Determine the approximate value that should replace the question mark (?) in the mathematical expression below.

$\frac{624.01}{?}+{12.98}^2=51.98\%\text{of}400.01$

The correct option is 16.

To find the approximate value that replaces the question mark (?), let us first simplify the given values to their nearest whole numbers for ease of calculation:

$624.01\approx 624$

$12.98\approx 13$

$51.98\%\approx 52\%$

$400.01\approx 400$

Now, substitute these approximate values back into the original expression:

$\frac{624}{?}+{13}^2=52\%\text{of}400$

Step 1: Calculate the right side of the equation:

$52\%\text{of}400=\frac{52}{100}\times 400=52\times 4=208$

Step 2: Simplify the square on the left side of the equation:

${13}^2=169$

Step 3: Rewrite the approximate equation with these computed values:

$\frac{624}{?}+169=208$

Step 4: Subtract 169 from both sides to isolate the term containing the question mark:

$\frac{624}{?}=208-169$

$\frac{624}{?}=39$

Step 5: Solve for (?):

$?=\frac{624}{39}$

$?=16$

Thus, the approximate value that replaces the question mark (?) is 16.