Question 17

Explanation

Let the four sorted natural numbers be $x_1, x_2, x_3, x_4$ where $x_1 \le x_2 \le x_3 \le x_4$.

Based on the problem information, we can form the following equations:

1. Median: Since there are an even number of data points (4), the median is the average of the two middle values.

   $$\frac{x_2 + x_3}{2} = 8 \implies x_2 + x_3 = 16$$

2. Range: The difference between the largest and smallest datum is 10.

   $$x_4 - x_1 = 10 \implies x_1 = x_4 - 10$$

3. Mean: The sum of all data divided by 4 is 8.

   $$\frac{x_1 + x_2 + x_3 + x_4}{4} = 8 \implies x_1 + x_2 + x_3 + x_4 = 32$$

We substitute the value $x_2 + x_3 = 16$ into the mean equation to find the relationship between $x_1$ and $x_4$:

Now we have a system of two equations for $x_1$ and $x_4$:

By adding both equations, we get:

Substituting the value of $x_4$ to get $x_1$:

Finally, we determine the values of $x_2$ and $x_3$. We know $x_2 + x_3 = 16$ and $3 \le x_2 \le x_3 \le 13$. Since the mode is unique, there must be one value that appears more than once. The only possibility satisfying the median condition and the order is if $x_2$ and $x_3$ are equal.

The complete data set is: $3, 8, 8, 13$. The mode of this data is $8$ (unique), which satisfies the problem condition.

Thus, the product of the second and fourth datum is: