Let's solve the given problem step by step.

Part (a)

We are asked to write an expression for the father's present age in terms of the son's age, $s$.

• Let $s$ represent the son's present age.
• The father is 32 years older than the son.

Thus, the father's present age can be expressed as: $\text{Father's age} = s + 32$

Part (b)

We are asked to form an equation based on the information that 10 years ago, the father was three times as old as his son.

• 10 years ago, the son's age was $s - 10$.
• 10 years ago, the father's age was $(s + 32) - 10 = s + 22$.

According to the problem, 10 years ago, the father was three times as old as his son: $s + 22 = 3(s - 10)$

Now, solve this equation: $s + 22 = 3(s - 10)$ Distribute the 3 on the right side: $s + 22 = 3s - 30$ Move all terms involving $s$ to one side and constants to the other: $22 + 30 = 3s - s$ Simplify: $52 = 2s$ Solve for $s$: $s = \frac{52}{2} = 26$

So, the son is 26 years old. Now, to find the father's present age: $\text{Father's age} = s + 32 = 26 + 32 = 58$

Thus, the father is 58 years old.

Would you like further details or clarification?

Here are five related questions:

1. How would the equation change if the age difference between father and son were 25 years instead of 32?
2. What if 5 years ago, the father was twice as old as the son—how would you adjust the equation?
3. Can you generalize an equation to relate father and son’s ages based on different time gaps?
4. What would the father's age be if the son were currently 30 years old?
5. If the son's current age is unknown, how can we express both ages algebraically?

Tip: When setting up word problems, always define variables clearly and translate each part of the statement into equations step by step.