 4$ to clear the fraction:

Distribute on the right:

Move the $6.4n$ to the right and $27.2$ to the left:

Divide both sides of $6.8 = 0.4n$ by $0.4$:

So there were $\boxed{17}$ students in the original group.