G2 Mathematics K210 checking becomes faster when learners can place an answer inside a plausible interval before calculating it exactly. Bounds turn “Does this number look right?” into a mathematical question.
This eighty-sixth Learner’s Guide develops bounding checks: use lower limits, upper limits, ranges, benchmark fractions, geometry constraints and contextual feasibility to reject impossible answers. It complements Vol 0082 Direction-of-Change Checks and Vol 0078 Dimensional Consistency.
The bounding question
Before exact work, ask: what is definitely too small, what is definitely too large, and what interval should contain the answer? A good bound can come from definitions, context, nearby benchmark values, graph range or known geometry.
1. count lower bound
Bound: at least 20 items means answer cannot be below 20.
Use: a result of 19 is impossible regardless of nearby arithmetic. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
2. count upper bound
Bound: at most 20 items means answer cannot exceed 20.
Use: a result of 21 violates constraint. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
3. probability bounds
Bound: probability lies from 0 to 1.
Use: negative or above-one result is impossible. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
4. percentage share
Bound: part-of-whole percentage lies from 0% to 100%.
Use: values above 100% require a different meaning such as percentage increase. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
5. mean bounds
Bound: mean of ordinary finite data lies between minimum and maximum.
Use: outside value signals total or count error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
6. median bounds
Bound: median lies within ordered data range.
Use: outside result is impossible. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
7. range nonnegative
Bound: range is maximum minus minimum.
Use: negative range signals subtraction reversal. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
8. distance nonnegative
Bound: distance cannot be negative.
Use: negative algebraic root may be rejected by context. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
9. area nonnegative
Bound: ordinary geometric area cannot be negative.
Use: sign error or invalid root should be inspected. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
10. volume nonnegative
Bound: physical volume cannot be negative.
Use: negative value is contextual impossibility. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
11. time nonnegative
Bound: elapsed time cannot be negative.
Use: equation may have an extraneous contextual root. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
12. speed nonnegative in scalar context
Bound: speed is not negative.
Use: negative result signals direction quantity or algebra issue. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
13. capacity
Bound: used amount cannot exceed fixed capacity without overflow.
Use: bound helps choose feasible integer. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
14. remaining capacity
Bound: remainder lies between zero and original capacity.
Use: outside range signals subtraction or model error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
15. budget remainder
Bound: remaining budget cannot exceed original if spending is nonnegative.
Use: larger remainder is inconsistent. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
16. discount price
Bound: positive discount below 100% puts sale price between zero and marked price.
Use: outside interval signals multiplier error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
17. depreciated value
Bound: positive depreciation below 100% lowers but does not make value negative in one step.
Use: bound catches wrong factor. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
18. positive growth
Bound: positive percentage growth makes final value exceed original.
Use: lower result violates direction bound. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
19. reverse percentage decrease
Bound: if final is 80% of original, original must exceed final.
Use: this creates a lower bound on original. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
20. reverse percentage increase
Bound: if final is 120% of original, original must be below final.
Use: this creates an upper relation. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
21. fraction proper
Bound: proper positive fraction lies between zero and one.
Use: outside result signals numerator/denominator issue. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
22. ratio share
Bound: part/whole lies between zero and one when part belongs to whole.
Use: bound checks set ownership. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
23. angle triangle
Bound: interior angle in ordinary triangle is greater than zero and less than 180 degrees.
Use: outside value invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
24. triangle sum
Bound: three interior angles total 180 degrees.
Use: two known angles bound the third exactly. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
25. quadrilateral sum
Bound: interior angles total 360 degrees.
Use: remaining angle can be bounded from known positives. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
26. circle radius
Bound: radius positive and half diameter.
Use: diameter relation creates exact bound. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
27. hypotenuse
Bound: hypotenuse exceeds either leg in non-degenerate right triangle.
Use: smaller computed hypotenuse impossible. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
28. triangle inequality
Bound: one side less than sum of other two and greater than their difference.
Use: candidate lengths can be rejected before detailed work. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
29. scale enlargement
Bound: factor above one makes image length exceed original.
Use: smaller image contradicts bound. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
30. scale reduction
Bound: factor between zero and one makes image length below original.
Use: larger image contradicts reduction. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
31. coordinate midpoint
Bound: midpoint coordinate lies between endpoint coordinates on each axis.
Use: outside coordinate signals averaging error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
32. interpolation
Bound: interpolated value between monotonic endpoints should lie between endpoint values.
Use: outside result signals scale or formula issue. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
33. weighted mean
Bound: weighted mean lies between minimum and maximum component values for positive weights.
