Question 1

Approximate the sum (to the nearest sensible option): 125.009 + 69.999 + 104.989 = ?

Accepted Answer

Introduction: Approximation problems reward quick rounding and sensible estimation. Here, the numbers are very close to integers, so rounding each to the nearest whole makes the arithmetic trivial while staying extremely accurate. Given Data / Assumptions: Values: 125.009, 69.999, 104.989 Goal: Fast approximate sum Options are coarse, so nearest-integer rounding suffices Concept / Approach: Round each term: 125.009 ≈ 125, 69.999 ≈ 70, 104.989 ≈ 105. Then add. This strategy minimizes error because each term is within about 0.01–0.011 of its rounded integer. Step-by-Step Solution: 125.009 ≈ 125 69.999 ≈ 70 104.989 ≈ 105 Approximate sum = 125 + 70 + 105 = 300 Verification / Alternative check: Exact sum = 299.997, which rounds to 300. The rounding error is only −0.003, negligible relative to options spacing. Why Other Options Are Wrong: 285 and 310 are too far from the exact 299.997. 415 and 420 are way off the feasible range of around 300. Common Pitfalls: Overcomplicating by keeping all decimals in mental math, or mis-rounding 69.999 down to 69 instead of up to 70. Final Answer: 300

Question 2

Approximate (nearest sensible option): 16.003 × 27.998 − 209.010 = ?

Accepted Answer

Introduction: Mixed multiply–subtract approximations are best handled by rounding each factor to a nearby integer and then performing the operations. Because the options are broad, nearest-integer rounding typically yields the correct choice. Given Data / Assumptions: Expression: 16.003 × 27.998 − 209.010 Options are spaced tens apart, so integer rounding is adequate. Concept / Approach: Approximate 16.003 ≈ 16, 27.998 ≈ 28, 209.010 ≈ 209. Compute product then subtract, comparing to options. Sensitivity is low because decimal tails are tiny relative to magnitudes here. Step-by-Step Solution: 16 × 28 = 448 448 − 209 ≈ 239 Nearest sensible option ≈ 240 Verification / Alternative check: Using slightly finer rounding (e.g., 16 × 28 − 209.01) still centers near 239, confirming 240 is the closest listed value. Why Other Options Are Wrong: 150 and 200 are far too small; 75 is not plausible; 260 overshoots the refined estimate by about 20+. Common Pitfalls: Rounding 27.998 to 27 (down) would bias low; standard practice is to round to nearest integer (28). Final Answer: 240

Question 3

Approximate the quotient (nearest sensible option): 840.003 ÷ 23.999 = ?

Accepted Answer

Introduction: Division approximations are quick when both numerator and denominator are near neat integers with a simple ratio. Here 840 and 24 are a known pair that simplify to 35 exactly, so tiny decimal tweaks will not change the nearest option. Given Data / Assumptions: Numerator ≈ 840 Denominator ≈ 24 Options include 35, which is 840/24 Concept / Approach: Round 840.003 → 840 and 23.999 → 24. Compute 840/24 = 35. Because the true numerator is slightly larger and denominator slightly smaller, the exact result is just over 35, reinforcing 35 as the closest option. Step-by-Step Solution: 840.003 ≈ 840 23.999 ≈ 24 Approximate quotient = 840 / 24 = 35 Verification / Alternative check: Exact value ≈ 35.000125... confirming that 35 is precise to within a tiny fraction relative to the option spacing. Why Other Options Are Wrong: 47, 38, 32, or 18 are not near the simple 35 ratio and deviate far from the refined exact quotient. Common Pitfalls: Overthinking minute decimals; near-integer quotients like 840/24 should be recognized instantly for speed. Final Answer: 35

Question 4

Approximate (nearest sensible option): 6885.009 − 419.999 − 94.989 = ?

