Evaluate 52000 ÷ 40 ÷ 65 × 30 step by step and hence find the value of ? in the equation 52000 ÷ 40 ÷ 65 × 30 = ? − √400.

Introduction / Context:
This question combines multi step arithmetic with a simple equation involving a square root. The left side involves division and multiplication of whole numbers, and the result is related to the unknown symbol ? through subtraction by √400. Many aptitude problems follow this pattern, where a straightforward numerical calculation is linked to a small algebraic step.

Given Data / Assumptions:

• Expression: 52000 ÷ 40 ÷ 65 × 30.

• Equation: 52000 ÷ 40 ÷ 65 × 30 = ? − √400.

• All operations are on real numbers and follow left to right rules for division and multiplication.

Concept / Approach:
First we simplify the left side expression in the natural order: divide 52000 by 40, then by 65, and finally multiply by 30. This gives a single numerical value. Then we recognise that √400 is 20, and substitute this into the equation. Solving for ? involves adding 20 to the calculated left side value. The resulting number is then matched to the options provided.

Step-by-Step Solution:
Step 1: Compute 52000 ÷ 40. Since 52000/40 = 5200/4 = 1300, the result is 1300. Step 2: Divide this result by 65: 1300 ÷ 65. Step 3: Note that 65 × 20 = 1300, so 1300 ÷ 65 = 20. Step 4: Multiply by 30: 20 × 30 = 600. Therefore 52000 ÷ 40 ÷ 65 × 30 = 600. Step 5: Compute the square root term: √400 = 20, since 20^2 = 400. Step 6: Substitute into the equation: 600 = ? − 20. Step 7: Solve for ?: add 20 to both sides to obtain ? = 600 + 20 = 620.

Verification / Alternative check:
We can verify by substituting ? = 620 back into the original equation. The right side becomes 620 − √400 = 620 − 20 = 600. This matches exactly the computed left side value. The arithmetic operations were simple multiples of 10 and common factors, so there is little room for error when carefully executed. This confirms that ? must equal 620.

Why Other Options Are Wrong:
Option 580 would give 580 − 20 = 560, which does not equal 600. Option 720 would give 720 − 20 = 700, again incorrect. Option 780 results in 780 − 20 = 760, and option 600 gives 600 − 20 = 580. None of these match the left side value 600. Only 620 produces the correct equality between both sides of the equation.

Common Pitfalls:
Common mistakes include performing operations in the wrong order, such as multiplying by 30 before completing both divisions, or miscomputing √400 as 400/2 instead of 20. Another error is forgetting to add the square root term back when solving for ?. Following left to right evaluation for division and multiplication, and treating the equation transformation step separately, helps avoid these issues.

Final Answer:
The value of the unknown symbol in the equation is 620.