A, B, C, and D all lie on a number line. C is the midpoint of `bar(AB)` and D is the midpoint of `bar(AC)`.
`{:("Quantity A","Quantity B"),("The ratio of "bar(AD)" to "bar(CB),"The ratio of "bar(AC)" to "bar(AB)):}`

Answer

To solve the problem, we need to analyze the relationships between the points A, B, C, and D on a number line, where C is the midpoint of segment AB and D is the midpoint of segment AC. We will calculate the two quantities and compare them.

### Step 1: Define the positions of A and B
Let's place point A at position 0 on the number line and point B at position 4x. This gives us a clear segment AB.

### Step 2: Find the position of point C
Since C is the midpoint of segment AB, we can calculate its position as follows:
\[ C = \frac{A + B}{2} = \frac{0 + 4x}{2} = 2x \]

### Step 3: Find the position of point D
D is the midpoint of segment AC. We need to find the position of D:
\[ D = \frac{A + C}{2} = \frac{0 + 2x}{2} = x \]

### Step 4: Calculate the lengths of segments
Now we can calculate the lengths of segments AD, CB, AC, and AB:
- Length of segment AD:
\[ AD = D - A = x - 0 = x \]
- Length of segment CB:
\[ CB = B - C = 4x - 2x = 2x \]
- Length of segment AC:
\[ AC = C - A = 2x - 0 = 2x \]
- Length of segment AB:
\[ AB = B - A = 4x - 0 = 4x \]

### Step 5: Calculate the ratios for Quantity A and Quantity B
- **Quantity A**: The ratio of \(AD\) to \(CB\):
\[ \text{Quantity A} = \frac{AD}{CB} = \frac{x}{2x} = \frac{1}{2} \]

- **Quantity B**: The ratio of \(AC\) to \(AB\):
\[ \text{Quantity B} = \frac{AC}{AB} = \frac{2x}{4x} = \frac{1}{2} \]

### Step 6: Compare the two quantities
Both quantities are equal:
\[ \text{Quantity A} = \frac{1}{2}, \quad \text{Quantity B} = \frac{1}{2} \]

### Conclusion
Since both quantities are equal, the answer is that the two quantities are equal.