Model drawing solved this "The present age of the son is 1/2 the difference between the ages of the father and the son. Four years ago, the son's age was 1/3 the difference between the ages of the father and the son. When the son's age is 1/4 the difference between the father's age and the son's age, what is the father's age?"

To solve this problem using a model-drawing or algebraic approach, let's define:

- Let **S** be the son's current age.
- Let **F** be the father's current age.

We are given the following relationships:

1. **The son's present age is half the difference between the father's and the son's ages:**

   \[
   S = \frac{1}{2}(F - S)
   \]

2. **Four years ago, the son's age was one-third the difference between the father's and son's ages at that time:**

   \[
   S - 4 = \frac{1}{3}(F - S - 4)
   \]

3. **In the future, the son's age will be one-quarter the difference between the father's age and the son's age:**

   \[
   S = \frac{1}{4}(F - S)
   \]

Let's now break this down and solve it step by step.

### Step 1: Equation from the first condition
From the first condition, \( S = \frac{1}{2}(F - S) \), multiply both sides by 2 to eliminate the fraction:

\[
2S = F - S
\]

Now, rearrange this to express \( F \) in terms of \( S \):

\[
F = 3S
\]

### Step 2: Equation from the second condition
From the second condition, \( S - 4 = \frac{1}{3}(F - S - 4) \), substitute \( F = 3S \) into this equation:

\[
S - 4 = \frac{1}{3}((3S) - S - 4)
\]

Simplify the right-hand side:

\[
S - 4 = \frac{1}{3}(2S - 4)
\]

Multiply both sides by 3 to eliminate the fraction:

\[
3(S - 4) = 2S - 4
\]

Expand both sides:

\[
3S - 12 = 2S - 4
\]

Now solve for \( S \):

\[
3S - 2S = -4 + 12
\]

\[
S = 8
\]

### Step 3: Find the father's age
Now that we know the son's age \( S = 8 \), substitute this into \( F = 3S \) to find the father's age:

\[
F = 3(8) = 24
\]

### Conclusion:
The father's current age is **24 years**.

Would you like to discuss further details or have any additional questions?

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Here are some related questions you might find useful:

1. How can the relationships between ages be expressed graphically?
2. What if the son's age was expressed as a fraction of the father's age?
3. How do algebraic techniques help in solving real-life word problems?
4. How would the problem change if the age difference was given explicitly?
5. Can this problem be solved using systems of equations involving more variables?

**Tip:** When solving word problems, identifying the key relationships between variables and expressing them algebraically helps break down the problem into manageable steps.

The present age of the son is 1/2 the difference between the ages of the father and the son. Four years ago, the son's age was 1/3 the difference between the ages of the father and the son. When the son's age is 1/4 the difference between the father's age and the son's age, what is the father's age?

Learn how to solve an age-related word problem where the son's age is related to the difference between his and his father's age. This problem uses algebraic techniques and linear equations to find the father's age. Suitable for students in Grades 8-10.

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