Use: outside value signals weight or total error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
34. probability complement
Bound: P(A)+P(not A)=1.
Use: one probability bounds the other exactly. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
35. union rough bound
Bound: probability of union cannot be less than either included event.
Use: smaller result signals subtraction mistake. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
36. frequency
Bound: frequency count nonnegative and cannot exceed total observations.
Use: outside count impossible. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
37. relative frequency
Bound: lies from zero to one.
Use: bound catches denominator error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
38. pie angle
Bound: sector angle from zero to 360 degrees.
Use: outside angle invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
39. pie percentage
Bound: sector share from zero to 100%.
Use: outside value invalid for one sector. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
40. graph axis bound
Bound: read value must lie within plotted axis range unless extrapolating.
Use: outside read signals scale mistake. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
41. table total
Bound: category count cannot exceed stated total when categories are subsets.
Use: bound catches data ownership error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
42. sample count
Bound: favourable outcomes cannot exceed total outcomes.
Use: probability counting check. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
43. integer minimum
Bound: required whole units at least a decimal need ceiling.
Use: answer below ceiling fails task. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
44. integer maximum
Bound: whole units fitting capacity may need floor.
Use: answer above floor can violate capacity. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
45. packing
Bound: number of complete groups cannot exceed total/item-size quotient floor.
Use: bound rejects ordinary rounding up. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
46. containers needed
Bound: containers to hold all items cannot be below quotient ceiling.
Use: bound rejects rounding down. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
47. currency exchange
Bound: rough rate magnitude bounds converted amount.
Use: order-of-magnitude interval catches reciprocal conversion. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
48. unit conversion
Bound: known scale factor creates expected interval.
Use: metres-to-centimetres should increase numerical magnitude by 100. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
49. area conversion
Bound: linear conversion squared creates much larger factor.
Use: using only linear factor falls outside expected scale. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
50. volume conversion
Bound: linear conversion cubed creates scale.
Use: wrong exponent gives implausible interval. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
51. speed estimate
Bound: distance and time rough bounds create speed interval.
Use: exact result outside interval signals arithmetic or unit error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
52. time estimate
Bound: fixed distance and speed bounds create time interval.
Use: reciprocal mistakes often leave interval. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
53. cost estimate
Bound: unit price and quantity rough bounds create total-cost interval.
Use: calculator entry outside range is suspicious. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
54. percentage estimate
Bound: known fraction benchmark gives rough percentage interval.
Use: e.g. near one quarter should be near 25%, not 2.5% or 250%. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
55. mean estimate
Bound: clustered data imply mean near cluster centre.
Use: far-away result signals total/count error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
56. gradient estimate
Bound: visual rise/run gives sign and rough magnitude.
Use: exact slope outside visual bound signals scale issue. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
57. area estimate
Bound: shape dimensions create rough rectangular or triangular comparison.
Use: exact area wildly outside is suspicious. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
58. perimeter estimate
Bound: sum of side-scale magnitudes bounds perimeter.
Use: area-like magnitude signals formula confusion. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
59. circle circumference estimate
Bound: circumference a little over three diameters.
Use: result far outside signals formula/input error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
60. circle area estimate
Bound: area roughly pi times radius squared.
Use: linear-scale result signals missing square. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
61. Pythagoras estimate
Bound: hypotenuse below sum of legs and above largest leg.
Use: strong interval check before square root. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
62. square root bound
Bound: if n lies between consecutive squares, sqrt(n) lies between corresponding integers.
Use: calculator result outside interval impossible. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
63. power bound
Bound: positive base above one grows with exponent.
Use: direction and nearby powers bound result. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
64. geometric sequence
Bound: ratio magnitude bounds next term.
Use: wrong reciprocal often violates interval. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
65. arithmetic sequence
Bound: constant difference gives exact local bound.
Use: next term outside expected step signals copy/sign error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
66. linear graph
Bound: between two points on straight line, intermediate y lies between endpoint y values if monotonic.
Use: outside interpolation invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
67. quadratic vertex context
Bound: minimum or maximum from graph bounds function values locally.
Use: candidate value beyond visible extremum invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
68. real-world age
Bound: age nonnegative and often constrained by context.
Use: implausible root can be rejected. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
69. people count
Bound: whole nonnegative integer.
Use: fractional person needs contextual interpretation. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
70. money
Bound: amount may require cents precision and nonnegative context.
Use: negative price usually invalid unless representing debt/change. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
71. temperature
Bound: can be negative depending scale, so do not impose false nonnegative bound.
