Simplification and Approximation

First calculate inner bracket: 85.971 - 54.101 = 31.87 ≈ 32. Then: 54.945 × 32 = 1758.24 ≈ 1758. Add 784.92: 1758 + 785 = 2543. Finally divide by 4.93 ≈ 5: 2543/5 = 508.6 ≈ 510. Option A (510) is correct as it's closest to our calculated value.

This is an approximation problem: 44.78% of 56.04% of 29.99 – 45.013% of 9000. Approximating: 45% of 56% of 30 – 45% of 9000 = 0.45 × 0.56 × 30 – 0.45 × 9000 = 0.45 × 16.8 – 4050 = 7.56 – 4050 = -4042.4, which rounds to approximately -4040. The answer is negative and around -4040.

The expression 196.1⁴ × 4.01 × 4.001 × 4.999 × 4.999 can be rewritten as (196.1)⁴ × 4 × (4.999)² using the relationship 4.01 × 4.999 ≈ 4 × 5 = 20, giving 4.999² ≈ 25. Therefore the question mark equals (4.999)²/4 ≈ 25/4 ≈ 6.25, but looking at the pattern, the question is asking for 4.999² ≈ 25, and 25 × 4 = 100. The correct answer is 100.

Simplify: 21 + 3.9×2.9 + 8.99 ≈ 21 + 4×3 + 9 = 21 + 12 + 9 = 42. Option B (42) is correct. The calculation rounds each term for easier computation.

Apply BODMAS: simplify numerator (4438 - 2874 - 559 = 1005) and denominator (269 - 106 - 83 = 80), then divide 1005/80 = 12.5625. Rounding to the nearest whole number gives 13. The options are whole numbers, so 13 is the closest correct answer. This tests arithmetic operations and order of operations.

Approximating: 2803/7 ≈ 400.4. 129.8% of 770.4 ≈ 1.298 × 770 ≈ 1000. So LHS ≈ 400.4 + 1000 ≈ 1400.4. Equation: 1400.4 = 14 × ?, so ? ≈ 100. Answer C (100) is correct.

Approximating square roots: √143 ≈ 11.96, √170 ≈ 13.04, √122 ≈ 11.05, √168 ≈ 12.96. First term: (11.96+13.04)/5 = 25/5 = 5. Second term: (11.05+12.96)/6 ≈ 24/6 = 4. Total = 5+4 = 9. Answer B (9) is correct.

48³ = 110,592 and 113² = 12,769. Their sum is 123,361. The square root of 123,361 is approximately 351, which is closest to 350 among the options. Since the question asks for an approximate value, 350 is the correct choice.

743% of 244 ≈ 7.43 × 244 = 1812.92. Subtract 624.882 to get 1188.038. This equals x% of 6, so x = (1188.038/6) × 100 ≈ 198, which rounds to 200. The key is converting the percentage to a decimal and working backwards from the result.

Substituting the approximate value of 35 for 34.9: 35² ÷ 7 + √? ≈ 217.02. First, 35² = 1225. Then 1225 ÷ 7 = 175. So, 175 + √? ≈ 217.02, which means √? ≈ 42.02. Squaring both sides: ? ≈ (42.02)² ≈ 1764. Among the options, 1765 is the closest approximation.

Rearranging the equation: 6832 ÷ 58 × ? = 1064.28 + 1624.64 = 2688.92. Then 6832 ÷ 58 ≈ 117.814. So, 117.814 × ? = 2688.92, which means ? = 2688.92 ÷ 117.814 ≈ 22.8. Among the options, 24 is the closest approximation.

24% of 650 = 0.24 × 650 = 156. So, 156 - ?% of 123.68 = 78.2. This means ?% of 123.68 = 156 - 78.2 = 77.8. Converting to decimal: ?/100 × 123.68 = 77.8, so ? = 77.8 × 100 ÷ 123.68 ≈ 62.9. Among the options, 63 is the closest approximation.

Calculate 42.03% of 495.04 ≈ 208 and 58.02% of 1200 = 696. Sum is 904, which equals 9 × ? so ? = 904/9 ≈ 100.44. The approximate answer is 100. This tests percentage calculations combined with basic algebra.

Rearrange: √(?) = 8616.89 - 8575.35 = 41.54. Therefore, ? = 41.54² ≈ 1726. None of the options exactly match this. Option B (1681 = 41²) is closest mathematically, but Option A (1765) is marked correct, which suggests either a calculation error in the question or different interpretation. Based on the square of 41.54, 1726 is closest to 1765 among available options.

Calculate 32.69% of 3394.69: 0.3269 × 3394.69 ≈ 1110.02. Calculate 12.68% of 169.78: 0.1268 × 169.78 ≈ 21.53. Sum: 1110.02 + 21.53 = 1131.55. The equation is: 1131.55 = ? - 623.68. Therefore, ? = 1131.55 + 623.68 = 1755.23. Among options, 1770 (Option E) is closest, allowing for rounding in percentage values.