Many Mathematics answers are correct as calculations but wrong as decisions. A student solves an equation, gets a neat value and stops, forgetting that the question offered only four possible choices, imposed a maximum distance or required the price to remain within a budget. The work becomes useful only when the solution meets the whole question. This chapter is about the discipline of checking every condition.
We use a fictional paper diagram of a crane boom as an unfamiliar setting. The Physics behind moments was introduced in the previous guide; here the central skills are inverse proportion, algebraic inequalities, unit conversion, discrete choices, budgets, rounding and explaining why an answer is or is not feasible. This is not engineering guidance and cannot be used to select or operate lifting equipment. All example values belong to a classroom mathematical model.
Quick route: translate the context into named variables; use the supplied relationship Fd = M; calculate a table; solve the force inequality; intersect it with distance and price bounds; restrict the answer to the choices given; check margins and rounding; and write a conclusion that cites all constraints. Progress from one condition to several without losing control of the units.
Check the actual 2027 G3 SEC Mathematics requirements
According to SEAB’s 2027 G3 syllabus directory, Mathematics is K310, replacing the earlier 4052 code. The official K310 syllabus lists Paper 1 and Paper 2 at 2 hours 15 minutes each, 90 marks each and 50% of the overall assessment each. Paper 1 consists of about 26 short-answer questions. Paper 2 has nine or ten questions of varying length, with its last question focused on mathematics in a real-world scenario. Candidates must show essential working. Unless a different accuracy is specified, non-exact numerical answers are ordinarily correct to three significant figures, and angles in degrees to one decimal place.
The syllabus emphasises Number and Algebra, Geometry and Measurement, and Statistics and Probability, as well as mathematical processes such as reasoning, communication and modelling. The SEC qualification begins in 2027 and, according to SEAB’s overview, retains the overall examination standards at each G level. Our fictional crane data are a teaching context for algebra and decision-making, not an official topic forecast or real engineering test.
Continuity: move from physical principles to mathematical decisions
Use Physics Volume 1067 to understand what a moment is and why the distance must be perpendicular. Mathematics Volume 1064 previously used force, distance and deadline constraints in a pulley context; this volume has a different mathematical centre: inverse relations, exact feasible intervals, discrete option tables and uncertainty bounds for an ideal turning-moment model. The EMS evidence review, Volume 1066 and English report guide, Volume 1065 show how to defend a decision in words. For early algebra repair, use the PSLE word-problem route, G3 Mathematics foundations owner and Secondary One transition guide.
One invented model with all values labelled
Imagine a simplified horizontal beam problem on paper. The question supplies a required balancing moment M = 60 N m. A balancing force F, measured in newtons, acts at a perpendicular distance d, measured in metres, from the pivot. The exercise gives the relation Fd = 60, with d positive. This relationship is conditional on the supplied model. It says nothing about wind, moving loads, foundations, structural ratings or actual equipment.
The exercise offers exactly five distance choices: 0.60 m, 0.80 m, 1.00 m, 1.20 m and 1.50 m. It imposes a numerical force threshold F ≤ 55 N and a maximum allowed distance d ≤ 1.25 m. A simple fictional price formula, unrelated to real crane costs, is C = 20 + 8d dollars; the budget is C ≤ 31 dollars. The question is which offered choice meets all three conditions.
Calculate F = 60/d. The five forces are respectively 100 N, 75 N, 60 N, 50 N and 40 N. The first three exceed the force threshold. The fifth passes the force test but exceeds the maximum distance. The fourth, d = 1.20 m, gives F = 50 N and price C = 20 + 8(1.20) = $29.60. It passes all three numerical restrictions. This is the only offered mathematical option.
Why algebra is faster than guessing
From 60/d ≤ 55 with d > 0, multiply by d and divide by 55 to obtain d ≥ 60/55 = 12/11 ≈ 1.0909 m. The distance condition gives d ≤ 1.25 m. Intersect them: 12/11 ≤ d ≤ 1.25. Any mathematical value inside this interval passes those two conditions. But the question restricts us to five offered distances. Only 1.20 m lies in the interval.
