There are four children in a family: two girls, Kate and Sally; and two boys, Tom and
Ben. Tom is two years older than Ben. Kate is twice as old as Sally. The combined
ages of the two boys is equal to the combined ages of the two girls. A year ago, Tom
was twice as old as Sally was then. How old are the four children?


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!

There are four children in a family: two girls, Kate and Sally; and two boys, Tom and Ben. Tom is two years older than Ben. Kate is twice as old as Sally. The combined ages of the two boys is equal to the combined ages of the two girls. A year ago, Tom was twice as old as Sally was then. How old are the four children?

Find out how to solve a word problem involving four children's ages. This problem uses algebraic techniques like systems of linear equations and the substitution method. Perfect for students in Grades 6-8.

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