Accepted Answer

Introduction: When subtracting near-integers, round each term to the nearest whole and compute. The residual decimals are small and typically cancel or adjust by less than 1–2 units, which the coarse options easily accommodate. Given Data / Assumptions: Start: 6885.009 Subtract: 419.999 and 94.989 Concept / Approach: Approximate each: 6885, 420, and 95. Compute 6885 − 420 − 95 quickly and select the closest listed option. This is the standard technique for fast mental arithmetic with slight decimal tails. Step-by-Step Solution: 6885 − 420 = 6465 6465 − 95 = 6370 Verification / Alternative check: Exact arithmetic: 6885.009 − 419.999 − 94.989 = 6370. (The decimals nearly cancel to an integer.) Why Other Options Are Wrong: 6650 and 6830 are too large; 6200 is too small; 6400 overshoots slightly. 6370 matches both the approximate and exact computations. Common Pitfalls: Subtracting in the wrong order or mis-rounding 419.999 down to 419; nearest rounding gives a cleaner computation path. Final Answer: 6370

Question 5

Compute the exact product: 503 × 201 = ?

Accepted Answer

Introduction: Even in an approximation section, sometimes exact multiplication is expected. Use distributive expansion to keep the arithmetic simple and error-free, especially when one factor is just over a round hundred or thousand. Given Data / Assumptions: Compute 503 × 201 exactly. Both numbers are close to “round” decompositions. Concept / Approach: Expand 201 as (200 + 1) or 503 as (500 + 3). Apply the distributive property: (a + b)(c + d) = ac + ad + bc + bd. This avoids long multiplication and reduces mental load. Step-by-Step Solution: 503 × 201 = 503 × (200 + 1) = 503 × 200 + 503 × 1 = 100600 + 503 = 101103 Verification / Alternative check: Alternatively, (500 + 3)(200 + 1) = 500*200 + 500*1 + 3*200 + 3 = 100000 + 500 + 600 + 3 = 101103. Same result confirms correctness. Why Other Options Are Wrong: 101100 is near but off by 3; 110000 and 100003 are far; 100500 is a mid-way distractor but incorrect. Common Pitfalls: Dropping small cross terms (like 3*200 or 500*1). Keep every term from the expansion to avoid small but fatal errors. Final Answer: 101103

Question 6

Evaluate the product: (2/3) × (6/8) × (2/3) × (3/4) = ?

Accepted Answer

Introduction: Multiply fractions by simplifying numerators and denominators before multiplying. This reduces computational effort and avoids large intermediate numbers. The final decimal form is requested by the options. Given Data / Assumptions: Expression: (2/3) × (6/8) × (2/3) × (3/4) Standard fraction simplification applies. Concept / Approach: Combine numerators and denominators first, then cancel common factors. Finally convert the simplified fraction to a decimal for the answer choice. Step-by-Step Solution: Numerator = 2 * 6 * 2 * 3 = 72 Denominator = 3 * 8 * 3 * 4 = 288 Value = 72 / 288 = 1 / 4 = 0.25 Verification / Alternative check: Incremental cancellations (e.g., 6/3 → 2; 3 in numerator with 3 in denominator) also lead to 1/4 quickly, confirming 0.25. Why Other Options Are Wrong: 0.5 and 0.125 correspond to 1/2 and 1/8, not the computed 1/4; 1.45 and 11.8 are irrelevant orders of magnitude larger. Common Pitfalls: Multiplying all factors first and then simplifying, which invites arithmetic errors; cancel early to stay accurate. Final Answer: 0.25

Question 7

Approximate (to the nearest integer): 15 7/8 × 23 1/5 + 3 4/5 × 5 1/8 = ?

