Age Word Problem Explained

This problem asks us to find the father's age based on two conditions relating his age and his son's age. We can solve this using algebra by setting up and solving a system of linear equations.

Setting Up Age Equations

First, let's define variables for the unknown ages:

• Let $f$ represent the father's age (in years).
• Let $s$ represent the son's age (in years).

Now, we translate the given information into mathematical equations:

Condition 1: "If 2 times a son's age is added to his father's age, the sum is 36 years."

This translates to the equation: $2s + f = 36$ (Equation 1)

Condition 2: "If 2.5 times the father's age is added to the son's age, the sum is 72 years."

This translates to the equation: $s + 2.5f = 72$ (Equation 2)

Our goal is to find the value of $f$.

Solving the System of Equations

We can solve this system using either substitution or elimination. Let's use the substitution method.

1. Isolate one variable: From Equation 1, we can easily express the son's age ($s$) in terms of the father's age ($f$): $2s = 36 - f$ $s = \frac{36 - f}{2}$ $s = 18 - 0.5f$
2. Substitute: Now substitute this expression for $s$ into Equation 2: $(18 - 0.5f) + 2.5f = 72$
3. Solve for $f$: Simplify and solve the resulting equation for $f$: $18 + (-0.5f + 2.5f) = 72$ $18 + 2f = 72$ Subtract 18 from both sides: $2f = 72 - 18$ $2f = 54$ Divide by 2: $f = \frac{54}{2}$ $f = 27$

So, the father's age is 27 years.

Alternative Method: Elimination

We can also use the elimination method for practice.

Our system is:

1: $2s + f = 36$

2: $s + 2.5f = 72$

To eliminate $s$, we can multiply Equation 2 by 2:

Now, subtract Equation 1 from Equation 3:

$(2s + 5f) - (2s + f) = 144 - 36$

$2s + 5f - 2s - f = 108$

$4f = 108$

Solve for $f$:

This method also gives the father's age as 27 years.

Verification of Ages

Let's check if our answer is correct by finding the son's age and plugging both ages back into the original conditions.

Using $s = 18 - 0.5f$: $s = 18 - 0.5 \times 27$ $s = 18 - 13.5$ $s = 4.5$

Check Condition 1: $2s + f = 2(4.5) + 27 = 9 + 27 = 36$. (Correct)

Check Condition 2: $s + 2.5f = 4.5 + 2.5(27) = 4.5 + 67.5 = 72$. (Correct)

Both conditions are satisfied, confirming our calculation.