The budget inequality gives 20 + 8d ≤ 31, so 8d ≤ 11 and d ≤ 1.375 m. This is less restrictive than the exercise’s 1.25 m distance limit. It does not change the final allowed set, but it must still be checked. The correct conclusion is not simply ‘1.20 works’; it is that all supplied restrictions have been considered and only the 1.20 m offered value satisfies them.
The first high-level habit: identify the domain
The relationship F = 60/d assumes d > 0. A calculator might return a negative F for negative d, but a negative perpendicular distance is not an allowed length in this exercise. The force inequality cannot be manipulated carelessly if the sign of d is unknown; multiplying an inequality by a negative number reverses its direction. State the positive domain first, then work.
Exam questions often impose integer, nonnegative or interval restrictions that the equation alone cannot reveal. A continuous algebraic solution may contain values that are mathematically valid but unavailable in the real-world scenario described. An option table is a clean way to enforce that last restriction. The student’s job is not to find every answer to an equation; it is to find the answer that satisfies the entire problem.
Direct and inverse proportion
For fixed M = 60, Fd = 60, so F is inversely proportional to d. Double d and F is halved; halve d and F doubles. The proportionality is not F ∝ d. The product of the two variables remains constant under the model. A table can show the pattern without a graph, but a graph makes the shape more visible.
If d is plotted on the horizontal axis and F on the vertical axis, the relation is a decreasing curve for positive d. Plotting F against 1/d instead gives a straight-line relation through the origin with gradient 60 in the units implied by the graph. This transformation is not required in every G3 question; it is a useful way to recognise inverse proportionality and diagnose why a straight line through the raw F–d points would be misleading.
Units, precision and the temptation to round early
Keep centimetres and metres separate. A force of 80 N at 75 cm corresponds to a moment of 80 × 0.75 = 60 N m, not 80 × 75 = 6000 N m. The arithmetic with 75 would give N cm if the unit were retained. Convert before substitution and write the unit in the final line.
Do not round 60/55 to 1.1 m before checking a candidate such as 1.095 m. The exact threshold is 12/11 ≈ 1.090909 m; 1.095 m is feasible under the force condition. Prematurely using 1.1 m would wrongly exclude it. Preserve the exact fraction or sufficient guard digits while deciding and round only at the final reporting step. This matters particularly near a boundary.
A constraint is not a preference
A device that needs the smallest force is not automatically the correct mathematical option if its required distance lies outside the permitted range. Our 1.50 m choice has the lowest calculated force, but it violates d ≤ 1.25. An answer that ranks options only by force has solved a different problem. First find the feasible set. Only after that may you compare other qualities among the survivors.
The word ‘at most’ includes the boundary. ‘Less than’ excludes it. ‘At least’ includes the lower boundary. ‘More than’ excludes it. If C ≤ 31, an exact cost of $31 passes; if the question says C < 31, it fails. Translate words into mathematical symbols before calculating so that language does not silently change the solution set.
How to use margins without claiming safety
In our ideal paper model, the 1.20 m option has a force margin of 55 − 50 = 5 N against the stated numerical threshold. It has a distance margin of 1.25 − 1.20 = 0.05 m and a price margin of 31 − 29.60 = $1.40. These margins show how far the arithmetic result is from each classroom limit. They do not constitute engineering safety factors or real operating margins; the scenario never supplied such standards.
A small margin may be worth noticing because measurement uncertainty could change the classification in a more realistic problem. But you must not invent uncertainty where the question gives none. If the exercise explicitly supplies a range of required moments, then use the worst relevant value for a robust comparison; otherwise solve the stated case only.
Uncertainty, tolerances and interval reasoning
Suppose an extension explicitly states that the required moment could lie between 59 and 61 N m, while the force limit stays 55 N. If we want an offered distance that meets the force condition for every moment in this range, use the largest possible moment: Fmax = 61/d ≤ 55. Thus d ≥ 61/55 ≈ 1.1091 m. The offered 1.20 m still passes. This is a useful introduction to worst-case reasoning, not a licence to assert that actual machines tolerate those loads.
The distinction between ‘possible’ and ‘guaranteed under the range’ matters. An option may work for M = 59 but fail for M = 61. If a question asks whether it could work, one scenario may be enough; if it asks whether it always meets the specified range, every allowed value must be considered. Answer the quantifier in the question, not merely the most convenient example.