Accepted Answer

Introduction: Mixed-number products can be computed exactly using improper fractions, then summed and rounded to the nearest integer (as requested). This balances exactness and the approximation nature of the problem to match the answer format. Given Data / Assumptions: 15 7/8 = 127/8; 23 1/5 = 116/5 3 4/5 = 19/5; 5 1/8 = 41/8 Sum the two products; report nearest integer. Concept / Approach: Convert to improper fractions, multiply numerators and denominators, add fractions with common denominator 40, then round the final decimal to the nearest whole number to match the options. Step-by-Step Solution: Term1 = (127/8) * (116/5) = 14732 / 40 = 368.3 Term2 = (19/5) * (41/8) = 779 / 40 = 19.475 Sum = (14732 + 779) / 40 = 15511 / 40 = 387.775 Nearest integer ≈ 388 Verification / Alternative check: Using decimal mixed forms (15.875 * 23.2) + (3.8 * 5.125) ≈ 368.3 + 19.475 = 387.775, same as above, confirming the rounding to 388. Why Other Options Are Wrong: 385 and 370 are farther from 387.775 than 388; 360 and 350 are much too small. Common Pitfalls: Rounding early (before addition) can shift the final rounding decision. Keep exact fraction sums until the end, then round once. Final Answer: 388

Question 8

Approximate the value (nearest sensible option): (2 3/5) × (15/26) × 283.75 = ?

Accepted Answer

Introduction: Converting mixed numbers to fractions or decimals first simplifies computation. Here, exact simplification yields a simple decimal, making the approximation trivial and precise relative to the listed options. Given Data / Assumptions: 2 3/5 = 13/5 = 2.6 Expression: (13/5) * (15/26) * 283.75 Concept / Approach: Simplify (13/5)*(15/26) by canceling 13 with 26 to get (1/5)*(15/2) = 15/10 = 1.5. Then multiply by 283.75 and choose the nearest option. Step-by-Step Solution: (13/5) * (15/26) = (1/5) * (15/2) = 15/10 = 1.5 Value = 1.5 * 283.75 = 425.625 Nearest sensible option = 425 Verification / Alternative check: Rounding 283.75 slightly (e.g., to 284) gives 1.5 * 284 = 426, still closest to 425 among options given typical rounding conventions. Why Other Options Are Wrong: 435 and 440 overshoot; 410 and 420 are too small relative to 425.6 exactness. Common Pitfalls: Not simplifying the fraction product first; computing directly with 2.6 * (15/26) is more error-prone than canceling 13/26 immediately. Final Answer: 425

Question 9

Approximate (nearest sensible option): 36.98276421 × 21.00002 = ?

Accepted Answer

Introduction: Products with one factor near an integer multiple lend themselves to quick rounding. Because 36.98… is close to 37 and 21.00002 is essentially 21, the result should be near 37 × 21, with negligible adjustment from tiny decimals. Given Data / Assumptions: First factor ≈ 37 Second factor ≈ 21 Options are spaced by 10 or more, so coarse rounding suffices. Concept / Approach: Compute 37 × 21 = 777. Adjust slightly downward because the first factor is a bit below 37 (36.98…) while the second is a hair above 21. The net effect keeps the product near 777, closer to 775 than 785. Step-by-Step Solution: 37 × 21 = 777 Refined product ≈ 36.9828 × 21.00002 ≈ 776.6–776.7 Closest option = 775 Verification / Alternative check: Exact multiplication (calculator) would yield about 776.64, which is 1.64 away from 775 and 8.36 from 785. Hence 775 is nearest. Why Other Options Are Wrong: 785, 800, and 805 are too large; 770 is slightly farther than 775 from the refined estimate. Common Pitfalls: Over-correcting for the tiny 0.00002 in the second factor; its impact is orders of magnitude smaller than the integer parts. Final Answer: 775

Question 10

Approximate (nearest sensible option): 14.995 × 8.001 × 20.991 = ?

