Use the median information to form an equation.

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Step 1: 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: $7m + 3n = 32 \quad (Equation 1)$

Step 2: Use the mean information to form a second equation. 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: $2m + n = 10 \quad (Equation 2)$

Step 3: Solve the system of linear equations. 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: $m = 32 - 30$ $m = 2$

Step 4: Find the value of $n$. 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 $n=6$.

Step 5: Verify the solution. 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: $m=2, n=6$ Drop the next question!