Four years ago, Marina was three times older than her daughter. Six years from now, Marina will be twice as old as her daughter. What are their current ages?

Let Marina's current age be \( M \), and her daughter's current age be \( D \).

### Step 1: Express the relationship from four years ago.
- Four years ago, Marina was three times older than her daughter.
- So, Marina's age four years ago was \( M - 4 \), and her daughter's age four years ago was \( D - 4 \).
- According to the problem, Marina was three times as old as her daughter: 
  \[
  M - 4 = 3(D - 4)
  \]
  Expanding this equation:
  \[
  M - 4 = 3D - 12
  \]
  Simplifying:
  \[
  M = 3D - 8 \tag{1}
  \]

### Step 2: Express the relationship six years from now.
- Six years from now, Marina will be twice as old as her daughter.
- Marina's age six years from now will be \( M + 6 \), and her daughter's age six years from now will be \( D + 6 \).
- According to the problem, Marina will be twice as old as her daughter:
  \[
  M + 6 = 2(D + 6)
  \]
  Expanding this equation:
  \[
  M + 6 = 2D + 12
  \]
  Simplifying:
  \[
  M = 2D + 6 \tag{2}
  \]

### Step 3: Solve the system of equations.
We now have two equations:
1. \( M = 3D - 8 \)
2. \( M = 2D + 6 \)

Equating both expressions for \( M \):
\[
3D - 8 = 2D + 6
\]
Solving for \( D \):
\[
3D - 2D = 6 + 8
\]
\[
D = 14
\]

### Step 4: Find Marina's age.
Substitute \( D = 14 \) into either equation for \( M \). Using \( M = 2D + 6 \):
\[
M = 2(14) + 6 = 28 + 6 = 34
\]

### Conclusion:
- Marina is currently 34 years old.
- Her daughter is currently 14 years old.

Would you like more details on any of the steps, or do you have any questions?

Here are some related questions:
1. What if the age ratio six years from now was different, say 1.5 times?
2. How would the equation change if the time frame was doubled?
3. Can you derive a general formula for such age problems?
4. How can you verify if the solution is correct?
5. How does the problem change if we adjust Marina's age difference to five years ago?

**Tip:** Always label variables clearly and keep equations organized to avoid confusion in solving systems.

Learn how to solve age-related word problems using algebra and systems of linear equations. This example shows how to determine the current ages of Marina and her daughter based on their age relationships four years ago and six years from now. Suitable for students in Grades 8-10.

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