A man and a woman are 2 kilometers apart. If a man starts walking directly away from a woman and she starts walking at the same time after him, she will overtake him in 1 hour. If they start walking at the same time toward each other, they will meet in 12 minutes. How fast does each walk?

To solve this problem, let's define the variables and analyze the scenarios:

1. Let \( v_m \) be the speed of the man in kilometers per hour (km/h).
2. Let \( v_w \) be the speed of the woman in kilometers per hour (km/h).

### Scenario 1: Walking Away from Each Other

- The man and woman start 2 kilometers apart.
- The man walks away from the woman, and she follows him.
- She overtakes him in 1 hour.

The distance the woman needs to cover to overtake the man is the initial 2 kilometers plus the extra distance the man has walked in 1 hour. Since she overtakes him in 1 hour:

\[
2 + v_m \cdot 1 = v_w \cdot 1
\]

Simplifying:

\[
2 + v_m = v_w \quad \text{(Equation 1)}
\]

### Scenario 2: Walking Toward Each Other

- The man and woman start 2 kilometers apart.
- They walk towards each other.
- They meet in 12 minutes, which is \( \frac{1}{5} \) of an hour.

The combined distance they cover is 2 kilometers:

\[
v_m \cdot \frac{1}{5} + v_w \cdot \frac{1}{5} = 2
\]

Multiply the entire equation by 5 to clear the fraction:

\[
v_m + v_w = 10 \quad \text{(Equation 2)}
\]

### Solving the System of Equations

Now, we have two equations:

1. \( v_w = 2 + v_m \)
2. \( v_m + v_w = 10 \)

Substitute Equation 1 into Equation 2:

\[
v_m + (2 + v_m) = 10
\]

Simplify:

\[
2v_m + 2 = 10
\]

Subtract 2 from both sides:

\[
2v_m = 8
\]

Divide by 2:

\[
v_m = 4
\]

Now substitute \( v_m = 4 \) into Equation 1:

\[
v_w = 2 + v_m = 2 + 4 = 6
\]

### Solution

- The man's speed \( v_m \) is **4 km/h**.
- The woman's speed \( v_w \) is **6 km/h**.

### Conclusion

The man walks at 4 km/h, and the woman walks at 6 km/h.

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Would you like more details, or do you have any questions?

Here are some related questions that might interest you:

1. How would the answer change if they met in a different time when walking towards each other?
2. What if the man and woman started at different speeds initially?
3. How would we solve this if we added a third person walking?
4. How can we model this problem using relative speed concepts?
5. Can we graph the positions of the man and woman over time?

**Tip:** When solving problems involving distances, speeds, and times, it’s helpful to keep track of the units and convert them to a common base when necessary.

Explore a mathematical problem involving speed and distance: a man and a woman start 2 kilometers apart. Learn how to calculate their walking speeds using equations and solve the problem step-by-step. Suitable for high school students.

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