Simplification and Approximation

Rearranging: √? = 7437.16 - 7365 - (5.4)² = 7437.16 - 7365 - 29.16 = 43. So ? = 43² = 1849. This is option A. The key is isolating the square root term by moving all other terms to the right side.

Approximating: 25.05 ≈ 25, 123.95 ≈ 124, 388.999 ≈ 389, 15.001 ≈ 15. So 25 × 124 + 389 × 15 = 3100 + 5835 = 8935. This is option C. The approximations slightly reduce the first term and slightly increase the second, balancing out to give approximately 8935.

Simplify by first doing the multiplication in the numerator: 47 × 562.58 ≈ 47 × 560 ≈ 26432. Then multiply in denominator: 23 × 112.23 ≈ 23 × 112 ≈ 2576. Now divide: 26432 ÷ 2576 ≈ 10.26. The question asks for approximate value, so 10 is the closest answer. Doing more precise calculation: (47 × 562.58) ÷ (23 × 112.23) = 26441.26 ÷ 2581.29 ≈ 10.24, which rounds to 10.

Step-by-step: 103.1% of 6401 ≈ 1.031 × 6400 = 6598.4. (3/7)% of 6300 ≈ 0.0043 × 6300 = 27.09. So 6598.4 - 27.09 + 12 = 6583.31. Refined calculation: 1.031×6401=6595.43, 0.0043×6300=27.09, 6595.43-27.09+12=6580.34. The closest option is 6557 (option B). The approximation is reasonable.

Calculate each percentage separately and add them: 17.1% of 725 = 0.171 × 725 ≈ 124.0, and 12.8% of 643 = 0.128 × 643 ≈ 82.3. Their sum is approximately 206.3, which rounds to 207. Option A (117) is too low, while C (272), D (180), and E (307) are either too high or don't match the approximation.

Approximate the percentages: 89.998% of 680.06 ≈ 90% of 680 = 612. 130.001% of 449.998 ≈ 130% of 450 = 585. 20.01% of 125.089 ≈ 20% of 125 = 25. Sum: 612 + 585 + 25 = 1222. Option A matches.

Approximate: 110.99 + 111.011 ≈ 222. 37.01 × 2.999 ≈ 37 × 3 = 111. 44.99% of 180 ≈ 45% of 180 = 81. Total: 222 + 111 + 81 = 414. Option D matches.

Let the unknown be x. The equation is x × 5 + 444.99 × 5.99 + 899.906 + 9.999 = 4579.99. Approximating: 5x + 445 × 6 + 900 + 10 = 4580. So 5x + 2670 + 910 = 4580, giving 5x = 1000 and x = 200. Option A is correct.

Approximate: 64^2 = 4096 = 2^12, 1024^(1/5) = 4 = 2^2, 34^2 ≈ 2^10, 16.97^2 ≈ 2^8. Expression = 2^12 × 2^2 × 2^10 ÷ (2^5 × 2^8) = 2^24 ÷ 2^13 = 2^11. So 2^(1/?) = 2^11, meaning 1/? = 11 or ? ≈ 0.09. Key distractor: 0.5 seems reasonable but doesn't satisfy the equation.

Approximating: 2904 ÷ 34.95 ≈ 2900 ÷ 35 = 82.86 ≈ 83. Then: (83 - 13) × 6 = 70 × 6 = 420. The answer is approximately 420.

Approximating: 139.93 ≈ 140, 24.102 ≈ 24, 27.89 ≈ 28, 7.53 ≈ 7.5. Then: (140 × 24) - (28 × 7.5) = 3360 - 210 = 3150. The closest option is D (3150). The calculation requires careful approximation and arithmetic to avoid errors. Options A, B, C, and E result from calculation errors or incorrect approximations.

Approximating: 67 + √9 + ? = 52 + 8² gives 67 + 3 + ? = 52 + 64, so 70 + ? = 116, therefore ? = 46. More precisely: 66.97 + 3.02 + ? = 52.114 + 64.704 gives 69.99 + ? = 116.818, so ? ≈ 46.83. The closest approximation is 46.

Calculate 144.8% of 1339 = 1.448 × 1339 ≈ 1939.27. Then add 42.02 × 18.484 ≈ 776.7. The total is approximately 2715.97, which rounds to 2720. The key is to compute each term accurately before adding.

Approximate each term to the nearest integer: 127.001 ≈ 127, 7.998 ≈ 8, 6.05 ≈ 6, 4.001 ≈ 4. Calculate 127 × 8 = 1016 and 6 × 4 = 24. Sum = 1040. The small decimal adjustments nearly balance out.

Follow order of operations: 4.9 × 7.9 ≈ 38.71. Then add 21 + 38.71 + 9.88 = 69.59. This rounds to 70, making option B correct.