What value should replace the question mark (?) in the following equation?\n\n45% of 400 + √(?) = 56% of 750 − 40% of 350

Introduction / Context:
This question combines basic percentage calculations with square roots and an equation. The goal is to find the unknown number under the square root so that the left hand side and right hand side of the equation balance. Such problems test your ability to compute percentages accurately, manipulate equations, and recall the relationship between a square and its square root.

Given Data / Assumptions:

• The equation is 45% of 400 + √(?) = 56% of 750 − 40% of 350.
• The unknown value under the square root is a non negative real number.
• All percentages are standard percentages, so 45% = 45 / 100, and so on.
• The operations follow normal arithmetic rules: calculate percentages first, then apply addition and subtraction.

Concept / Approach:
First, compute the known percentage expressions on both sides using the formula (percentage / 100) * number. Once you have numerical values, simplify the right hand side to a single number. Then subtract the known term on the left hand side to isolate the square root term. Finally, square both sides (or recognize a perfect square) to find the unknown value. Checking with familiar squares (like 100^2, 99^2, 101^2) helps confirm the answer.

Step-by-Step Solution:
Step 1: Compute 45% of 400: (45 / 100) * 400 = 0.45 * 400 = 180. Step 2: Compute 56% of 750: (56 / 100) * 750 = 0.56 * 750 = 420. Step 3: Compute 40% of 350: (40 / 100) * 350 = 0.40 * 350 = 140. Step 4: Simplify the right hand side: 56% of 750 − 40% of 350 = 420 − 140 = 280. Step 5: Now the equation becomes: 180 + √(?) = 280. Step 6: Subtract 180 from both sides to isolate the square root term: √(?) = 280 − 180 = 100. Step 7: To remove the square root, square both sides: ? = 100^2 = 10000.

Verification / Alternative check:
Substitute ? = 10000 back into the original equation. Then √(10000) = 100. The left side becomes 45% of 400 + 100 = 180 + 100 = 280. The right side, as computed earlier, is 56% of 750 − 40% of 350 = 420 − 140 = 280. Both sides match exactly, so ? = 10000 is confirmed as correct.

Why Other Options Are Wrong:
1000: √1000 is approximately 31.6, which would make the left side 180 + 31.6 ≈ 211.6, not 280.
10201: √10201 is 101, giving a left side of 180 + 101 = 281, which is slightly larger than 280.
9801: √9801 is 99, giving a left side of 180 + 99 = 279, slightly less than 280.
6400: √6400 is 80, giving a left side of 180 + 80 = 260, again not equal to 280.

Common Pitfalls:
Common errors include miscalculating one of the percentages, especially 56% of 750 or 40% of 350, or forgetting to subtract properly when isolating the square root term. Another pitfall is squaring too early instead of first simplifying the equation, which can create unnecessary complexity. Finally, always remember that √x denotes the principal non negative square root, so you only consider the positive value when matching it to a perfect square.

Final Answer:
The missing value is 10000, because √(10000) = 100 balances the equation.