To determine Emily’s and Derek’s current ages, we start by defining variables to represent their ages based on the relationships described in the problem.
Step 1: Define Variables
Let:
- $ D $ be Derek’s current age.
- Since Emily is currently twice as old as Derek, we write:
$$
E = 2D
$$
Step 2: Use the Second Condition
Four years ago, Emily’s age was:
$$
E - 4 = 2D - 4
$$
Four years ago, Derek’s age was:
$$
D - 4
$$
According to the problem, at that time, Emily was three times as old as Derek:
$$
2D - 4 = 3(D - 4)
$$
Step 3: Solve the Equation
Start by expanding the right-hand side:
$$
2D - 4 = 3D - 12
$$
Now subtract $2D$ from both sides:
$$
-4 = D - 12
$$
Add 12 to both sides:
$$
8 = D
$$
So Derek is currently 8 years old.
Step 4: Find Emily's Age
Using the relationship $ E = 2D $:
$$
E = 2 \times 8 = 16
$$
So Emily is currently 16 years old.
Step 5: Verify the Solution
Current Ages:
- Emily: 16
- Derek: 8
- Check: 16 = 2 × 8 ✅
Four Years Ago:
- Emily: 16 - 4 = 12
- Derek: 8 - 4 = 4
- Check: 12 = 3 × 4 ✅
Both conditions are satisfied.
Final Answer
Emily’s current age is $\boxed{16}$ and Derek’s current age is $\boxed{8}$.
