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 option is 32 years.

Step-by-Step Explanation:

Step 1: Define the variables based on the given ratio.
Let the current age of the local library be $x$ years.
Since the ratio of the current age of the library to that of the courthouse is 1:3, the current age of the courthouse is $3x$ years.

Step 2: Express their ages four years from now.
In four years:
- 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 using the given relation.
We are given that four years from now, the age of the courthouse will be $\frac{5}{2}$ times the age of the library. Therefore, we can write the equation as:

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

Step 4: Solve for $x$.
Multiply both sides of the equation 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:

$x=12$

So, the current age of the library is 12 years.

Step 5: Determine the current age of the courthouse.
Current age of the courthouse = $3x=3\times 12=36$ years.

Step 6: Calculate the age of the courthouse four years ago.
Age of the courthouse four years ago = $36-4=32$ years.

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