Question 1

Approximate sum: 85% of 225 + 32.91 × 5.01 ≈ ?

Accepted Answer

Introduction / Context: We approximate a percentage of a fixed number and add a product of two decimals. The problem belongs to an approximation set, so we round to the nearest offered integer. Given Data / Assumptions: 85% of 225. 32.91 × 5.01. Concept / Approach: Compute each component with two-decimal accuracy and sum. The closest integer option among the choices is selected. Step-by-Step Solution: 85% of 225 = 0.85 * 225 = 191.25.32.91 × 5.01 ≈ 32.91 * (5 + 0.01) ≈ 164.55 + 0.3291 = 164.8791.Total ≈ 191.25 + 164.8791 ≈ 356.1291.Nearest option: 355. Verification / Alternative check: If we rounded 5.01 to 5.0, we would get 32.91 * 5 = 164.55 and a total ≈ 355.80, still pointing to 355–356. With available choices, 355 is best. Why Other Options Are Wrong: 340 or 345: Both too low compared to ~356. 375: Too high for the computed sum. Common Pitfalls: Multiplying 32.91 by 5.01 incorrectly (e.g., adding 5.01 and 32.91) or mishandling the small 0.01 adjustment. Break the product into two parts for clarity. Final Answer: 355

Question 2

Approximate total: (15.96)^2 + 75% of 285 ≈ ?

Accepted Answer

Introduction / Context: This combines a squared decimal with a percentage of an integer. We compute each part accurately enough to pick the closest integer option. Given Data / Assumptions: (15.96)^2. 75% of 285 = 0.75 * 285. Concept / Approach: Calculate the square carefully, compute the percentage exactly, add them, and then round to the nearest choice if needed. Step-by-Step Solution: (15.96)^2 ≈ 254.7216.75% of 285 = 0.75 * 285 = 213.75.Total ≈ 254.7216 + 213.75 = 468.4716.Nearest option: 470. Verification / Alternative check: Rounding the square to 255 would give 255 + 213.75 = 468.75, still near 470 and far from the other options, strengthening the choice of 470. Why Other Options Are Wrong: 435 or 440: Too low relative to ≈468.5. 485: Slightly high; our total is well under 480. Common Pitfalls: Approximating the square too coarsely (e.g., using 16^2 = 256 without recognizing the small reduction from 16 to 15.96), which can introduce a larger rounding error. Compute the square or use a close estimate like 16^2 minus a small correction. Final Answer: 470

Question 3

Fill the missing value by approximation: 0.0005 × 76 × 10^3 + ? ≈ 170% of 29.98

Accepted Answer

Introduction / Context: We match a left-hand expression to a percentage of a number. One term involves scientific notation. After computing both sides approximately, we solve for the missing addend and select the nearest integer option. Given Data / Assumptions: Compute: 0.0005 × 76 × 10^3 + ? ≈ 170% of 29.98. 170% = 1.7 as a multiplier. Concept / Approach: Evaluate the fixed product first, then the right-hand side, and finally isolate the unknown ?. Round to the nearest feasible option as customary for approximation problems. Step-by-Step Solution: 0.0005 × 10^3 = 0.5.0.5 × 76 = 38.Right side: 1.7 × 29.98 ≈ 50.966 (≈ 51).So 38 + ? ≈ 50.966 ⇒ ? ≈ 12.966.Nearest option: 12 (among 12, 15, 16, 18). Verification / Alternative check: If we round 29.98 to 30, RHS ≈ 51 exactly; then ? ≈ 13. The closest available integer is still 12 (off by 1), which is acceptable within approximation tolerance given the choices provided. Why Other Options Are Wrong: 15 or 16 or 18: Overshoot the required ~13 by a wider margin than 12 does. Common Pitfalls: Misinterpreting 10^3 or distributing 0.0005 incorrectly. Always compute the power-of-ten part first to stabilize place values, then multiply the remaining factors. Final Answer: 12

Question 4

Find the missing percentage: 39.8% of 400 + ?% of 350 = 230 (approximate to the nearest option).

Accepted Answer

Introduction / Context: This is a reverse-percentage problem. We compute the first term, subtract it from the total 230, and then back out the percentage of 350 that remains. Given Data / Assumptions: First term: 39.8% of 400. Second term: ?% of 350. Sum ≈ 230. Concept / Approach: Compute the known percentage value, find the required remainder, convert that remainder to a percentage of 350, and finally select the nearest whole-number percentage from the options. Step-by-Step Solution: 39.8% of 400 = 0.398 × 400 = 159.2.Remainder needed = 230 − 159.2 = 70.8.Let x% of 350 ≈ 70.8 ⇒ (x/100) × 350 ≈ 70.8.x ≈ (70.8 / 350) × 100 ≈ 20.23%.Nearest listed percentage: 20%. Verification / Alternative check: Check: 20% of 350 = 70; total = 159.2 + 70 = 229.2, close to 230. Given approximation framing, 20% is the most reasonable choice. Why Other Options Are Wrong: 15%: Gives 52.5, too small. 18%: Gives 63, still small. 25%: Gives 87.5, too large. Common Pitfalls: Mixing bases (e.g., applying the second percentage to 400 instead of 350) or forgetting to convert the percentage remainder correctly. Always identify the correct base for each percentage. Final Answer: 20

Approximation Questions