When several constraints appear, check them one at a time
A clear candidate table can have columns for distance, predicted force, model price, force pass, distance pass and price pass. This makes it difficult to forget a restriction. In a written answer, you need not reproduce a huge table if inequalities settle the matter quickly. But for small discrete choice sets, a compact table is often the clearest route from conditions to a decision.
If no candidates satisfy the restrictions, do not invent a compromise and announce a winner. Write that there is no feasible option under the given choices and constraints. If the prompt asks for a next step, identify which constraint would have to change and by how much, purely within the fictional model. An empty feasible set is a valid mathematical conclusion.
Thirty-five worked checkpoints, from direct calculation to advanced modelling
Checkpoint 1: Given M = 60 N m and d = 0.60 m, find F.
Worked answer: F = 60/0.60 = 100 N. Interpretation: The inverse relationship means a short lever arm requires greater force. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 2: For d = 0.80 m, find F.
Worked answer: F = 60/0.80 = 75 N. Interpretation: Do not describe this as 75 N m; the unknown is a force. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 3: For d = 1.00 m, find F.
Worked answer: F = 60 N. Interpretation: The unit length makes the product equal the numerical force. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 4: For d = 1.20 m, find F.
Worked answer: F = 50 N. Interpretation: The calculation must be tested against each available limit. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 5: For d = 1.50 m, find F.
Worked answer: F = 40 N. Interpretation: A numerically attractive force still violates the stated distance cap. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 6: Find the minimum d so that F ≤ 55 N.
Worked answer: 60/d ≤ 55 gives d ≥ 12/11 m, for positive d. Interpretation: Solve the inequality before rounding or selecting a candidate. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 7: A teacher writes d ≥ 1.1 m after rounding the threshold.
Worked answer: The exact bound is approximately 1.090909 m, not 1.1 m. Interpretation: Rounding an inequality boundary can wrongly exclude valid near-boundary choices. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 8: Test d = 1.095 m against the force limit.
Worked answer: 60/1.095 ≈ 54.8 N, so it passes 55 N. Interpretation: Carry sufficient precision through the classification. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 9: Test d = 1.08 m against the same limit.
Worked answer: 60/1.08 ≈ 55.6 N, so it fails. Interpretation: A small change in distance can cross the threshold. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 10: Convert 75 cm to metres.
Worked answer: 75 cm = 0.75 m. Interpretation: Unit conversions should precede substitution. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 11: Compute a moment from 80 N and 75 cm.
Worked answer: M = 80 × 0.75 = 60 N m. Interpretation: Using 75 without converting silently changes the numerical unit. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 12: Find the model price at d = 1.20 m if C = 20 + 8d.
Worked answer: C = 20 + 9.6 = $29.60. Interpretation: The price formula is invented for an algebra exercise, not an actual quote. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 13: Use the budget C ≤ 31 to bound d.
Worked answer: 20 + 8d ≤ 31 gives d ≤ 11/8 = 1.375 m. Interpretation: Convert each narrative condition into its own inequality. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 14: Intersect d ≥ 12/11 and d ≤ 1.25.
Worked answer: The continuous feasible interval is 12/11 ≤ d ≤ 1.25. Interpretation: An intersection means all inequalities must hold simultaneously. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 15: Which of 0.60, 0.80, 1.00, 1.20 and 1.50 lies in that interval?
Worked answer: Only 1.20 m. Interpretation: The available choices restrict the algebraic solution. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 16: Find the model force margin for d = 1.20.
Worked answer: 55 − 50 = 5 N. Interpretation: This is a mathematical margin, not a verified engineering safety allowance. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 17: Find the distance margin for d = 1.20.
Worked answer: 1.25 − 1.20 = 0.05 m = 5 cm. Interpretation: Label margins with quantities and units. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 18: Find the model price margin at d = 1.20.
Worked answer: 31 − 29.60 = $1.40. Interpretation: A price margin is not interchangeable with a force margin. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 19: If M increases to 66 N m, find F for d = 1.20.
Worked answer: F = 66/1.20 = 55 N. Interpretation: This reaches the inclusive force threshold exactly. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 20: If M = 66.6 N m at d = 1.20, test the limit.
