Find the approximate value that should replace the question mark (?) in the given expression.

$20.09\%\text{of}?+70.03\%\text{of}699.92=\sqrt{676.09}÷12.97+499.99$

The correct answer is 60.

We are given the following mathematical expression:

$20.09\%\text{of}?+70.03\%\text{of}699.92=\sqrt{676.09}÷12.97+499.99$

Step 1: Round off each term to its nearest convenient value:
- 20.09% ≈ 20%
- 70.03% ≈ 70%
- 699.92 ≈ 700
- 676.09 ≈ 676
- 12.97 ≈ 13
- 499.99 ≈ 500

Step 2: Simplify the square root term:

$\sqrt{676.09}\approx \sqrt{676}=26$

Step 3: Substitute the rounded values into the expression:

$20\%\text{of}?+70\%\text{of}700=26÷13+500$

Step 4: Calculate the Right-Hand Side (RHS):

$\text{RHS}=(26÷13)+500$

$\text{RHS}=2+500=502$

Step 5: Calculate individual components of the Left-Hand Side (LHS):

$70\%\text{of}700=\frac{70}{100}\times 700=490$

$20\%\text{of}?=\frac{20}{100}\times ?=0.2\times ?$

Step 6: Form the linear equation and solve for the question mark (?):

$0.2\times ?+490=502$

$0.2\times ?=502-490$

$0.2\times ?=12$

$?=\frac{12}{0.2}$

$?=60$

Therefore, the approximate value that replaces the question mark (?) is 60.