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We are tasked with finding the location of point \( B \) such that the ratio of distances \( AB \) to \( BC \) is 2:1, where \( A = -16 \) and \( C = 5 \).

### Step-by-Step Solution:
Let \( B = x \) represent the unknown location of point \( B \) on the number line.

1. The distance \( AB \) is \( |x - (-16)| = |x + 16| \).
2. The distance \( BC \) is \( |5 - x| \).

We know from the problem that the ratio \( \frac{AB}{BC} = \frac{2}{1} \), so:

\[
\frac{|x + 16|}{|5 - x|} = 2
\]

### Solving the Equation:
Now, solve for \( x \) by considering two cases based on the absolute values.

#### Case 1: \( x + 16 \geq 0 \) and \( 5 - x \geq 0 \)
This implies \( x \geq -16 \) and \( x \leq 5 \). The absolute values drop out, and we have:

\[
\frac{x + 16}{5 - x} = 2
\]

Cross-multiply:

\[
x + 16 = 2(5 - x)
\]

Expanding the right side:

\[
x + 16 = 10 - 2x
\]

Solve for \( x \):

\[
x + 2x = 10 - 16
\]
\[
3x = -6
\]
\[
x = -2
\]

Since \( -2 \) is within the range \( -16 \leq x \leq 5 \), this is a valid solution.

#### Case 2: \( x + 16 \geq 0 \) and \( 5 - x \leq 0 \)
This would imply \( x > 5 \), but that contradicts the given that \( B \) lies between \( A \) and \( C \).

Thus, the location of point \( B \) is \( \boxed{-2} \).

### Conclusion:
The correct location of point \( B \) is at \( x = -2 \).

Would you like further clarification on any steps, or should I expand on a particular part of this solution?

Here are some related questions:
1. How would the problem change if the ratio were \( 3:1 \) instead of \( 2:1 \)?
2. What is the general formula for finding a point dividing a segment in a given ratio?
3. What is the total length of segment \( AC \), and how can this be used in related problems?
4. Can you solve a similar problem where the points are in 2D space?
5. What happens if the point \( B \) were outside the segment \( AC \)?

**Tip**: When solving problems involving ratios on a number line, consider both the positive and negative cases that come with absolute values to ensure you don't miss a solution.

Find the location of B so that the ratio of AB to BC is 2:1, given that A = -16 and C = 5.

Learn how to find the location of point B on a number line when given that the ratio of AB to BC is 2:1, with A = -16 and C = 5. This problem involves key algebraic concepts and distance calculations, suitable for students in Grades 8-10.

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