Worked answer: F = 55.5 N, above 55 N, so it fails. Interpretation: A design choice may switch status when a supplied model input changes. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 21: For M between 59 and 61 N m, find worst-case F at d = 1.20.
Worked answer: Fmax = 61/1.20 ≈ 50.8 N. Interpretation: Use the range endpoint that creates the most demanding force requirement. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 22: For the same M range, find the minimum robust d.
Worked answer: d ≥ 61/55 ≈ 1.1091 m. Interpretation: ‘For every case in the range’ is stronger than ‘for one value’. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 23: Suppose d may be any length from 0.90 to 1.00 m.
Worked answer: No d in that range meets F ≤ 55 for M = 60, since 60/d ≥ 60 N. Interpretation: A correct problem can have no feasible solution. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 24: Suppose the allowed d interval becomes 1.10 to 1.20 m.
Worked answer: The continuous force-feasible interval is 1.10 to 1.20 m, but check any discrete options separately. Interpretation: Keep the continuous interval distinct from the option list. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 25: Find F if d doubles from 0.60 m to 1.20 m.
Worked answer: Force falls from 100 N to 50 N, a halving. Interpretation: Inverse proportionality works when M remains fixed. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 26: Find d if F halves from 75 N to 37.5 N.
Worked answer: Distance doubles from 0.80 m to 1.60 m. Interpretation: A force improvement can breach a distance constraint. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 27: If a candidate costs $29.60 and has a $31 cap, how much budget is left?
Worked answer: The remaining budget is $1.40. Interpretation: A constraint answer must carry the right currency. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 28: If price becomes C = 23 + 8d, test d = 1.20.
Worked answer: C = 32.60, which fails the $31 cap. Interpretation: Changing one input can empty the feasible set even when forces are unchanged. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 29: If the distance cap becomes d ≤ 1.00 m, is there a feasible offered option?
Worked answer: No; force requires d ≥ 12/11 > 1.00. Interpretation: Write ‘none’ rather than choosing the least bad invalid option. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 30: A question requires C < 31 rather than C ≤ 31.
Worked answer: An exact $31 is now excluded; $29.60 still passes. Interpretation: Strict and inclusive inequalities have different boundaries. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 31: Find the product of F and d for the five example choices.
Worked answer: Each product equals 60 N m. Interpretation: A constant product is the signature of this inverse model. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 32: Explain the graph of F against d.
Worked answer: It is a decreasing reciprocal curve F = 60/d for d > 0. Interpretation: Do not join data as a straight line just because a ruler is available. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 33: Explain the graph of F against 1/d.
Worked answer: It is linear with proportionality constant 60 in the transformed variables. Interpretation: A suitable transformation can reveal the underlying model. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 34: A student claims the 1.50 m choice is best because F = 40 N.
Worked answer: Reject it under the original d ≤ 1.25 rule. Interpretation: Feasibility is a prior test; preference among invalid options is not a solution. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
Checkpoint 35: A draft concludes ‘the model proves the crane is safe’.
Worked answer: Change it to ‘the 1.20 m option alone satisfies the supplied numerical classroom constraints’. Interpretation: Mathematical feasibility under fictional assumptions is not a real-world safety assessment. Recheck the domain of the variables, the direction of the inequality and the physical unit. If the problem provides an option list, compare the result with that list before finishing. A helpful examination answer shows enough working for the reader to follow the calculation and ends with a sentence addressing the actual decision requested, not just a number from a calculator.
A complete extended Paper 2-style modelling question
A fictional question offers five discrete distances: 0.60, 0.80, 1.00, 1.20 and 1.50 m. The supplied model is F = 60/d for positive d, with force in newtons. It also requires F ≤ 55, d ≤ 1.25 and C = 20 + 8d ≤ 31. Show all calculations needed to select the only option meeting the stated numerical conditions. Then determine whether your conclusion survives an explicitly supplied range 59 ≤ M ≤ 61, where F = M/d.
First solve the force constraint: d ≥ 60/55 ≈ 1.0909. Combine with d ≤ 1.25. The price constraint gives d ≤ 1.375, which does not narrow the interval further. Only 1.20 m from the five offered options lies in the final interval. It gives force 50 N and price $29.60, both valid. To test the specified moment range, take M = 61 for maximum force. At d = 1.20, F = 61/1.20 ≈ 50.8 N, below 55 N. Thus the same choice survives this stipulated range under the simplified formula.
