Let's define variables for the ages of each child currently:

• Let $t$ be Tom's current age.
• Let $b$ be Ben's current age.
• Let $k$ be Kate's current age.
• Let $s$ be Sally's current age.

Step 1: Translate the information into equations.

From the problem, we can extract the following relationships:

1. Tom is two years older than Ben: $t = b + 2$

2. Kate is twice as old as Sally: $k = 2s$

3. The combined ages of the boys is equal to the combined ages of the girls: $t + b = k + s$

4. A year ago, Tom was twice as old as Sally was then: $t - 1 = 2(s - 1)$ Simplifying this equation: $t - 1 = 2s - 2 \quad \Rightarrow \quad t = 2s - 1$

Step 2: Solve the system of equations.

We now have the following system of equations:

1. $t = b + 2$
2. $k = 2s$
3. $t + b = k + s$
4. $t = 2s - 1$

Substitution:

From equation (4), we have $t = 2s - 1$. Substituting this into equation (1) gives: $2s - 1 = b + 2$ Solving for $b$: $b = 2s - 3$

Now substitute $t = 2s - 1$ and $b = 2s - 3$ into equation (3): $(2s - 1) + (2s - 3) = k + s$ Simplifying: $4s - 4 = k + s$ Substitute $k = 2s$ from equation (2): $4s - 4 = 2s + s$ Simplifying further: $4s - 4 = 3s$ $s = 4$

Step 3: Find the ages of all the children.

Now that we know $s = 4$, we can substitute back to find the other ages:

• From $k = 2s$, $k = 2(4) = 8$ (Kate's age).
• From $t = 2s - 1$, $t = 2(4) - 1 = 7$ (Tom's age).
• From $b = 2s - 3$, $b = 2(4) - 3 = 5$ (Ben's age).

Final Answer:

• Tom is 7 years old.
• Ben is 5 years old.
• Kate is 8 years old.
• Sally is 4 years old.

Would you like more details on any part of the solution? Here are 5 related questions to deepen your understanding:

1. What are some methods to solve systems of equations?
2. How do substitution and elimination methods compare in solving systems?
3. Can you explain how to check if the solution satisfies all conditions of the problem?
4. How does this problem demonstrate the concept of age difference relationships?
5. What is a general strategy for solving word problems with multiple unknowns?

Tip: Always start by clearly defining variables and translating word problems into equations!