Solve the following mathematical word problem.

The ratio of the current age of a local library to that of a historic courthouse is 1:3 respectively. Four years from now, the age of the courthouse will be $\frac{5}{2}$ times the age of the library. Determine the age of the courthouse four years ago.

Correct Answer: The correct answer is 32 years.

Step-by-Step Explanation:

Let's break down the problem step-by-step to find the age of the courthouse four years ago.

Step 1: Define variables for the current ages

The problem states that the ratio of the current age of the library to that of the courthouse is 1:3.

Let the current age of the library be $x$ years.

Then, the current age of the courthouse is $3x$ years.

Step 2: Express ages four years from now

Four years from now:

The age of the library will be $x+4$ years.

The age of the courthouse will be $3x+4$ years.

Step 3: Set up the equation based on the given condition

We are given that four years from now, the age of the courthouse will be $\frac{5}{2}$ times the age of the library.

So, we can write the equation:

$3x+4=\frac{5}{2}(x+4)$

Step 4: Solve for $x$

Multiply both sides by 2 to clear the fraction:

$2(3x+4)=5(x+4)$

Expand both sides:

$6x+8=5x+20$

Subtract $5x$ from both sides:

$x+8=20$

Subtract 8 from both sides:

$x=12$

Step 5: Calculate the current age of the courthouse

Since the current age of the courthouse is $3x$:

$\text{Current age of courthouse}=3\times 12=36\text{years}$

Step 6: Determine the age of the courthouse four years ago

To find the age four years ago, subtract 4 from the current age of the courthouse:

$\text{Age four years ago}=36-4=32\text{years}$

Thus, the age of the courthouse four years ago was 32 years.