The final sentence must be precise: ‘The 1.20 m option is the only listed distance satisfying all three stated constraints; it also meets the force limit for every moment in the supplied 59–61 N m interval.’ It must not read ‘This crane is safe for all weights’ or ‘The tested equipment will work,’ because no equipment has been tested in this mathematical exercise.
A two-option comparison and the issue of optimisation
Consider an adjusted, separate exercise where both d = 1.20 and d = 1.50 are permitted and both satisfy the same force cap. Which is preferable? The algebra alone cannot answer without an objective. If the objective is minimising the model force, 1.50 is better. If minimising the invented model cost C = 20 + 8d, 1.20 is cheaper. If the problem values compactness, the shorter option again differs from the smallest-force choice.
This is the distinction between feasibility and optimisation. Feasibility asks whether conditions are met. Optimisation selects a best value according to an explicitly stated objective within the feasible set. If a question never defines what ‘best’ means, you should not silently invent your own criterion. Identify the ambiguity or use the stated preference.
How to write a mathematically complete explanation
A strong answer gives the formula and domain, obtains the exact or sufficiently precise bounds, shows their intersection, checks the allowed choices, and states the final option with its relevant values. It does not need a dramatic introduction. Clarity matters because a marker must distinguish evidence of reasoning from guesses, and because a real-life reader needs to see why the option meets all conditions.
Use words for the conceptual steps: ‘Since d is positive…’ before multiplying an inequality; ‘therefore the feasible interval is…’ after intersection; and ‘among the available choices…’ before the discrete decision. These phrases expose reasoning without bloating the solution.
A short recovery plan for students missing the basics
If division with decimals causes errors, practise 60 ÷ 1.20 separately. If inequality directions are uncertain, use simple number-line examples and insist on stating the sign before multiplying. If interval intersection is unfamiliar, draw two number lines and shade their overlap. If real-world questions feel overwhelming, turn the paragraph into a small table: variable, unit, formula, constraint, candidate. The point is to rebuild a sequence of reliable actions rather than memorise an entire answer.
A stronger student should attempt the extended question with the answer hidden, then explain aloud why 1.50 m is invalid under the original distance cap. Explaining a rejected candidate is often more revealing than explaining the survivor. Try modifying one constraint and predicting whether the feasible set grows, shrinks or disappears before calculating.
Frequently asked questions
Is crane design an official Mathematics topic? No. The mathematical ideas of inverse relations, algebra, units, measurement and modelling are being practised in an invented setting.
Does G3 Mathematics have two papers? Yes. The 2027 K310 syllabus lists two 90-mark papers, each 2 hours 15 minutes and worth 50%.
Why must I show working? The official syllabus says omission of essential working loses marks. A number without its reasoning may not meet the question.
Can I round 1.090909 to 1.1 before deciding? Not when a candidate lies close to that boundary. Preserve exact or guard digits until the comparison is complete.
Why is 1.50 m rejected when it gives only 40 N? Because it violates the stated 1.25 m maximum distance in the original exercise.
What if the solution interval contains several options? Check any additional objective or constraint. If none is given, state the feasible choices rather than inventing one winner.
What if none works? State that the feasible set is empty under the given restrictions. Do not choose an invalid compromise unless asked to discuss changes.
Is a force margin a safety factor? No. Our fictional margin is the numerical gap to a classroom threshold, not an engineering certification.
Why can I not use negative d? The model defines d as a positive perpendicular distance, so negative values are outside its stated domain.
What should I study next? Return to the Physics moments foundation or practise a fresh multi-constraint scenario that uses a different context; the method should transfer without memorising the numbers.
A final checklist before leaving the page
Name every variable and its unit. Record whether quantities are measured, supplied or calculated. Check positivity before manipulating inverse inequalities. Keep exact thresholds where possible. Intersect constraints, then filter the options. State all margins only when requested, and distinguish mathematical success from real-world approval. If you need the source principle, revisit Physics Volume 1067; for a related but different constraint model use Volume 1064; for rebuilding algebra use the G3 Mathematics foundations route. A clear decision is the end of the calculation, not a substitute for it.