Accepted Answer

Introduction: When all three factors are very close to round integers, multiply the rounded values to get a near-exact estimate. The tiny deviations (±0.005, ±0.001, etc.) have minimal influence relative to the coarse answer options provided. Given Data / Assumptions: 14.995 ≈ 15 8.001 ≈ 8 20.991 ≈ 21 Concept / Approach: Approximate product ≈ 15 * 8 * 21. Multiply stepwise and compare to options. This approach is standard in approximation questions involving nearly-integer factors. Step-by-Step Solution: 15 * 8 = 120 120 * 21 = 2520 Verification / Alternative check: A slightly more exact check: 14.995 * 8.001 ≈ 120.0 (to within about 0.01), then 120 * 20.991 ≈ 2518.9, which rounds to 2520 and matches the nearest option. Why Other Options Are Wrong: 1950, 2200, 2100, and 1520 are far from the 2520 region implied by near-integer rounding. Common Pitfalls: Attempting exact multi-decimal multiplication under time pressure. Recognize near-integers and round sensibly first. Final Answer: 2520

Question 11

Approximation and quick calculation: Evaluate 26.823 × 27.923 × 4.5001 and select the closest option (use sensible rounding/estimation where appropriate).

Accepted Answer

Introduction / Context: This approximation problem tests number sense and the ability to round sensibly to get a near-exact product quickly. By grouping close-to-integer factors and using small adjustments, you can arrive at a highly accurate estimate without long multiplication. Given Data / Assumptions: Expression: 26.823 * 27.923 * 4.5001 Answer choices are coarse (hundreds-level), so a near-integer rounding is appropriate. Select the closest option to the true value. Concept / Approach: Round each factor to a convenient nearby value, compute, and then refine if needed. Because options are spaced by 50–400, rounding to 27, 28, and 4.5 or using a tighter pair like 26.8 and 27.9 still lands very close to the exact product. Use compensations only if choices are close together. Step-by-Step Solution: Approximate 26.823 ≈ 26.8 and 27.923 ≈ 27.9; 4.5001 ≈ 4.5 Compute first product: 26.8 * 27.9 ≈ (27 * 28) − small ≈ 756 − small ≈ about 748 Multiply by 4.5: 748 * 4.5 ≈ 3366 Using a slightly tighter calc, the exact value is about 3370.48 (for reference), so the nearest listed option is 3400. Verification / Alternative check: Use midpoint method: (26.823 * 27.923) ≈ 27.373 * 27.373 ≈ 749.3; then 749.3 * 4.5001 ≈ 3371.9. Again, closest to 3400. Why Other Options Are Wrong: 3500 and 3450: Slightly high versus the refined estimate near 3370. 3100: Too low given both partial estimates. None of these: A reasonable close option exists (3400). Common Pitfalls: Over-rounding one factor too far (e.g., 30) without compensating others can push the product well above the true range. Keep rounding balanced across factors. Final Answer: 3400

Question 12

Approximation and quick calculation: Evaluate 3739 + (164 × 27) and choose the closest option.

Accepted Answer

Introduction / Context: This checks quick multiplication and addition. Since options are widely spaced, an accurate but mental computation is enough to reach the nearest choice without a calculator. Given Data / Assumptions: Compute 3739 + 164 * 27. Pick the closest available option. Concept / Approach: Multiply 164 by 27 using a breakup such as 164 * (30 − 3) or (20 + 7) * 164, then add 3739. Because answer choices differ by thousands, a precise integer computation is fast and safest. Step-by-Step Solution: 164 * 27 = 164 * (30 − 3) = 4920 − 492 = 4428 Total = 3739 + 4428 = 8167 Nearest listed option to 8167 is 8200. Verification / Alternative check: Compute 164*20 = 3280 and 164*7 = 1148; sum 3280 + 1148 = 4428; add 3739 → 8167. Same result, confirming the nearest option is 8200. Why Other Options Are Wrong: 105400: Off by two orders of magnitude; misread or misplace zeros. 4000: Below even the product 164*27; impossible. 690: Far smaller than either component. None of these: A very close option (8200) exists. Common Pitfalls: Misplacing a zero or using 164*30 without subtracting 164*3. Keep track of the compensation step. Final Answer: 8200

Question 13

Approximation and quick calculation: Evaluate 21 + (3.7 × 2.9) and select the closest option.

