Father and Son Age Problem Analysis

This problem requires us to determine the father's age based on two specific relationships given between his age and his son's age. We'll translate these relationships into algebraic equations and solve them systematically.

Formulating Age Equations

Let's denote the son's current age as $S$ and the father's current age as $F$.

The first condition states: "4 times a son's age is added to his father's age, the sum is 100 years." This translates to the equation:

$4S + F = 100$

The second condition states: "2.5 times the father's age is added to the son's age, the sum is 97 years." This gives us the equation:

$S + 2.5F = 97$

Solving for Father's Age

We now have a system of two linear equations:

1. $4S + F = 100$
2. $S + 2.5F = 97$

To solve for $F$ (the father's age), we can use the method of elimination. First, let's eliminate $S$. We can multiply the second equation by 4 so that the coefficient of $S$ matches the first equation:

$4 \times (S + 2.5F) = 4 \times 97$

$4S + 10F = 388$ (Equation 3)

Now, we subtract the first equation ($4S + F = 100$) from Equation 3:

$(4S + 10F) - (4S + F) = 388 - 100$

Simplifying the left side:

$4S + 10F - 4S - F = 288$

$9F = 288$

To find $F$, we divide both sides by 9:

$F = \frac{288}{9}$

$F = 32$

Therefore, the father's age is 32 years.

Solution Verification

To ensure our calculation is correct, let's find the son's age using $F = 32$ and the first equation $4S + F = 100$:

$4S + 32 = 100$

$4S = 100 - 32$

$4S = 68$

$S = \frac{68}{4}$

$S = 17$

Now, let's check if these ages ($S = 17$, $F = 32$) satisfy the second condition $S + 2.5F = 97$:

$17 + (2.5 \times 32)$

$17 + 80$

$97$

Since $97 = 97$, our calculated ages satisfy both conditions, confirming the father's age is correct.

Final Father's Age Answer

The father's age is 32 years.