Simplification and Approximation

Round 81.56 ≈ 82, 5.14 ≈ 5, 3.78 ≈ 4. Then 82 ÷ 5 × 4 = 16.4 × 4 = 65.6. The closest option is D (60). More precisely: 81.56 ÷ 5.14 ≈ 15.87, then 15.87 × 3.78 ≈ 59.99. Option D is the best approximation.

11.49² = (11.5 - 0.01)² ≈ 11.5² = 132.25. Since 11.49 is slightly less than 11.5, the actual value is slightly less than 132.25. Among the options, 130 is the closest approximation. Option A is correct.

825% of 330 = (825/100) × 330 = 8.25 × 330 = 2722.5. Then 2722.5 ÷ 507 ≈ 5.37. Approximating gives 5. Option A is correct.

Factor out 11: (2222 + 3333 + 4444) × 11 = 9999 × 11 = 109989. Among the options, 110000 is the closest approximation.

Approximate: 58² ≈ 3364, 33² ≈ 1089, 12² ≈ 144. Then 3364 - 1089 - 144 = 2131, which rounds to 2000. The difference of squares formula a² - b² = (a+b)(a-b) is also useful.

Approximate and simplify: 628.905 ÷ 9.003 ÷ 14.01. First: 629 ÷ 9 ≈ 70. Then: 70 ÷ 14 = 5. The calculation is approximately 629 ÷ 9 ÷ 14 = 629 ÷ 126 ≈ 5. Consecutive division means dividing by the product of divisors: 9 × 14 = 126, and 629 ÷ 126 ≈ 5.

Rearrange: 798.991 - 498.052 - 99.76 = ?. Calculate: 799 - 498 - 100 = 301 - 100 = 201. Approximating: 798.991 ≈ 799, 498.052 ≈ 498, 99.76 ≈ 100. So 799 - 498 - 100 ≈ 200. The exact calculation gives 798.991 - 498.052 - 99.76 = 201.179, which is approximately 200.

Calculate: 1069 + 1996.96 + 1966.66 = 5032.62. Approximating: 1069 ≈ 1070, 1996.96 ≈ 2000, 1966.66 ≈ 1970. Sum ≈ 1070 + 2000 + 1970 = 5040. The exact sum is 5032.62, which is closest to 5040. Option E is correct.

To find 177% of 14591, multiply 14591 by 1.77. Approximating: 14591 × 1.77 ≈ 14600 × 1.77 = 25842. The closest option is 25800. This tests percentage calculations with large numbers.

Divide 14675 by 11 to get approximately 1334. Then set up 1334 = 57.96 × ?, so ? = 1334 ÷ 57.96 ≈ 23. This tests division and solving for an unknown in an equation.

First calculate 280% of 1525: (280/100) × 1525 = 2.8 × 1525 = 4270. Then divide by 16.96: 4270 ÷ 16.96 ≈ 4270 ÷ 17 = 251.76. Among the options, 260 is the closest approximate value. More precisely: 4270 ÷ 16.96 ≈ 251.7, which rounds to approximately 250, but 260 is the best available option.

We need to approximate 21 + 63 ÷ 17. Following order of operations (PEMDAS/BODMAS), division comes before addition. 63 ÷ 17 ≈ 3.71 (since 17 × 3 = 51 and 17 × 4 = 68, it's closer to 4). Then 21 + 3.71 ≈ 24.71, which rounds to approximately 25. Option D is correct.

11.25 is very close to 11. So (11.25)^3 is approximately equal to 11^3 = 1331. Option B matches this approximation.

Round the numbers to nearest hundreds: 6888 ≈ 6900, 417 ≈ 400, 97 ≈ 100. So 6900 - 400 - 100 = 6400. Option D matches.

Approximating: 15.28 × 3.56 ÷ 2.99 ≈ 15 × 3.5 ÷ 3 = 52.5 ÷ 3 = 17.5. The closest option is 20. Using better approximation: 15.28 × 3.56 ≈ 54.4, then 54.4 ÷ 2.99 ≈ 18.2, still closest to 20 among given options.