Accepted Answer

Introduction / Context: Small decimal multiplications can be handled mentally by rounding to one decimal place and then adjusting slightly. Here, the choices are far apart, so a near-exact evaluation is straightforward and safe. Given Data / Assumptions: Expression: 21 + 3.7 * 2.9 Choose the closest option to the true value. Concept / Approach: Approximate 3.7 * 2.9 as 3.7 * 3.0 − 3.7 * 0.1. Then add 21. Because the product is under 11 and options jump by 5 or more, closeness is guaranteed. Step-by-Step Solution: 3.7 * 2.9 = 3.7 * (3.0 − 0.1) = 11.1 − 0.37 = 10.73 Total = 21 + 10.73 = 31.73 Nearest listed option is 32. Verification / Alternative check: Compute exactly: 3.7 * 2.9 = (37/10)*(29/10) = 1073/100 = 10.73; add 21 → 31.73. Same conclusion. Why Other Options Are Wrong: 74, 70: Too large for a small decimal multiplication added to 21. 27: Below the base 21 + 6; not possible. None of these: 32 is clearly the closest. Common Pitfalls: Rounding 2.9 to 3 and forgetting to subtract 0.37, which slightly overstates the result but still keeps 32 as the nearest choice. Final Answer: 32

Question 14

Approximation and quick calculation: Evaluate (21 + 99) × (30 − 19.02) and select the closest option.

Accepted Answer

Introduction / Context: This problem mixes a clean integer sum with a near-integer difference. Sensible rounding turns it into one quick multiplication. The options are spaced by large gaps, so a tight estimate identifies the correct choice. Given Data / Assumptions: (21 + 99) = 120 (30 − 19.02) ≈ 10.98 Compute 120 * 10.98 approximately. Concept / Approach: Multiply 120 by 11 and subtract 120 * 0.02. This avoids full decimal multiplication and produces a near-exact result suitable for matching to options. Step-by-Step Solution: 30 − 19.02 = 10.98 120 * 10.98 = 120 * (11 − 0.02) = 1320 − 2.4 = 1317.6 Nearest listed integer choice is 1290 (difference ≈ 27.6). Verification / Alternative check: Exact multiplication confirms 1317.6. Since the next higher provided choice 3581 is far off and 131 is an order of magnitude too small, 1290 is the closest available. Why Other Options Are Wrong: 3581: Way too large. 131: Ten times too small relative to 1317.6. None of these: Although 1318 would be exact rounding, 1290 is the closest provided choice. Common Pitfalls: Multiplying 120 by 11 without subtracting the small correction for 0.02 leads to 1320, still closer to 1290 than any other option but slightly less precise than 1318. Final Answer: 1290

Question 15

Approximation and quick calculation: Evaluate 198.001 × 25 + 112.05 × 24.998 and choose the closest option.

Accepted Answer

Introduction / Context: When factors are very close to round numbers, tiny adjustments yield near-exact sums. Here both multipliers are near 25, making mental arithmetic efficient and reliable for the nearest option. Given Data / Assumptions: Compute 198.001*25 and 112.05*24.998, then add. Pick the closest option. Concept / Approach: Use 25 = 100/4 to multiply quickly, and treat 24.998 ≈ 25 with a very small correction. Because answer choices differ by 20–250, maintaining the correction keeps the choice unambiguous. Step-by-Step Solution: 198.001 * 25 = 4950.025 112.05 * 24.998 ≈ 112.05 * (25 − 0.002) = 2801.25 − 0.2241 = 2801.0259 Sum ≈ 4950.025 + 2801.026 ≈ 7751.051 Nearest given option: 7750 Verification / Alternative check: Exact calculation gives about 7751.05, still closest to 7750 compared with other options in the 7500–7570 range. Why Other Options Are Wrong: 7570, 7550, 7500: All are at least ~180 away from the refined sum near 7751. None of these: 7750 fits best among provided choices. Common Pitfalls: Rounding both terms down to 25 or 24 can understate the result by ~200; keep the small correction for 24.998. Final Answer: 7750

Question 16

Approximation and quick calculation: Evaluate 17.995 × 16.005 + 15.999 × 15.001 and select the closest option.

