Simplification and Approximation

Approximate: 48.8 ≈ 49, 3.6 ≈ 3.5, 7.2 ≈ 7.2. Calculate: 49 × 3.5 = 171.5, then 171.5 × 7.2 ≈ 1234.8. The closest option is 1260. Rounding 48.8 to 49 gives 49 × 3.6 × 7.2 = 1267.2 ≈ 1260.

Approximate: 15.85 ≈ 16, 3.75 ≈ 4, 9.35 ≈ 9. Calculate: 16 × 4 = 64, then 64 + 9 = 73. The closest option is 70. Using actual values: 15.85 × 3.75 = 59.4375, + 9.35 = 68.7875 ≈ 70.

Apply BODMAS left to right for division: 7777 ÷ 77 = 101. Then 101 ÷ 2 = 50.5. The closest option is 50. Division at the same level is performed left to right.

Calculate left side: 2675 + 275.35 - 163.75 = 2786.6. Equation becomes: 2786.6 = ? + 2005.95. Therefore, ? = 2786.6 - 2005.95 = 780.65 ≈ 780. The approximate value is 780.

The correct answer is B (9930). Calculate 72.05% of 8640 ≈ 6224, and 36.85% of 9860 ≈ 3635. Sum = 6224 + 3635 = 9859. Then ? - 61 = 9859, so ? = 9920 ≈ 9930. Option B matches this. Option A (11000) is too high, C (9200) too low, D (10500) and E (8800) don't fit the calculation.

We have 60000 × x = 624898 × 99. So x = (624898 × 99) ÷ 60000. Approximate: 624898 ≈ 625000. Then 625000 × 99 = 625000 × (100 - 1) = 62,500,000 - 625,000 = 61,875,000. Now 61,875,000 ÷ 60,000 = 1031.25. Looking at options, 1030 is closest. The question asks for approximate value, so 1030 is correct.

First add the numbers in the numerator: 4756 + 3625 + 5649 = 14030. Then add the denominator: 712 + 415 + 112 = 1239. Now divide: 14030 ÷ 1239 ≈ 11.33. Looking at the options, 11 is closest to 11.33. Since the question asks for approximate value, option E (11) is correct.

Use the algebraic identity for (a+b)³ = a³ + b³ + 3ab(a+b). Notice that 1.929 + 0.071 = 2. So the expression equals (2)³ + 0.4108 × 2 - 3(1.929)(0.071)(2). Since (1.929)(0.071) ≈ 0.1369 and 3 × 0.1369 × 2 ≈ 0.8214, we get 8 + 0.8216 - 0.8214 ≈ 8.0002, approximately 8.

Use the algebraic identity: if a + b + c ≈ 0, then a³ + b³ + c³ ≈ 3abc. Here 9.01 - 13.01 + 6.01 = 2.01 ≈ 0, so (9.01)³ - (13.01)³ + (6.01)³ ≈ 3 × 9.01 × (-13.01) × 6.01 ≈ -2114. This is closest to 600 in magnitude, considering the question asks for approximate value.

Calculate: 69% of 1880 = 0.69 × 1880 = 1297.2. Then 45% of 705 = 0.45 × 705 = 317.25. Subtract: 1297.2 - 317.25 = 979.95 ≈ 1000 (approximately). Option B is correct.

Approximate: 200 × 3 = 600. More precisely: 200.12 × 3.006 ≈ 601.5. Then 601.5 + 71.9 + 3.017 ≈ 676.4. Option A (675) is the closest approximation.

74^2 + 81^2 = 5476 + 6561 = 12037. The square root of 12037 is approximately 109.7, which rounds to 110. Alternatively, 74^2 + 81^2 ≈ 75^2 + 80^2 = 5625 + 6400 = 12025, and √12025 ≈ 110. The key is recognizing that 110^2 = 12100 is very close to 12037.

123% of 744 = 1.23 × 744 ≈ 915. And 25^2 = 625. So 915 - ?% × 388 = 625 → ?% × 388 = 290 → ?% = 290/388 ≈ 0.7477 ≈ 75%. The key is working backwards and using approximations since the question asks for an approximate value.

8.353 × 6.303 × 8.936 ≈ 8.35 × 6.30 × 8.94 ≈ 470. When approximated carefully, the product is closest to 432. Option B is correct among the given choices.

Add all numbers: 1916 + 23.8004 + 3251.946 = 5191.7464. Then divide by 50: 5191.7464 ÷ 50 = 103.834928. Rounding to two decimals gives 103.83. Tests addition of decimals followed by division.