Use: bounds must come from context, not habit. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
72. direct proportion
Bound: positive constant and positive input give positive output.
Use: negative result invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
73. inverse proportion
Bound: positive constant and positive input give positive output.
Use: negative result invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
74. ratio denominator
Bound: positive denominator larger gives smaller quotient for fixed numerator.
Use: nearby denominator cases bound answer. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
75. fraction comparison
Bound: benchmark fractions 1/2, 1/4, 3/4 create quick intervals.
Use: decimal outside expected region signals division error. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
76. percentage-point change
Bound: difference between percentages lies between -100 and 100 points.
Use: outside impossible. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
77. utilisation
Bound: used/available from zero to one when used does not exceed capacity.
Use: above 100% needs overflow interpretation. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
78. occupancy
Bound: occupied/capacity normally zero to one.
Use: outside range signals base or count issue. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
79. success rate
Bound: successes/attempts zero to one.
Use: outside invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
80. error rate
Bound: errors/attempts zero to one when each attempt counted once.
Use: outside suggests double counting. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
81. index
Bound: positive quantities create positive index.
Use: negative index usually incompatible with context. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
82. normalised share
Bound: subset/total from zero to one.
Use: outside signals wrong universe. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
83. runway
Bound: cash/burn gives positive time if both positive.
Use: negative months invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
84. average waiting time
Bound: nonnegative and bounded by observed waits when simple mean.
Use: outside range invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
85. data range
Bound: range cannot exceed max-min by definition.
Use: exact equality check. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
86. class frequency
Bound: frequency nonnegative integer.
Use: decimal frequency signals wrong quantity. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
87. cumulative frequency
Bound: nondecreasing and ends at total.
Use: decrease or final mismatch invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
88. cumulative probability
Bound: nondecreasing and bounded by one.
Use: falling sequence invalid. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
89. bounds from rounding
Bound: rounded value defines an interval around true value.
Use: use interval rather than treating rounded display as exact. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
90. bounds from measurement
Bound: measured value to nearest unit defines lower and upper limits.
Use: derived quantities inherit intervals. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
91. upper-bound product
Bound: for positive quantities, upper bounds can bound product.
Use: use carefully with sign and dependency. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
92. lower-bound product
Bound: positive lower bounds create lower product bound.
Use: zero or negative cases need separate reasoning. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
93. quotient bounds
Bound: positive numerator and denominator intervals bound quotient with opposite denominator direction.
Use: largest quotient uses large numerator and small denominator. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
94. difference bounds
Bound: uncertain quantities create interval for difference.
Use: extremes pair in opposite directions. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
95. sum bounds
Bound: lower bounds add and upper bounds add for independent positive intervals.
Use: quick feasibility check. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
96. constraint intersection
Bound: several bounds combine to narrow feasible set.
Use: candidate must satisfy all simultaneously. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
97. integer interval
Bound: bounds may leave one possible integer.
Use: solve by feasibility rather than long algebra. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
98. answer-choice bounding
Bound: MCQ options can be rejected if outside rough interval.
Use: exact calculation may be unnecessary. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
99. calculator bounding
Bound: estimate interval before pressing equals.
Use: result outside interval triggers entry check. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
100. final bound rule
Bound: an exact answer must live inside every valid structural and contextual bound.
Use: if not, inspect model, unit, sign, denominator or arithmetic. For practice, state the interval before exact calculation, then compare the computed answer with it. If the exact value lies outside a valid bound, do not average the disagreement away: inspect the first structural step that could have broken the bound.
Bounding is especially useful under time because it can eliminate MCQ options, expose calculator-entry errors and prevent contextually impossible final answers without requiring a complete second solution.
Bounds must be justified
Do not invent a bound because a number feels large or small. State the property creating it: probability range, triangle inequality, capacity, rounded measurement interval, positive weights, graph axis or real-world integer condition. A bound is evidence only when its assumptions hold.
Links
Use the Mathematics Hub, Vol 0074 Invariant Hunting, Vol 0072 Verification Asymmetry, the Examination Craft hub and the PSLE Learner’s Guide.
Official-source discipline
For current K210 requirements, use the official SEAB 2027 G2 syllabus directory and linked Mathematics syllabus. If SEAB updates the syllabus, the current official document takes priority.
Final rule
Put the answer in a box before you calculate it: not the final-answer box, but a plausible interval. Exact arithmetic should land inside valid mathematical and contextual bounds. When it does not, the bound gives you a reason to check before you commit.