Accepted Answer

Introduction / Context: This expression features pairs of numbers around convenient integers (16, 18) and (15, 16). Using near-square and symmetry ideas speeds up the calculation and leads to a precise estimate for the closest option. Given Data / Assumptions: Compute A = 17.995 * 16.005 Compute B = 15.999 * 15.001 Required: A + B Concept / Approach: Note 18*16 = 288; the tiny deviations keep A near 288. Also, B ≈ 16*15 = 240 with minimal adjustment. Summing gives a value around 528; detailed correction refines this by a fraction only. Step-by-Step Solution: A = (18 − 0.005)*(16 + 0.005) ≈ 288 + 18*0.005 − 16*0.005 − 0.005^2 ≈ 288 + 0.09 − 0.08 ≈ 288.01 B = (16 − 0.001)*(15 + 0.001) ≈ 240 + 16*0.001 − 15*0.001 − 0.001^2 ≈ 240 + 0.001 ≈ 240.001 Sum ≈ 288.01 + 240.001 ≈ 528.011 Nearest option: 528 Verification / Alternative check: Direct multiplication yields about 528.011, matching the above reasoning and reinforcing 528 as closest. Why Other Options Are Wrong: 513 and 440 are far below; 818 is far above the balanced near-16 and near-18 products. Common Pitfalls: Roughly using 18*16 + 16*15 = 528 without considering tiny corrections actually already lands on the correct neighborhood; avoid overcomplicating adjustments that change nothing at the hundreds place. Final Answer: 528

Question 17

Approximation and quick calculation: Evaluate 127.007 × 7.998 + 6.05 × 4.001 and choose the closest option.

Accepted Answer

Introduction / Context: Both products have one factor near an integer: 8 and 4. Using those nearby integers and applying small corrections yields a near-exact sum, perfect for choosing among options spaced by tens. Given Data / Assumptions: P1 = 127.007 * 7.998 ≈ 127.007 * (8 − 0.002) P2 = 6.05 * 4.001 ≈ 6.05 * (4 + 0.001) Concept / Approach: Compute with the nearest integer then adjust by the tiny residual. Since both residuals are very small, the nearest hundred or ten remains stable. Step-by-Step Solution: P1 ≈ 127.007*8 − 127.007*0.002 ≈ 1016.056 − 0.254 ≈ 1015.802 P2 ≈ 6.05*4 + 6.05*0.001 = 24.2 + 0.00605 ≈ 24.206 Sum ≈ 1015.802 + 24.206 ≈ 1040.008 Closest listed option: 1040 Verification / Alternative check: Direct calculation gives about 1040.008, confirming the chosen option. Why Other Options Are Wrong: 1090, 1200, 1120: All are 80–160 away from the refined sum ~1040; too far for these tiny corrections. Common Pitfalls: Skipping the small −0.002 adjustment in the first product leads to ~1016.06, but the second product still keeps the sum very near 1040; the same nearest choice results. Final Answer: 1040

Question 18

Approximation and quick calculation: Evaluate 198.995 × 12.005 + 16.25 × 6.95 and select the closest option.

