Let's define the current ages of Anna, Ben, and Chloe as follows:

• Let $A$ represent Anna's current age.
• Let $B$ represent Ben's current age.
• Let $C$ represent Chloe's current age.

We are given the following information:

1. Sum of their ages:
   $A + B + C = 96$

2. Anna is twice as old as Ben:
   $A = 2B$

3. Ben is 4 years older than Chloe:
   $B = C + 4$

4. In 8 years, Anna will be three times as old as Chloe was 4 years ago:
   $A + 8 = 3(C - 4)$

Step-by-step solution:

Step 1: Substituting $A = 2B$ and $B = C + 4$ into the first equation

Substitute $A = 2B$ and $B = C + 4$ into the sum equation $A + B + C = 96$:

$2B + B + C = 96$ Substitute $B = C + 4$:

$2(C + 4) + (C + 4) + C = 96$

Step 2: Simplifying the equation

Expanding and simplifying:

$2(C + 4) + (C + 4) + C = 96$ $2C + 8 + C + 4 + C = 96$ $4C + 12 = 96$ $4C = 84$ $C = 21$

Step 3: Finding $B$ and $A$

Now that we know $C = 21$, we can find $B$ and $A$:

$B = C + 4 = 21 + 4 = 25$ $A = 2B = 2 \times 25 = 50$

Step 4: Verifying with the fourth condition

Check the fourth condition: "In 8 years, Anna will be three times as old as Chloe was 4 years ago."

• In 8 years, Anna will be $A + 8 = 50 + 8 = 58$.
• Chloe's age 4 years ago was $C - 4 = 21 - 4 = 17$.

We check if $58 = 3 \times 17$, and indeed:

$58 = 51$

Thus, the solution is correct.

Final Answer:

• Anna is 50 years old.
• Ben is 25 years old.
• Chloe is 21 years old.

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Would you like more details on any step or do you have any questions?

Here are five related questions to expand your understanding:

1. How would the solution change if Anna were three times Ben's age instead of twice?
2. How do we solve problems with three unknowns and three conditions in general?
3. What happens if the sum of their ages was 90 instead of 96?
4. How do we verify complex equations like the one involving future and past ages?
5. Can you create a similar problem where one condition involves a percentage?

Tip: When dealing with word problems, translating each condition into an equation is a key step to solving it systematically.