Evaluate 65% × 700 + √196 − 9 × 3 by carefully applying percentage, square root and multiplication operations.

Introduction / Context:
This arithmetic problem combines percentage calculation, a square root, and integer multiplication. The aim is to evaluate the expression step by step: 65% × 700 + √196 − 9 × 3. Such mixed operations are common in aptitude tests and require careful attention to order of operations and basic numerical skills.

Given Data / Assumptions:

• Expression: 65% × 700 + √196 − 9 × 3.

• Percentages are interpreted as parts per hundred.

• √196 denotes the positive square root of 196.

• Operations follow the usual order: roots and multiplications first, then additions and subtractions.

Concept / Approach:
We first convert 65% into a fraction or decimal and multiply by 700 to find that term. Next, we evaluate the square root √196. We also compute 9 × 3 as a simple product. Finally, we perform the addition and subtraction in the correct order. Doing each step separately with clear intermediate results helps avoid sign mistakes and misplacement of decimal points.

Step-by-Step Solution:
Step 1: Compute 65% of 700. Since 65% = 65/100, we have (65/100) × 700 = 0.65 × 700. Step 2: Multiply 0.65 × 700 = 65 × 7 = 455. So 65% × 700 = 455. Step 3: Evaluate the square root: √196 = 14, because 14^2 = 196. Step 4: Compute the product 9 × 3 = 27. Step 5: Substitute these values into the expression: 65% × 700 + √196 − 9 × 3 = 455 + 14 − 27. Step 6: Add 455 and 14: 455 + 14 = 469. Step 7: Subtract 27: 469 − 27 = 442.

Verification / Alternative check:
We can check by re grouping. First compute 455 − 27 = 428, then add the root term 14 to get 428 + 14 = 442. This is consistent with the earlier calculation. Another mental check is to approximate: 65% of 700 is a little less than 70% of 700, which would be 490, so a value in the mid 400s after subtracting and adding small integers is reasonable. The exact arithmetic confirms 442 without ambiguity.

Why Other Options Are Wrong:
Option 522 would require 455 + 14 − 27 to be 522, which would mean adding instead of subtracting 27. Options 402, 362 and 472 result from different miscalculations, such as subtracting both 14 and 27 from 455, or miscomputing the percentage. Since the step by step arithmetic clearly leads to 442, these alternatives are incorrect. Only 442 matches the correctly evaluated expression.

Common Pitfalls:
A frequent error is confusing 65% with 0.65 and misplacing the decimal when multiplying by 700, sometimes giving 45.5 or 4550 instead of 455. Another mistake is miscomputing √196 as 196/2 or 98. Finally, some candidates may forget the order of operations and perform the subtraction before the multiplication, changing the result. Handling each operation in the correct sequence avoids these mistakes and leads directly to the correct answer.

Final Answer:
The value of 65% × 700 + √196 − 9 × 3 is 442.