Use the median information to form an equation.

The numbers $2, 9, 7m, 3n, 21, 24$ are arranged in ascending order. There are 6 numbers. For an even set of numbers, the median is the average of the two middle terms. The 3rd term is $7m$ and the 4th term is $3n$. The median is given as 16.

$$\frac{7m + 3n}{2} = 16$$

Multiply both sides by 2:

$$7m + 3n = 32 \quad \text{(Equation 1)}$$

The numbers are $4, 8, 6, 4m, 19, 2n$. There are 6 numbers. The mean is given as $9\frac{1}{2}$, which is $\frac{19}{2}$ or 9.5.
The mean is the sum of the numbers divided by the count of numbers.

$$\frac{4 + 8 + 6 + 4m + 19 + 2n}{6} = 9.5$$

Sum the constant terms: $4 + 8 + 6 + 19 = 37$.

$$\frac{37 + 4m + 2n}{6} = 9.5$$

Multiply both sides by 6:

$$37 + 4m + 2n = 9.5 \times 6$$ $$37 + 4m + 2n = 57$$

Subtract 37 from both sides:

$$4m + 2n = 57 - 37$$ $$4m + 2n = 20$$

Divide the entire equation by 2 to simplify:

$$2m + n = 10 \quad \text{(Equation 2)}$$

We have two equations:

1. $7m + 3n = 32$
2. $2m + n = 10$

From Equation 2, express $n$ in terms of $m$:

$$n = 10 - 2m$$

Substitute this expression for $n$ into Equation 1:

$$7m + 3(10 - 2m) = 32$$ $$7m + 30 - 6m = 32$$ $$m + 30 = 32$$

Subtract 30 from both sides:

$$m = 32 - 30$$ $$m = 2$$

Substitute the value of $m=2$ back into the expression for $n$:

$$n = 10 - 2(2)$$ $$n = 10 - 4$$ $$n = 6$$

The values are $m=2$ and $n=6$.

For the first set of numbers: $2, 9, 7m, 3n, 21, 24$.
Substitute $m=2$ and $n=6$: $2, 9, 7(2), 3(6), 21, 24 \implies 2, 9, 14, 18, 21, 24$.
The median is $\frac{14+18}{2} = \frac{32}{2} = 16$. This is correct.

For the second set of numbers: $4, 8, 6, 4m, 19, 2n$.
Substitute $m=2$ and $n=6$: $4, 8, 6, 4(2), 19, 2(6) \implies 4, 8, 6, 8, 19, 12$.
The sum is $4+8+6+8+19+12 = 57$.
The mean is $\frac{57}{6} = 9.5 = 9\frac{1}{2}$. This is correct.

The values are:

$$m=2, n=6$$