Accepted Answer

Introduction / Context: Two near-integer multiplications combined: keep one factor per product near a simple integer and adjust slightly. Option spacing (50–180) permits a straightforward nearest-choice selection after a tight estimate. Given Data / Assumptions: P1 = 198.995 * 12.005 ≈ 199 * 12 P2 = 16.25 * 6.95 ≈ 16.25 * 7 Concept / Approach: Compute with rounded integers then account for tiny deviations if options appear close. Here, even the simple rounding lands at the correct choice neighborhood. Step-by-Step Solution: Rough P1 ≈ 199 * 12 = 2388 Rough P2 ≈ 16.25 * 7 = 113.75 Sum rough ≈ 2388 + 113.75 ≈ 2501.75 → very close to 2500 Refined exact is about 2501.87; still nearest to 2500 Verification / Alternative check: Using 199 * 12.005 ≈ 2388 + 0.995 ≈ 2388.995 and 16.25 * 6.95 ≈ 112.94; sum ≈ 2501.94. Nearest option remains 2500. Why Other Options Are Wrong: 2580, 2550: Too high by ~50–80; 2400 is too low by ~100; thus 2500 is best. Common Pitfalls: Rounding both products upward can bias the sum toward 2550; balance rounding or include a small correction. Final Answer: 2500

Question 19

Approximation and quick calculation: Evaluate 2375.85 ÷ 18.01 − (4.525 × 8.05) and choose the closest option.

Accepted Answer

Introduction / Context: A division near 18 and a product near 4.5*8 provide easy mental approximations. Subtract the second from the first and select the nearest choice. Options differ by only a few units, so keep approximations tight. Given Data / Assumptions: Term1 ≈ 2375.85 / 18.01 Term2 ≈ 4.525 * 8.05 Concept / Approach: Approximate Term1 with 2376/18 ≈ 132. Then compute Term2 around 4.5*8 = 36, with a small upward correction. Subtract to obtain the final estimate and compare to options. Step-by-Step Solution: Term1 ≈ 2376 / 18 = 132 Term2 ≈ 4.525 * 8.05 ≈ 36.4 (since 4.525*8 ≈ 36.2 and +0.05*4.525 ≈ 0.226) Difference ≈ 132 − 36.4 ≈ 95.6 Exact calculation gives about 95.49; nearest listed option = 96. Verification / Alternative check: Use 2375.85/18.01 ≈ 131.99 and 4.525*8.05 ≈ 36.50; difference ≈ 95.49; nearest to 96. Why Other Options Are Wrong: 103 is too high; 88 and 90 are low versus the computed mid-90s result. Common Pitfalls: Dividing by 18.01 as if 18.1 (which would understate Term1 more) or rounding the product too coarsely to 36 without the 0.4–0.5 bump. Final Answer: 96

Question 20

Approximation and quick calculation: Solve for the missing value (?) in 2508 ÷ 15.02 + ? × 11 = 200.

Accepted Answer

Introduction / Context: This linear equation features a near-integer division combined with an unknown multiplied by 11. Compute the division accurately enough, transfer it across the equation, and isolate the unknown. The choices are integers, so pick the nearest integer solution. Given Data / Assumptions: Equation: 2508 / 15.02 + 11*? = 200 Compute the division to a couple of decimals to get a precise integer ?. Concept / Approach: Let D = 2508 / 15.02. Then 11*? = 200 − D, and ? = (200 − D) / 11. Because D is about 167, the result is near 3, and integer options make selection straightforward. Step-by-Step Solution: D ≈ 2508 / 15.02 ≈ 166.98 200 − D ≈ 33.02 ? ≈ 33.02 / 11 ≈ 3.00 Nearest integer option: 3 Verification / Alternative check: Compute more precisely: 2508/15.02 ≈ 166.98; then (200 − 166.98)/11 ≈ 3.002; still rounds to 3 and matches the given options. Why Other Options Are Wrong: 4 or 8 or 13 overshoot the target sum when back-substituted; they make the left-hand side exceed 200 by large margins. Common Pitfalls: Using 15 instead of 15.02 or dividing by 11 before subtracting the division term, which mixes the operations incorrectly. Final Answer: 3

Approximation Questions